0486656098
TRANSCRIPT
Lectures onClassical Differential
GeometrySECOND EDITION
Dirk J. StruikMASSACHUSETTS INSTITUTE OF TECHNOLOGY
DOVER PUBLICATIONS, INC.New York
Copyright © 1950, 1961 by Dirk J. Struik.All rights reserved under Pan American and International
Copyright Conventions.
Published in Canada by General Publishing Company,Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Com-pany, Ltd.
This Dover edition, first published in 1988, is an un-abridged and unaltered republication of the second edition(1961) of the work first published in 1950 by the Addison-Wesley Publishing Company, Inc., Reading, Massachusetts.
Manufactured in the United States of AmericaDover Publications, Inc., 31 East 2nd Street, Mineola, N.Y.
11501
Library of Congress Cataloging-in-Publication Data
Struik, Dirk Jan, 1894-Lectures on classical differential geometry / Dirk J.
Struik. - 2nd ed.p. cm.
Reprint. Originally published: Reading, Mass. : Addi-son-Wesley Pub. Co., 1961.
Bibliography: p.Includes index.
ISBN 0-486-65609-81. Geometry, Differential. I. Title.
QA641.S72 1988516.3'602-dc 19 87-34903
CIP
CONTENTS
PREFACE . . . . . . . . . . . . . . . . . . . . . . . V
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . Vii
CHAPTER 1. CURVES . . . . . . . . . . . . . . . . . . 1
1-1 Analytic representation . . . . . . . . . . . . . . . 1
1-2 Arc length, tangent . . . . . . . . . . . . . . . . . 51-3 Osculating plane . . . . . . . . . . . . . . . . . . 101-4 Curvature . . . . . . . . . . . . . . . . . . . . 131-5 Torsion . . . . . . . . . . . . . . . . . . . . . 15
1-6 Formulas of Frenet . . . . . . . . . . . . . . . . . 18
1-7 Contact . . . . . . . . . . . . . . . . . . . . . 231-8 Natural equations . . . . . . . . . . . . . . . . . 261-9 Helices . . . . . . . . . . . . . . . . . . . . . 331-10 General solution of the natural equations . . . . . . . . . 361-11 Evolutes and involutes . . . . . . . . . . . . . . . . 391-12 Imaginary curves . . . . . . . . . . . . . . . . . 441-13 Ovals . . . . . . . . . . . . . . . . . . . . . . 471-14 Monge . . . . . . . . . . . . . . . . . . . . . 53
CHAPTER 2. ELEMENTARY THEORY OF SURFACES . . . . . . . 55
2-1 Analytical representation . . . . . . . . . . . . . . . 552-2 First fundamental form . . . . . . . . . . . . . . . 582-3 Normal, tangent plane . . . . . . . . . . . . . . . . 622-4 Developable surfaces . . . . . . . . . . . . . . . . 662-5 Second fundamental form. Meusnier's theorem . . . . . . . 732-6 Euler's theorem . . . . . . . . . . . . . . . . . . 772-7 Dupin's indicatrix . . . . . . . . . . . . . . . . . 832-8 Some surfaces . . . . . . . . . . . . . . . . . . . 862-9 A geometrical interpretation of asymptotic and curvature lines . . 932-10 Conjugate directions . . . . . . . . . . . . . . . . 962-11 Triply orthogonal systems of surfaces . . . . . . . . . . . 99
CHAPTER 3. THE FUNDAMENTAL EQUATIONS . . . . . . . . . 105
3-1 Gauss . . . . . . . . . . . . . . . . . . . . . 1053-2 The equations of Gauss-Weingarten . . . . . . . . . . . 1063-3 The theorem of Gauss and the equations of Codazzi . . . . . . 1103-4 Curvilinear coordinates in space . . . . . . . . . . . . 1153-5 Some applications of the Gauss and the Codazzi equations . . . 1203-6 The fundamental theorem of surface theory . . . . . . . . . 124
iii
1V CONTENTS
CHAPTER 4. GEOMETRY ON A SURFACE . . . . . . . . . . . 127
4-1 Geodesic (tangential) curvature . . . . . . . . . . . . . 1274-2 Geodesics . . . . . . . . . . . . . . . . . . . . 1314-3 Geodesic coordinates . . . . . . . . . . . . . . . . 1364-4 Geodesics as extremals of a variational problem . . . . . . . 1404-5 Surfaces of constant curvature . . . . . . . . . . . . . 1444-6 Rotation surfaces of constant curvature . . . . . . . . . . 1474-7 Non-Euclidean geometry . . . . . . . . . . . . . . . 1504-8 The Gauss-Bonnet theorem . . . . . . . . . . . . . . 153
CHAPTER 5. SOME SPECIAL SUBJECTS . . . . . . . . . . . 162
5-1 Envelopes . . . . . . . . . . . . . . . . . . . . 1625-2 Conformal mapping . . . . . . . . . . . . . . . . 1685-3 Isometric and geodesic mapping . . . . . . . . . . . . 175
5-4 Minimal surfaces . . . . . . . . . . . . . . . . . . 1825-5 Ruled surfaces . . . . . . . . . . . . . . . . . . 1895-6 Imaginaries in surface theory . . . . . . . . . . . . . 196
SOME PROBLEMS AND PROPOSITIONS . . . . . . . . . . . . 201
APPENDIX: The method of Pfaffians in the theory of curves and surfaces. 205
ANSWERS TO PROBLEMS . . . . . . . . . . . . . . . 217
INDEX . . . . . . . . . . . . . . . . . . . . 226
PREFACE
This book has developed from a one-term course in differential geometrygiven for juniors, seniors, and graduate students at the MassachusettsInstitute of Technology. It presents the fundamental conceptions of thetheory of curves and surfaces and applies them to a number of examples.Some care is given to historical, biographical, and bibliographical material,not only to keep alive the memory of the men to whom we owe the mainstructure of our present elementary differential geometry, but also to allowthe student to go back to the sources, which still contain many preciousideas for further thought.
No author on this subject is without primary obligation to the twostandard treatises of Darboux and Bianchi, who, around the turn of thecentury, collected the result of more than a century of research, themselvesadding greatly to it. Other fundamental works constantly consulted bythe author are Eisenhart's Treatise, Scheffers' Anwendung, and Blaschke'sVorlesungen. Years of teaching from Graustein's Differential Geometryhave also left their imprint on the presentation of the material.
The notation used is the Gibbs form of vector analysis, which after yearsof competition with other notations seems to have won the day, not only inthe country of its inception, but also in many other parts of the world.Those unfamiliar with this notation may be aided by some explanatoryremarks introduced in the text. This notation is amply sufficient for thosemore elementary aspects of differential geometry which form the subjectof this introductory course; those who prefer to study our subject withtensor methods and thus to prepare themselves for more advanced researchwill find all they need in the books which Eisenhart and Hlavaty havedevoted to this aspect of the theory. Some problems in the present bookmay serve as a preparation for this task.
Considerable attention has been paid to the illustrations, which may behelpful in stimulating the student's visual understanding of geometry. Inthe selection of his illustrations the author has occasionally taken his in-spiration from some particularly striking pictures which have appeared inother books, or from mathematical models in the M.I.T. collection.* The
* In particular, the following figures have been wholly or in part suggested byother authors: Fig. 2-33 by Eisenhart, Introduction; Figs. 2-32, 2-33, 3-3, 5-14by Scheffers, Anwendung; Figs. 2-22, 2-23, 5-7 by Hilbert-Cohn Vossen; andFig. 5-9 by Adams (loc. cit., Section 5-3). The models, from L. Brill, Darmstadt,were constructed between 1877 and 1890 at the Universities of Munich (underA. Brill) and Gottingen (under H. A. Schwarz).
V
vi PREFACE
Art Department of Addison-Wesley Publishing Company is responsiblefor the excellent graphical interpretation and technique.
The problems in the text have been selected in such a way that most ofthem are simple enough for class use, at the same time often conveying aninteresting geometrical fact. Some problems have been added at the endwhich are not all elementary, but reference to the literature may here behelpful to students ambitious enough to try those problems.
The author has to thank the publishers and their adviser, Dr. EricReissner, for their encouragement in writing this book and Mrs. Violet Haasfor critical help. He owes much to the constructive criticism of his classin M 442 during the fall-winter term of 1949-50, which is responsible formany an improvement in text and in problems. He also acknowledges withappreciation discussions with Professor Philip Franklin, and the help ofMr. F. J. Navarro.
DIRK J. STRUM
PREFACE TO THE SECOND EDITION
In this second edition some corrections have been made and an appendixhas been added with a sketch of the application of Cartan's method ofPfaffians to curve and surface theory. This sketch is based on a paperpresented to the sixth congress of the Mexican Mathematical Society,held at Merida in September, 1960.
A Spanish translation by L. Bravo Gala (Aguilar, Madrid, 1955) isonly one of the many tokens that the book has been well received. Theauthor likes to use this opportunity to express his appreciation and tothank those who orally or in writing have suggested improvements.
D.J.S.
BIBLIOGRAPHY
ENGLISH
EISENHART, L. P., A treatise on the differential geometry of curves and surfaces.Ginn & Co., Boston, etc., 1909, xi + 476 pp. (quoted as Differential Geometry).
EISENHART, L. P., An introduction to differential geometry with use of the tensorcalculus. Princeton University Press, Princeton, 1940, 304 pp.
FORSYTH, A. R., Lectures on the differential geometry of curves and surfaces. Uni-versity Press, Cambridge, 1912, 34 + 525 pp.
GRAUSTEIN, W. C., Differential geometry. Macmillan, New York, 1935, xi + 230PP.
KREYSZIG, E., Differential geometry. University of Toronto Press, Toronto,1959, xiv + 352 pp. Author's translation from the German: Differentialgeometry.Akad. Verlagsges, Leipzig, 1957, xi + 421 pp.
LANE, E. P., Metric differential geometry of curves and surfaces. University ofChicago Press, Chicago, 1940, 216 pp.
POGORELOV, A. V., Differential geometry. Noordhoff, Groningen, 1959,ix + 171 pp. Translated from the Russian by L. F. BORON.
WEATHERBURN, C. E., Differential geometry of three dimensions. UniversityPress, Cambridge, I, 1927, xii + 268 pp.; II, 1930, xii + 239 pp.
WILLMORE, T., An introduction to differential geometry. Clarendon Press,Oxford, 1959, 317 pp.
FRENCH
DARBOUx, G., Legons sur la Worie genkrale des surfaces. 4 vols., Gauthier-Villars, Paris (2d ed., 1914), I, 1887, 513 pp.; II, 1889, 522 pp.; III, 1894, 512 pp.;IV, 1896, 548 pp. (quoted as Legons).
FAVARD, J., Cours de geometrie differentielle locale. Gauthier-Villars, Paris,1957, viii + 553 pp.
JULIA, G., Elements de geometrie infcnitksimale. Gauthier-Villars, Paris, 2d ed.,1936, vii + 262 pp.
RAFFY, L., Legons sur les applications geom triques de l'analyse. Gauthier-Villars, Paris, 1897, vi + 251 pp.
GERMAN
BLASCHKE, W., Vorlesungen fiber Differentialgeometrie I, 3d ed., Springer, Berlin,1930, x + 311 pp. (quoted as Differentialgeometrie).
BLASCHKE, W., Einfuhrung in die Differentialgeometrie. Springer-Verlag, 1950,vii + 146 pp.
vii
Viii BIBLIOGRAPHY
HAACK, W., Differential-geometrie. Wolfenbiittler Verlagsanstalt. Wolfen-biittel-Hannover, 2 vols., 1948, I, 136 pp.; II, 131 pp.
HLAVATY, V., Differentialgeometrie der Kurven and Flachen and Tensorrechnung.tbersetzung von M. Pinl. Noordhoff, Groningen, 1939, xi + 569 pp.
KOMMERELL, V. UND K., Allgemeine Theorie der Raumkurven and Flachen. DeGruyter & Co., Berlin-Leipzig, 3e Aufl., 1921, I, viii + 184 pp.; II, 196 pp.
SCHEFFERS, G., Anwendung der Differential- and Integralrechnung auf Geometrie.De Gruyter, Berlin-Leipzig, 3e Aufl. I, 1923, x + 482 pp.; II, 1922, xi + 582 pp.(quoted as Anwendung).
STRUBECKER, K., Differentialgeometrie. Sammlung Goschen, De Gruyter,Berlin, I, 1955, 150 pp.; II, 1958, 193 pp.; III, 1959, 254 pp.
ITALIAN
BIANCHI, L., Lezioni di geometria differenziale. Spoerri, Pisa, 1894, viii + 541pp.; 3e ed., I, 1922, iv + 806 pp.; II, 1923, 833 pp. German translation by M.LuKAT; Vorlesungen caber Differentialgeometrie. Teubner, Leipzig, 2e Aufl., 1910,xviii + 721 pp. (Lezioni I quoted as Lezioni.)
RUSSIAN
FINIKOV, S. P., Kurs differencial'noi geometrii. Moscow, 1952, 343 pp.RASEVSKIr, P. K., Kurs differencial'noi geometrii. Moscow, 4th ed., 1956,
420 pp.VYGODSKII, M. YA., Differential'naya geometriya. Moscow-Leningrad, 1949,
511 pp.
There are also chapters on differential geometry in most textbooks of advancedcalculus, such as:
GOURSAT, E., Cours d'analyse mathematique. Gauthier-Villars, Paris, I, 5e ed.,1943, 674 pp. English translation by E. R. HEDRICK, Course in mathematicalanalysis. Ginn & Co., Boston, I, 1904, viii + 548 pp.
The visual aspect of curve and surface theory is stressed in Chapter IV ofD. HILBERT & S. COHN-VossEN, Anschauliche Geometrie. Springer, Berlin, 1932,viii + 310 pp.
The best collection of bibliographical notes and references in: Encyklopadie derMathematischen Wissenschaften. Teubner, Leipzig, Band III, 3 Teil (1902-'27),606 pp., article by H. V. MANGOLDT, R. V. LILIENTHAL, G. SCHEFFERS, A. Voss,H. LIEBMANN, E. SALKOWSKI.
Also :
PASCAL, E., Repertorium der Hoheren Mathematik. 2e Aufl. Zweiter Band,Teubner, Leipzig. Berlin, I (1910), II (1922), article by H. LIEBMANN andE. SALKOWSKI.
The history of differential geometry can be studied in COOLIDGE, J. L., A historyof geometrical methods. Clarendon Press, Oxford, xviii + 451 pp., especially pp.318-387.
Lectures onClassical Differential
Geometry
CHAPTER 1
CURVES
1-1 Analytic representation. We can think of curves in space as pathsof a point in motion. The rectangular coordinates (x, y, z) of the pointcan then be expressed as functions of a parameter u inside a certain closedinterval:
x=x(u), y=y(u), z=z(u); u1<u<u2. (1-la)
It is often convenient to think of u as the time, but this is not necessary,since we can pass from one parameter to another by a substitution u =f(v) without changing the curve itself. We select the coordinate axes insuch a way that the sense OX -+ OY -+ OZ is that of a right-handed screw.We also denote (x, y, z) by (x1, x2, x3), or for short, x;, i = 1, 2, 3. Theequation of the curve then takes the form
x; = x;(u); u1 < u < u2. (1-lb)
We use the notation P(x;) to indicate a point with coordinates x;.
EXAMPLES. (1) Straight line. A straight line in space can be given bythe equation
x; = a; + ub;,
where a;, b; are constants and at least one of the b; P, 0.
(1-2)
This equation represents a line passing through the point (as) with itsdirection cosines proportional to b;. Eq. (1-2) can also be written:
x, - a, x2 - a2 x3 - a3
bl= b2 = b3
(2) Circle. The circle is a planecurve. Its plane can be taken as
z = 0 and its equation can then be
written in the form:
x=acosu, y=asinu, z=0;0 < u < 21r. (1-3)
Here a is the radius, u = L POX(Fig. 1-1).
(3) Circular helix. The equationis FIG. 1-1
1
2 CURVES [CH. 1
x = a cos u, y = a sin u, z = bu; a, b constants. (1-4)
This curve lies on the cylinder x2 + y2 = a2 and winds around it in sucha way that when u increases by 2ir the x and y return to their original value,while z increases by 2irb, the pitch of the helix (French: pas; German:Ganghohe). When b is positive the helix is right-handed (Fig. 1-2a); whenb is negative it is left-handed (Fig. 1-2b). This sense of the helix is inde-pendent of the choice of coordinates or parameters; it is an intrinsic prop-erty of the helix. A left-handed helix can never be superimposed on aright-handed one, as everyone knows who has handled screws or ropes.
The functions x;(u) are not all constants. If two of them are constantsEqs. (1-1) represent a straight line parallel to a coordinate axis. We alsosuppose that in the given interval of u the functions x;(u) are single-valuedand continuous, with a sufficient number of continuous derivatives (firstderivatives in all cases, seldom more than three). It is sufficient for thispurpose to postulate that there exists at a point P of the curve, where
au
i
P/
+---------
ILL:
BFic. 1-2(a) FIG. 1-2(b)
1-1] ANALYTIC REPRESENTATION 3
u = uo, a set of finite derivatives x4°+1)(uo), n sufficiently large. Then wecan express xi(uo + h) as follows in a Taylor development:
xi(uo + h) = xi(u)
= xi(uo) + 1 xi(uo) +h' xi(uo) + ... + ni xjn)(uo) + o(h'), (1-5)
where the zi, zi, . . . xi(°) represent derivatives with respect to u and o(h*)is a term such that
A
lim o h = 0.")h^
This is always satisfied, for all values of n, n > 0, when the xi(u) arecomplex functions of a complex u and the first derivatives 1i exist. Thefunctions xi(u) are then analytic. However, we usually consider the xi as realfunctions of a real variable u. The curve (1-1) with the conditions (1-5) isbetter called an arc of curve, but we shall continue to use the term curve aslong as no ambiguity occurs. Points where all ii vanish are called singular,otherwise regular with respect to u. When speaking of points, we meanregular points. When we replace the parameter u by another parameter,
u = J (u1), (1-6)
* See e.g. P. Franklin, A treatise on advanced calculus, John Wiley and Sons,New York, 1940, p. 127; Ch. J. de la Vall a Poussin, Cours d'analyse infcnitoWmale,Dover Publications, New York, 1946, Tome I, 8th ed., p. 80.
4 CURVES [cHi. 1
we postulate that f(u1) be differentiable; when du/dul 0 0 regular pointsremain regular.
Curves can also be defined in ways different from (1-1). We can use theequations
F1(x, y, z) = 0, F2(x, y, z) = 0, (1-7)or
y = fi(x), z = f2(x), (1-8)
to define a curve. The type (1-8) can be considered as a special form of(1-1), x being taken as parameter. We obtain it from (1-1) by eliminatingu from y and x, and also from z and x. This is always possible whendx/du 0 0, so that u can be expressed in x. Type (1-8) expresses thecurve C as the intersection of two projecting cylinders (Fig. 1-3). As toEqs. (1-7), they define two implicit functions y(x) and z(x) when thefunctional determinant
F1F2 _ aFl aF2 _ aF2 aF1 54 0.*Y z) az ay az ay
This brings us to (1-8) and thus to (1-1).The representations (1-7) and (1-8) define the space curve as the inter-
section of two surfaces. But such an intersection may split into severalcurves. If, for instance, F1 and F2 represent two cones with a commongenerating line, Eqs. (1-7) define this line together with the remainingintersection. And if we eliminate u from the equations
x=u, y=u2, z=u3, (1-9)
which represent a space curve C of
the third degree (a cubic parabola),
we obtain the equations
y=x2, xz-y2=0,which represent the intersection of acylinder (Fig. 1-4) and a cone. Thisintersection contains not only thecurve (1-9), but also the Z-axis.
The complete intersection of analgebraic surface of degree m and analgebraic surface of degree n is aspace curve of degree mn, which may
* Franklin, loc. cit., pp. 340-341. FIG. 1-4
1-21 ARC LENGTH, TANGENT 5
split into several curves, the degree of which adds up to mn. We say that a sur-face is of degree k when it is intersected by a line in k points or, what is thesame, by a plane in a curve of degree k. A space curve is of degree 1 if a planeintersects it in l points. The points of intersection may be real, imaginary,coincident, or at infinity. In the case (1-9) we substitute x, y, z into theequation of a plane ax + by + cz + d = 0, and obtain a cubic equation for u,which has three roots, indicating three points of intersection.
This explains why we often prefer to give a curve by equations of theform (1-1). Moreover, this presentation allows a ready application of theideas of vector analysis.
1zl tF hi e1, e2, e3i oror t s purpose, efor short e; (i = 1, 2, 3) be unitvectors in the direction of the positiveX, Y, and Z-axes. Then we can givea curve C by expressing the radiusvector OP = x of a generic point Pas a function of u (Fig. 1-5) in thefollowing way:
x = x1e1 + x2e2 + x3e3, (1-10) e2
e1where the x; are given by Eq. (1-1). /We indicate P not only by P(xi), xbut also by P(x) or P(u); we shallalso speak of "the curve x(u)."
Fia. 1-5
The xi are the coordinates of x, the vectors x1e1, x2e2, x3e3 are the com-ponents of x along the coordinate axes. We shall often indicate a vectorby its coordinates, as x(x, y, z), or as x(xi).
The length of the (real) vector x is indicated by
xi = -1141 + 42 + xs (1-11)
1-2 Arc length, tangent. We suppose in this and in the next sections(until Sec. 1-12) that the curve C is real, with real u. Then, as shown inthe texts on calculus,* we can express the arc length of a segment of thecurve between points A (uo) and P(u) by means of the integral
s(u) = fv±2 + y2 + du = J V du. (2-1)+b
* See e.g. P. Franklin, A treatise on advanced calculus, John Wiley and Sons,New York, 1940, pp. 284, 294; Ch. J. de la Vallee Poussin, Cours d'analyse infini-tesimale, Dover Publications, New York, 1946, Tome I, 8th ed., p. 272.
6 CURVES
The dot will always indicate differentiation with respect to u:
a = dx/du, i; = dx1/du. (2-2)
The square root is positive. The expression g g is the scalar product of awith itself; it is always >0 for real curves. x is assumed to be nowherezero in the given interval (no singular points, see Sec. 1-1).
We define the scalar product of two vectors v(vi) and w(w;) by the formula:
V W = W V = V1W1 + V2W2 + V8w3
It can be shown that, V being the anglebetween v and w (Fig. 1-6),
v w = JvJJwJ cos 'p. (2-4)
This shows that v w = 0 means that vand w are perpendicular. A unit vectoru satisfies the equations
Jul=1,FIG. 1-6
(2-3)
The arc length s increases with increasing u. The sense of increasingarc length is called the positive sense on the curve; a curve with a sense onit is called an oriented curve. Most of our reasoning in differential geom-etry is with oriented curves; our space has also been oriented by the intro-duction of a right-handed coordinate system. However, our results areoften independent of the orientation.
When we change the parameter on the curve from u to ul the arc lengthretains its form, with ul instead of u. We can express this invarianceunder parameter transformations by replacing Eq. (2-1) with the equation
ds2 = dx2 + dye + dz2 = dx dx, (2-5)
which is independent of u.When we now introduce s as parameter instead of u - which is always
legitimate, since ds/du 0 0 - then Eq. (2-5) shows us that
dx dx ='
1
d ds (2-6)
The vector dx/ds is therefore a unit vector. It has a simple geometricalinterpretation. The vector Ax joins two points P(x) and Q(x + .x) onthe curve. The vector Ax/As has the same direction as Ax and for As -+ 0
1-21 ARC LENGTH, TANGENT
passes into a tangent vector at P (Fig. 1-7).
Since its length is 1 we call the vector
t = dx/ds (2-7)
the unit tangent vector to the curve at P. Itssense is that of increasing s. Since
dx dx dsdu ds du'
(2-8)
we see that 1i = dx/du is also a tangent vector,though not necessarily a unit vector.
We often express the fact that the tangentis the limiting position of a line through Pand a point Q in the given interval of u, whenQ -> P, by saying that the tangent passes 0through two consecutive points on the curve.This mode of expression seems unsatisfactory,but it has considerable heuristic value and canstill be made quite rigorous.
7
Ax
FIG. 1-7
THEOREM. The ratio of the arc and the chord connecting two points P andQ on a curve approaches unity when Q approaches P.
Indeed, when As is the are PQ and c is the chord PQ, then for Q -' P(Fig. 1-8):
As Aslim =1im
c N/(AX)2 + (Ay)2 + (AZ)2
0s/Du= lim
(yu)z+ (A/Jfy\z + (Qu\z
s
)
x2 + y2 + j2= 1, (2-9)
FIG. 1-8
which proves the theorem. It also proves that the ratio of Au and c isfinite.
The ratios Ox/Ds, Ay/Os, t z/Os (Fig. 1-8) therefore approach, for As -> 0,the cosines of the angles which the oriented tangent at P makes with thepositive X-, Y-, and Z-axes. This means that t can be expressed in theform:
t = e1 cos a, + e2 cos a2 + e3 cos a3, (2-10)
S CURVES
which is in accordance with the identity
cost al + cost a2 + cost a3 = 1. (2-11)
A generic point A (X) on the tangent line at P isdetermined by the equation (Fig. 1-9) :
X = x + vt, v = PA, (2-12)
in coordinates (supposing all dxi P` 0)
X1 - X1 _ X2 - X2 _ X3 - x3 (2-13)COS al COS a2 cos a3
or
X1 - x1 _ X2 - x2 _ X3 - xdxl dx2 dx3
ExAMPLES. (1) Circle (Fig. 1-10).
(2-14)
x = a cos u, y = a sin u, z = 0; (2-15)z =-a sin u, y = a cos u;s = au + const, take s = au;
x = a cos (s/a), y = a sin (s/a).
Unit tangent vector: t(-sin u, cos u).
Equation of tangent line:
X1-acosuX2-asinu- sin u coS u
Fm. 1-10B
FIG. 1-9
FIG. 1-11
[CH. I
1-2] ARC LENGTH, TANGENT
or, writing for (X1X2) again (xy):
9
x cos u + y sin u = a.
(2) Circular helix (Fig. 1-11).
x = a cos u, y = a sin u, z = bu; (2-16)
.t =-a sin u, y= a cos u, z=b, s = ua2+b2=cu
tI--sinu,acosu,b)\\ c c C
The tangent vector makes a constant angle a3 with the Z-axis:
cos a3 = b/c, hence tan a3 = a/b.
If B is the intersection of the tangent at P with the XOY-plane, and P3the projection of P on this plane, then
P3B = PP3 tan a3 = bub = au = are AP3.
The locus of B is therefore the involute of the basic circle of the cylinder(see Sec. 1-11).
(3) A space curve of degree four
(Fig. 1-12).
x = all -+- cos u), y = a sin u,z = 2a sin u/2.
For 0 < u < 2ir we obtain thepoints above the XOY-plane; for- 27r < u < 0 those below this plane.The whole curve is described for - 27r< u < 27r. Elimination of u gives
x2 + y2 + 32 = 4a2,(x - a)2 + y2 = a2.
The curve is the full intersection ofthe sphere with center at 0 and radius2a, and the circular cylinder withradius a and axis in the XOZ-plane FIG. 1-12
10 CURVES [CH. 1
at distance a from OZ. The substitution u = tan u/4 allows us to writethe coordinates of the curve by means of rational functions:
2a(1 - u2)2 4au(1 - u2) 4auX
(1-+2)2' l' (1+u2)2 z=1+u2.
Substitution of these expressions into the equation of a plane,
ax + by + cz + d = 0,
gives a biquadratic equation, showing that the degree of the curve is fourindeed. The are length
s=aJ 1+cos22du0
is an elliptic integral.
The reader will recognize in Fig. 1-12 the so-called temple of Viviani (1692),well known in the theory of multiple integration, remarkable because both thearea and the volume of the hemisphere x > 0 outside the cylinder is rational in a.
1-3 Osculating plane. The tangent can be defined as the line passingthrough two consecutive points of the curve. We shall now try to find theplane through three consecutive points, which means the limiting positionof a plane passing through three nearby points of the curve when two ofthese points approach the third. For this purpose let us consider a plane
X a = p; X generic point of the plane, a 1 plane, p a constant, (3-1)
passing through the points P, Q, R on the curve given by X = x(uo),X = x(u1), X = X(U2). Then the function
f (u) = x a - p, x = x(u) (3-2)
satisfies the conditions
f (uo) = 0, f(U1) = 0, f (U2) = 0.
Hence, according to Rolle's theorem, there exist the relations
f(Vi) = 0, f'(v2) = 0, uo < v1 < u1, u1 < V2 < U2,and
f"(v3) = 0, v1 < V3 <, v2.
When Q and R approach P, u1, u2, v1, v2, v3 -+ uo, and writing u for uo,we obtain, for the limiting values of a and p, the conditions:
f(u)f'(u) = x a = 0, (z, it at P(u)) (3-3)f"(u)=2-a=0.
1-31 OSCULATING PLANE 11
Eliminating a from Eqs. (3-3) and (3-1) we obtain a linear relationbetween X - x, it, it (all 1 a) :
X = x + XI + µit, X, µ arbitrary constants, (3-4a)in coordinates
X1-x1 X2-x2 X3-x3x1 x2 t3 = 0 (3-4b)xl z2 t3
This is the equation of the plane through three consecutive points of thecurve, to which John Bernoulli has given the pleasant name of osculatingplane (German: Schmiegungsebene). It passes through (at least) three con-secutive points of the curve. It also passes through the tangent line (givenby A = 0 in Eq. (3-4a)). The osculating plane is not determined whenit = 0 or when it is proportional to it.
We can express Eqs. (3-4) in another way by the introduction of the vectorproduct of two vectors v and w:
e1 e2 e3
V1 V2 V3
w1 w2 W3
= -w x v. (3-5)
It can be shown that, being the angle of v and w,
v x w = Ivl lwl sin (p u, (3-6)
where u is a unit vector perpendicular to v and w in such a way that thesense v -+ w --+ u is the same as that of OX -> 0 Y -+ OZ, hence a right-handedone. We always have v x v = 0. When v x w = 0, then w has the directionof v, or w = Xv; in this case we say that v and w are collinear.
The triple scalar product (or parallelepiped product) of three vectors (v, w, u) is:
ul u2 u3
V1 V2 V3
W1 W2 W3
(vwu) = (wuv) = (uvw) = etc. (3-7)
It is zero when the three vectors (each supposedly 54 0) are coplanar, that is,can be moved into one plane.
The following formula is also useful:
(a x b) (c x d) = (a c) (b d) - (a d) (b c), (3-8)
with the special case
(a x b) (a x b) = (a a) (b b) - (a b)2. (3-9)
12 CURVES [CH. 1
The equation of the osculating plane can also be written (comp. Eq. (3-7)with Eq. (3-4b)):
(X-x,z,x)=0. (3-10)
As to the two exceptional cases, if they are valid at all points of thecurve then they both are satisfied for straight lines and only for those:
(a) x=0, x=a, x=ua+b(b) it = Xi, x = cex' = cfl(t), x = cf2(t) + d,
where a, b, c, d are fixed vectors.If it = X i ('ii Fl- 0) at one point of the curve, then we call this point
a point of inflection. The tangent at such a point has three consecutivepoints in common with the curve (see Section 1-7).
Since PQ passes into the tangent at P, andQR into the tangent at Q, we say that theosculating plane contains two consecutive tangentlines. This indicates that we may facilitateour understanding of the nature of the oscu- p,lating planes by taking (Fig. 1-13) the pointsP1, P2, P3, ... on the curve and consideringthe polygonal line P1P2.P3... The sidesP1P2, P2P3, ... are all very short, and repre-sent the tangent lines; the planes P1P2P3,P2P3P4, ... represent the osculating planes.Two consecutive tangent lines P1P2j P2P3 liein the osculating plane P1P2P3; two consecu-tive osculating planes P1P2P3, P2P3P4 intersectin the tangent line P2P3, etc.
P1
FIG. 1-13
ExAMPLES. (1) Plane curve. Since in this case the lines PQ and QR, con-sidered in Eq. (3-1), lie in the plane of the curve, the osculating plane coin-cides with the plane of the curve. This is also clear from Fig. 1-13. Whenthe curve is a straight line the osculating plane is indeterminate and may beany plane through the line.
(2) Circular helix. The equation of the osculating plane is
X1- acosu X2- asinu X3- bu- a sin u acosu b
- acosu - a sin u 0
or, writing (x, y, z) for (X1, X2, X3):
= 0, (3-10)
bx sin u - by cos u + az = abu.
1-41 CURVATURE 13
This equation is satisfied by x = X cosu, y = X sin u, z = bu for all values ofX, which (by fixed u) shows that theosculating plane at P contains theline PA parallel to the XOY-planeintersecting the Z-axis. The planethrough PA and the tangent at P isthe osculating plane at P (Fig. 1-14).The locus of the lines PA, indicatedby P1A 1, P2A 2, P3A ...... is a sur-face called the right helicoid (see Sec.2-2, 2-8).
1-4 Curvature. The line in theosculating plane at P perpendicularto the tangent line is called the prin-cipal normal (e.g., the lines AP inFig. 1-14). In its direction we placea unit vector n, the sense of which maybe arbitrarily selected, provided it iscontinuous along the curve. If wenow take the are length as parameter:
FIG. 1-14
x = x(s), t = dx/ds = x', t t = 1, (4-1)
where the prime signifies differentiation with respect to s, then we obtainby differentiating t t = 1:
t- t' = 0. (4-2)
This shows that the vector t' = dt/ds is perpendicular to t, and since
t = x' = ±u', t' = ii(u')2 + zu", (4-3)
we see that t' lies in the plane of z and it, and hence in the osculating plane.We can therefore introduce a proportionality factor K such that
k = dt/ds = Kn. (4-4)
The vector k = dt/ds, which expresses the rate of change of the tangentwhen we proceed along the curve, is called the curvature vector. The factorK is called the curvature; SKI is the length of the curvature vector. Althoughthe sense of n may be arbitrarily chosen, that of dt/ds is perfectly deter-mined by the curve, independent of its orientation; when s changes sign, t
14 CURVES [Cxi. 1
also changes sign. When n (as is often done) is taken in the sense of dt/ds,then K is always positive, but we shall not adhere to this convention (seebelow, small type).
When we compare the tangent vectors t(u) at P 4'---Atand t + zt(u + h) at Q (Fig. 1-15) by moving tfrom P to Q, then t, At and t + At form an isoscelestriangle with two sides equal to 1, enclosing theangle Op, the angle of contingency. Since
jOtI = 2 sin Dp/2= A(P + terms of higher order in Ago,
we find for i,p->0:
1.1 = Idt/dsl = jkj = Idp/dsl, (4-5)FIG. 1-15
which is the usual definition of the curvature in the case of a plane curve.From Eq. (4-4) follows:
K2 = x" . x" (4-6)
We define R as K-1. The absolute value of R is the radius of curvature,which is the radius of the circle passing through three consecutive points of thecurve, the osculating circle. To prove it, we first observe that this circlelies in the osculating plane. Let a circle be determined in this plane asintersection of the plane and the sphere given by
(X- c) (X- c) - r2 = 0 (X generic point of the sphere, c center, r radius).
This sphere must pass through the points P, Q, R on the curve given byX = x(so), X = x(s1), X = x(s2); the vector c points from 0 to a point inthe osculating plane so that x - c lies in the osculating plane. Reasoningas in Sec. 1-3 on the function
f (s) = (x - c) (x - c) - r2, x = x(s), c, r constants,
we find, for the limiting values of c and r, the conditions
As) = 0,f(s) = 0, or (x - c) x' = 0, (4-7)
f"(s) = 0, or (x - c) x"+c lies in the osculating plane, it is a linear combination of x' and
x". Hence
t+ At
x-C=Xx'+µx",
1-5] TORSION 15
where X and µ are determined by Eq. (4-7). We find X = 0, -1 = µx" x",so that
c=x "- µx
or, in consequence of Eqs. (4-4) and (4-6) :
C = x + Kn/K2 = x + Rn. (4-8)
This shows that the center of the osculating circle lies on the principalnormal at distance RI from P. Though R = K 1 may be positive or nega-tive, the vector Rn is independent of the sense of n, having the sense of thecurvature vector. Its end point is also called the center of curvature.
Eq. (4-6) shows algebraically that the equation of the curve determines K2but not K uniquely. So long as we consider only one radius of curvature, it doesnot make much difference what sign we attach to K. The simplest way is to
take K > 0, that is, to take the sense of the curvatureKn _ vector as the sense of n. But the sign of K is of some
importance when we consider a family of curvaturevectors. For instance, if we take a plane curve (withcontinuous derivatives) with a point of inflection at A(Fig. 1-16), then the curvature vectors are pointed indifferent directions on both sides of A and it may beconvenient to distinguish the concave and the convexsides of the curve by different signs of K. The field ofunit vectors n along the curve is then continuous. We
Kn shall meet another example in the theory of surfacesFIG. 1-16 where we have many curvature vectors at one point
(Sec. 2-6).When the curve is plane we can remove the ambiguity in the sign of K by
postulating that the sense of rotation from t to n is the same as that from OX toOY. Then K can be defined by the equation K = dsp/ds, where rp is the angle ofthe tangent vector with the positive X-axis.
1-5 Torsion. The curvature measures the rate of change of the tangentwhen moving along the curve. We shall now introduce a quantity measur-ing the rate of change of the osculating plane. For this purpose we intro-duce the normal at P to the osculating plane, the binormal. In it we placethe unit binormal vector b in such a way that the sense t -' n -p b isthe same as that of OX - OY -- OZ; in other words, since t, n, b aremutually perpendicular unit vectors, we define the vector b by the formula :
b=txn. (5-1)
CURVES [CH. 1
These three vectors t, n, b can be taken as a new frame of reference.
16
They satisfy the relations
(5-2)
This frame of reference, moving along the curve, forms the moving trihedron.The rate of change of the osculating plane is expressed by the vector
b' = db/ds.
This vector lies in the direction of the principal normal, since, according tothe equation b t = 0,
b b = 1,
0,
so that, introducing a proportionality factor r,
db/ds = - rn. (5-3)
We call r the torsion of the curve. It may be positive or negative, likethe curvature, but where the equation of the curve defines only K2, it doesdefine r uniquely. This can be shown by expressing r as follows:
(t x n')= - K1x if. (x, x (K 'x ),)=
or
(x'x"x"')r = x x , x' = dx/ds. (5-4)
This formula expresses r in x(s) and its derivatives independently of theorientation of the curve, since change of s into -s does not affect the right-hand member of (5-4). The sign of r has therefore a meaning for thenonoriented curve. We shall discuss this further below.
The equations
x'(s) = dx/ds = (dx/du)(du/ds) = au' = x(x a)-1/2,
x" = %(u')2 + xu" = [(x x)x - (x . x)±](± . x)-2,x", = %(u')8 + 3%u'u" + ku'",
allow us to express K2 and r in terms of an arbitrary parameter. We findthe formulas
1-5] TORSION
K2 =(x x) 3
(xxx)
17
g = dx/du (5-5a)
(5-5b)
We see that K and T have the dimension L-1. Where IK 1I = SRI iscalled the radius of curvature, IT 'I = ITI is called the radius of torsion.However, this quantity ITI does not admit of such a ready and elegantgeometrical interpretation as IRI.
EXAMPLES. (1) Circular helix. From Eq. (2-16) we derive, if a2+b2=c,
(- a aX,c sin u, c cos u,
c,
a" - c cos u, - sin u, 0 I, (s parameter, u = C)
X... ac3
sin u, - ac3
cos u, 0 .
Hence0 K=±a/c2,
(x'x"x`__ bx" x" e
a a--cosu --sinu c'__ba2C2a . aWsin u - Wcosu
Hence T is positive when b is positive, which is the case when (see Sec. 1-1)the helix is right-handed; r is negative for a left-handed helix. We also seethat K and T are both constants, and from the equations
F K Ta
K2 + T2, b K2 + T
we can derive one and only one circular helix with given K, r and with givenposition with respect to the coordinate axes (change of a into -a does notchange the helix; it only changes u into u +7r).
(2) Plane curve. Since b is constant, T = 0. If, conversely, r = 0,0, or z + Xii + µz = 0; A, A functions of s. This is a linear
homogeneous equation in g, which is solved by an expression of the form
x = clfl(S) + c2f2(s),*
* See e.g. P. Franklin, Methods of advanced calculus, New York: McGraw-HillBook Co., 1941, p. 351. A vector equation is equivalent to three scalar equations,so that the result reached for scalar differential equations can immediately betranslated into vector language.
18
hence
CURVES ICH. 1
x = c1F1(s) + c2F2(s) + c3,
where the c,, i = 1, 2, 3, are constant vectors and the F;, j = 1, 2, functionsdetermined by the arbitrary X and A. This shows that the curve x(s)lies in the plane through the end point of c3 parallel to cl and C2. Thismeans that x(s) can be any plane curve. For straight lines the torsion isindeterminate.
Curvature and torsion are also known as first and second curvature, andspace curves are also known as curves of double curvature.
The name torsion is due to L. I. Vallt e, Traite de geometrie descriptive, p. 295of the 1825 edition. The older term was flexion. The name binormal is due toB. de Saint Venant, Journal Ecole Polytechnique 18, 1845, p. 17.
1-6 Formulas of Frenet. We have found that t' = Kn and b' = - rn.Let us complete this information by also expressing n' = do/ds in termsof the unit vectors of the moving trihedron. Since n' is perpendicular to n,n n' = 0, and we can express n' linearly in terms of t and b:
n' = alt + alb.
Since according to Eq. (5-2)
al =and
we find for do/ds:
The three vector formulas,
dt
ds
b' = r,
n' _ - Kt + rb.
Kn
+ rb (6-1)
- rn
together with dx/ds = t, describe the motion of the moving trihedron alongthe curve. They take a central position in the theory of space curves andare known as the formulas of Frenet, or of Serret-Frenet.
1-6] FORMULAS OF FRENET 19
They were obtained in the Toulouse dissertation of F. Frenet, 1847, of whichan abstract appeared as "Sur les courbes h double courbure," Journal de Mat hem.17 (1852), pp. 437-447. The paper of J. A. Serret appeared Journal de Mathkm.16 (1851), pp. 193-207; it appeared after Frenet's thesis, but before Frenet madehis results more widely known.
The coordinates of t, n, and b are the cosines of the angles which theoriented tangent, principal normal, and binormal make with the positivecoordinate axes. When we indicate this by t(cos a$), n(cos F+i), b(cos tii),i = 1, 2, 3, the Frenet formulas take the following coordinate form:
d3COS ai = K COS $j,
dsCOS Ni =-K Cos ai + r Cos yi,
s cos yi = - T COS Pi. (6-1a)
The three planes formed by the three sides of the moving trihedron(Fig. 1-17) are called:
the osculating plane, through tangentand principal normal, with equation
(y-the normal plane, through principalnormal and binormal, with equation
(y-x)-t=0,the rectifying plane, through binormaland tangent, with equation
(y -
Rectifyingplane
Normal
LP--2lane
- -1COsculating plane
FIG. 1-17
If we take the moving trihedron at P as the trihedron of a set of new Car-tesian coordinates x, y, z, then the behavior of the curve near P is expressedby the formulas (6-la) in the form (x" = Kn, x"' = - K2t + K'n + Krb):
x'1, y'=0, z'0,x" = 0, y" = K, z" = 0, (6-2)
X111 =-K2 y111 = K, Z... = KT.
From these equations we deduce for s --> 0
lim y2 = lim 22xx'2x = 2' (6-3)
z z' z" z"' KTlim z = lim
322x'= lim 6x(x,)2 = lim
6(x')s = 6'
20 CURVES [CH. 1
hence
lim z2 T2 8 221C 1 =
2RT2.
y3= -- = 7-2K-1
36 K3 9 9
b-z
N t=x
FIG. 1-19
This shows that the projections of the curve on the three planes of the
moving trihedron behave near P like the curves
y = 2 x2 (projection on the osculating plane),
z = 6 x3 (projection on the rectifying plane), (6-4)
z2 =
9
r2Ry3 (projection on the normal plane).
Fig. 1-18 shows this behavior in an orthographic projection, takingK > 0, T > 0. If the sign of -r is changed, the projection on the rectifyingplane changes to that of Fig. 1-19. This again shows the geometricalmeaning of the sign of T.
Fig. 1-20 gives a representation of the curve and its trihedron in space.
EXERCISES
1. Find the curvature and the torsion of the curves:(a) x u, y=v2, z=u3.
U1.(b) x=u, y= uu z=1u (Why isr=0?)
1-6] FORMULAS OF FRENET 21
Fla. 1-20
(c) y = f(x), z = 9(x)(d) x = a(u - sin u), y = all - cos u), z = bu.(e) x = a(3u - u3), y = 3aus, z = a(3u + u3) (Here K' = r2.)
2. (a) When all the tangent lines of a curve pass through a fixed point, thecurve is a straight line. (b) When they are parallel to a given line the curve is alsoa straight line.
3. (a) When all the osculating planes of a curve pass through a fixed point,the curve is plane. (b) When they are parallel to a given plane the curve is alsoplane.
4. The binormal of a circular helix makes a constant angle with the axis of thecylinder on which the helix lies.
5. Show that the tangents to a space curve and to the locus C of its centers ofcurvature at corresponding points are normal.
6. The locus C of the centers of curvature of a circular helix is a coaxial helixof equal pitch.
7. Show that the locus of the centers of curvature of the locus C of problem 6 isthe original circular helix and that the product of the torsions at correspondingpoints of C and the helix is equal to K', the square of the curvature of the helix.
22 CURVES [CH. I
8. All osculating planes to a circular helix passing through a given point notlying on the helix have their points of contact with the helix in the same plane.
9. Prove that the curve
x = a sine u, y = a sin U COs u, z=acosu
lies on a sphere, and verify that all normal planes pass through the origin. Showthat this curve is of degree four.
10. When x = x(t) is the path of a moving point as a function of time, showthat the acceleration vector lies in the osculating plane.
11. Determine the condition that the osculating circle passes through fourconsecutive points of the curve.
12. Show that for a plane curve, for which x = x(s), y = y(s), and the sign of Kis determined by the assumption of p. 15, K = x'y" - x"y'.
13. Starting with the Eq. (1-5) for the curve a(s), derive expansions for the pro-jections on the three planes of the moving trihedron, and compare with Eq. (6-4).
14. Determine the form of the function o(u) such that the principal normals ofthe curve x = a cos u, y = a sin u, z = p(u) are parallel to the XOY-plane.
15. (a) The binormal at a point P of the curve is the limiting position of thecommon perpendicular to the tangents at P and a neighboring point Q, when Q P.Also find (b) the limiting position for the common perpendicular to the binormals.
16. Find the unit tangent vector of the curve given by
F,(x, y, z) = 0, F2(x, y, z) = 0-
17. Transformation of Combescure. Two space curves are said to be obtainablefrom each other by such a transformation if there exists a one-to-one correspondencebetween their points so that the osculating planes at corresponding points areparallel. Show that the tangents, principal normals, and binormals are parallel.(Following L. Bianchi, Lezioni I, p. 50, we call such transformations after E. J. C.Combescure, Annales Ecole Normale 4, 1867, though the transformations discussedin this paper are more specifically qualified, and deal with certain triply orthogonalsystems of curves.)
18. Cinematical interpretation of Frenet's formulas. When a rigid body rotatesabout a point there exists an axis of instantaneous rotation, that is, the locus of thepoints which stay in place. Show that this axis for the moving trihedron (we donot consider the translation expressed by dx = t ds) has the direction of the vectorR = rt + Kb, so that the Frenet formulas can be written in the form
t'=Rxt, n'=Rxn, b'=Rxb.
This constitutes the approach to the theory of curves (and surfaces) typical ofG. Darboux and E. Cartan, the "m6thode du tribdre mobile." (Compare G. Dar-boux, Legons I; E. Cartan, La methode du repere mobile, Actualitess scientifiques194, Paris, 1935.)
1-71 CONTACT 23
19. Spherical image. When we move all unit tangent vectors t of a curve C to apoint, their end points will describe a curve on the unit sphere, called the sphericalimage (spherical indicatrix) of C. Show that the absolute value of the curvature isequal to the ratio of the arc length dst of the spherical image and the arc length ofthe curve ds. What is the spherical image (a) of a straight line; (b) of a planecurve; (c) of a circular helix?
20. Third curvature. When we extend the operation of Exercise 19 to the vectorsn and b, we obtain the spherical image of the principal normals and of the binormals.If ds and dsb represent the elements of are of these two images respectively, showthat =+1/K2 + r2 and Idsb/dsj = ITI. The quantity K2 V+ r2 is some-times called the third (or total) curvature.
1-7 Contact. Instead of stating that figures have a certain number ofconsecutive points (or other elements) in common, we can also state thatthey have a contact of a certain order. The general definition is as follows(Fig. 1-21):
Let two curves or surfaces 2;1, 2;2 have a regular point P in common. Takea point A on 2;1 near P and let AD be its distance to 12 Then 12 has a con-tact of order n with F+1 at P, when for A -' P along E1
lim (Ap k
is finite (V-0) for k = n + 1, but =0for k = n. [AD = o((AP)k) for k = 1,2, .., nJ
When M1 is a curve x(u) and 12 asurface (Fr, F,,, Fs not all zero)
FIG. 1-21
E1
F(x, y, z) = 0, (7-1)
we make use of the fact that the distance AD of a point A (x1, y1, z1) near Pis of the same order as F(xi, yi, z1). The general proof of this fact requiressome surface theory, but in the case of the plane and the sphere, the onlycases we discuss in the text, it can be readily demonstrated (see Exercise 4,Sec. 1-11).
Let us now consider the function obtained by substituting the x; of thecurve E1 into Eq. (7-1):
f(u) = F[x(u), y(u), z(u)]. (7-2)
This procedure is simply a generalization of the method used in Sees. 1-3and 1-4 to obtain the equations of the osculating plane and the osculatingcircle. Let f (u) near P(u = uo) have finite derivatives f (') (uo), i = 1, 2,
24 CURVES [CH. 1
..., n + 1. Then if we take u = u, at A and write h = ul - uo, thenthere exists a Taylor development of f(u) of the form (compare Eq. (1-5)):
2 hn+lf(ul) = f(uo) + hf'(uo) +
h2)f(uo) + . + (n +
1)iP+1) (uo) + o(hn+1).
(7-3)
Here f(uo) = 0, since P lies on 2;2, and h is of order AP (see theoremSec. 1-2) ; f (u,) is of the order of AD. Hence necessary and sufficient condi-tions that the surface has a contact of order n at P with the curve are that at Pthe relations hold:
f(u) = f'(u) = f"(u) = . . . = f(n)(u) = 0; f(n+l)(u) T 0. (7-4)
In these formulas we have replaced uo by u.In the same way we find, if E2 is a curve defined by
Fi(x, y, z) = 0, F2(x, y, z) = 0,
that necessary and sufficient conditions for a contact of order n at P betweenthe curves are that at P
AM = fi(u) = ... = fi")(u) = 0, (7-5)f2(u) = f2(u) = ... = fr)(u) = 0,
where
fl(u) = FI[x(u), y(u), z(u)], f2(u) = F2[x(u), y(u), z(u)]
and at least one of the two derivatives fJn+I)(u), fP+I)(u) at P does not vanish.We can develop similar conditions for the contact of two surfaces.
Instead of AD we can use segments of the same order, making, for in-stance, PD = PA (then L D no longer 90°).
If we compare these conditions with our derivation of osculating planeand osculating circle, then we see that they are, in these cases, identicalwith the condition that E, and E2 have n + 1 consecutive points in common.And so we can say that, in general:
Two figures M, and 12, having at P a contact of order n, have n + 1 consecu-tive points in common.
Indeed, following again a reasoning analogous to that of Secs. 1-3 and1-4, and confining ourselves to the case expressed by Eq. (7-1), let usdefine F with n + 1 independent parameters. These are just enough to letsurface E2 pass through (n + 1) points (uo, ul, . . ., un). If these n + 1points come together in point u = uo, then the n + 1 equations (7-4) aresatisfied; if there were more such equations (7-4), then we could determinethe parameters in F so that F+2 would pass through more than n + 1 points.
1-7] CONTACT 25
A similar reasoning holds for the other cases of contact. From this andthe theorems of Sees. 1-3 and 1-4 follows:
A tangent has a contact of (at least) order one with the curve.An osculating plane and an osculating circle have a contact of (at least)
order two with the curve.
The study of the contact of curves and surfaces was undertaken in considerabledetail in Lagrange's Traite des fonctions analytiques (1797) and in Cauchy'sLesons sur les applications du calcul infinitesimal a la geometric I (1826).
We shall now apply this theory to find a sphere passing through fourconsecutive points of the curve, the osculating sphere. Let this sphere begiven by the equation
(X- c) (X-c)-r2=0, (X generic point of the sphere, c its center, r radius).
Consider, in accordance with Eq. (7-2) :
f(s) = (x - c) (x - c) - r2, x = x(s).
Then the Eqs. (7-4) take the form, apart from f(s) = 0:
f(s) = 0, or (x -f"(s) = 0, or (x - c) Kn + 1 = 0,f"'(s) = 0, or (x - C) (K'n - K2t + Krb) = 0,
or (r 516 0)
(x - (x - (x -The center 0 of the osculating sphere is thus uniquely determined by
c = x + Rn + TR'b. (7-6)
This sphere has a contact of order three with the curve. Its intersectionwith the osculating plane is the osculating circle. Its center lies in thenormal plane (Fig. 1-22) on a line parallel tothe binormal, the polar axis. The radius of b
the osculating sphere isO
r = R2 ++ (TR')2. (7-7) R,When the curve is of constant curvature (not -Ra circle), the center of the osculating sphere co- P O' n
incides with the center of the osculating circle.
The result expressed by Eq. (7-6) is due toMonge (1807), see his Applications (1850), p. 412. tMonge's notation is quite different. FIG. 1-22
26 CURVES [CH. 1
1-8 Natural equations. When a curve is defined by an equationx = x(s), its form depends on the choice of the coordinate system. When acurve is moved without change in its shape, its equation with respect to thecoordinate system changes. It is not always immediately obvious whethertwo equations represent the same curve except for its position with respectto the coordinate system. Even in the simple case of equations of thesecond degree in the plane (conics) such a determination requires somework. The question therefore arises: Is it possible to characterize a curveby a relation independent of the coordinates? This can actually be ac-complished; such an equation is called natural or intrinsic.
It is easily seen that a relation between curvature and arc length gives anatural equation for a plane curve. Indeed, if we give an equation of theform
K = K(S),
then we find, using the relations
R-1 = K = dcp/ds, cos p = dx/ds,
that x and y can be found by two quadratures:
(8-1)
sin ' = dy/ds,
N V 8
x = R cos s dcp, y = R sin (p djp, lp = K ds. (8-2)wu Wo so
Change of integration constant in x and y means translation, change ofintegration constant in p means rotation of the curve, and thus we canobtain all possible equations in rectangular coordinates, selecting in eachcase the most convenient one for our purpose.
This representation of a curve by means of K (or R) and s goes back to Euler,who used it for special curves: Comment. Acad. Petropolit. 8, 1736, pp. 66-85.The choice of K and s as natural coordinates can be criticized, since s still con-tains an arbitrary constant and K is determined but for the sign. G. Scheffershas therefore developed a system of natural equations of a plane curve in whichd(K2)/drp is expressed as a function of K2. See Anwendung 1, pp. 84-91.
EXAMPLES. (1) Circle: K = a = const.
x=Rsinp, y= -Rcossp, ifu=gyp-2,x = R cos u, y = R sin u. (Fig. 1-23)
When a = 0 we obtain a straight line.(2) Logarithmic spiral.
R = as + b = s cot a + p csc a (a, p constants),
1-8] NATURAL EQUATIONS 27
FIG. 1-24
FIG. 1-23
s cosa + p, (selecting s = 0 for ' = a),p = a -1- (tan a In
R = p(csc a) exp (0 cot a), 0 = - a. (Fig. 1-24)
Introducing polar coordinates x + iy = re', we find
r = p exp (B cot a) (x = p, y = 0 for s = 0, rp = a).
(3) Circle involute:
R2 = gas; rp = 2 a N/s, s =
R = arp, x = a cos w d(p = a(cos rp + p sin (p),
y= aJ rp sin (p dip = a(sin p - p cos gyp). (Fig. 1-25)
(4) Epicycloid. We start with the equations in x and y:
x = (a + b) cos - b cos b bb
y = bsin a b+b
s=4b(a+b)',a cos-,2bR
= 4b(a + b) sin a'a + 2b 2b
28 CURVES [CH. 1
FIG. 1-25
FIG. 1-26
FIG. 1-27
Hence the natural equation is:
zs _ _A2+ B2 - 1'
where
(8-3)
A = 4b(a + b), B = 4b(a + b).A > B. (Fig. 1-26) (8-4)
a ' a + 2b '
When A < B we obtain a hypocycloid. An epicycloid is the locus of apoint on the circumference of a circle when rolling on a fixed circle on theoutside; a hypocycloid when rolling on the inside. When A/B is rational,the curve is closed. When a - oo, A -, 4b, B -- 4b, and we obtain
s2 + R2 = 16b2,
the equation of a cycloid, obtained by rolling a circle of radius b on astraight line (Fig. 1-27). We now prove the
1-81 NATURAL EQUATIONS 29
FUNDAMENTAL THEOREM for space curves:If two single-valued continuous functions K(s) and r(s), s > 0, are given,
then there exists one and only one space curve, determined but for its positionin space, for which s is the arc length (measured from an appropriate point onthe curve), K the curvature, and r the torsion.
The equations K = K(s), r = r(s) are the natural or intrinsic equations ofthe space curve.
The proof is simple when we confine ourselves to analytic functions.Then we can write, in the neighborhood of a point s = so, h = s - so:
x(s) = x(so) + x'(so) + h2 X"(8-) + ...
provided the series is convergent in a certain interval s1 < so < s2. Then,substituting for x', x", etc., their values with respect to the moving tri-hedron at P(so), we obtain
x' = t, x" = Kn, x,,, = -K2t + K'n + Krb,
so that
x,,,,
x(s) = x(so) + ht + jKh2n + jh°(- K2t + K 'n + Krb) + , (8-5)
where all terms can be successively found by differentiating the Frenetformulas, and all successive derivatives of K and r taken, as well as t, n, byat P(so) are supposed to exist because of the analytical character of thefunctions. If we now choose at an arbitrary point x(so) an arbitrary setof three mutually perpendicular unit vectors and select them as t, n, bythen Eq. (8-5) determines the curve uniquely (inside the interval of con-vergence).
It is, however, possible to prove the theorem under the sole assumptionthat K(s) and r(s) are continuous. In this case we apply to the system ofthree simultaneous differential equations of the first order in y,
da= K0i
d= - ica + 7-f, L = - r#, (8--6)
TS ds ds
the theorem concerning the existence of solutions.
This theorem is as follows. Given a system of differential equations
y,), i= 1,2,....n,dx
where the f; are single-valued and continuous in their n + 1 arguments inside agiven interval (with a Lipschitz condition, satisfied in our case). Then there
30 CURVES [CH. 1
exists a unique set of continuous solutions of this system which assumes givenvalues y°, yz..... y°° when x = x0.*
We deduce from this theorem that we can find in one and in only one waythree continuous solutions ai(s), 01(s), yi(s) which assume for s = so thevalues 1, 0, 0 respectively. We can similarly find three continuous solu-tions a2, 02, 72, so that
a2(so) = 0, l32(so) = 1, 'Y2(50) = 0,
and three more continuous solutions a3, 03, 73, so that
a3(80) = 0, 03(60) = 0, 73(50) = 1.
The Eqs. (8-6) lead to the following relations between the a, 0,,y:
2 d (a2 + F+1 + 71) = Kola, - ICalo1 + r7'iF3l - TYlyl = 0,
or
+ 01 + 71 = const = 1 + 0 + 0 = 1. (8-7a)
Similarly, we find two more relations of the same form:
a2 + 02 + 72 = 1, a32 2+ 03 + 73 = 1, (8-7b)
and the three additional relations:
ala2 + 0q102 + 7172 = Ofa1a3 + 13103 + 7173 = 0,a2a3 + 92/33 + 7273 = 0. (8-8)
We have thus found a set of mutually perpendicular unit vectors
t(ala2a3), n(/31$2$3), b(7172y3),
where the ai, 13i, yi all are functions of the parameter s (i = 1, 2, 3).
This is the consequence of the theorem that if the relations (8-7) and (8-8)hold, the relations
Eai = 2;#i = Zy, = 1, Mai3i = Lai7i = ZQi7i = 0also hold. This means geometrically that, when t, n, b are three mutuallyorthogonal unit vectors defined with reference to the set ei, e2, e3, then el, e2, e3are three mutually orthogonal unit vectors defined with reference to the sett, n, b.
* Compare E. L. Ince, Ordinary differential equations, Longmans, Green and Co.,London, 1927, p. 71.
1-8] NATURAL EQUATIONS 31
There are oo I trihedrons (t, b, n). If we now integrate t, then theequation
x =ft ds (8-9)
determines a curve which has not only t as unit tangent vector field, butbecause of Eq. (8-6) also has (t, n, b) as its moving trihedron, K and T beingits curvature and torsion, and s, because of Eq. (8-9), its arc length.Hence there exists one curve C with given K(s) and T(s) of which the movingtrihedron at P(so) coincides with the coordinate axes.
We now must show that every other curve C which can be brought into aone-to-one correspondence with C such that at corresponding points, givenby equal s, the curvature and torsion are equal, is congruent to C. Thismeans that C can be made to coincide with C by a motion in space. Let usmove the point s = 0 of C to the point s = 0 of C (the origin) in such a waythat the trihedron (t, n, b) of C coincides with the trihedron (t, n, b) of C(the system e1, e2, e3). Let (x;, «;, i3,, Ti) and (x;, a;, j3 , -y;) now denote thecorresponding elements of the moving trihedron of C and C respectively.Then Eq. (8-6) holds for («;, 4j, y;) and for (ai, 13i, -y;) with the same K(s)and T(s). Hence (we omit the index i for a moment):
da+ada+jdO-+0do+Y +y = K«$+Ka/3- K/3a + 15y
ds ds ds ds ds ds/ /- KN« + T - T--yFiq - TyFi = 0,
or
a« + 0n + yy = const.
This constant is 1, since it is 1 for s = 0. For the a;, ... , -y; the equationshold:
aid; + 0th. + y:y. = 1, .,2 + 0,1 + y1 = 1, ai + + Y, = 1,
which is equivalent to the statement that the three vector pairs (a;, /3j, y;),(«;, 0;, -y;) make the angle zero with one another. Hence a; = «;, O; = Vii;,yc = Ti for all values of s, so that
dds
(x; - x;) = 0.
This shows that x; - x; = const, but this constant is zero, since it is zerofor s = 0. The curves C and C coincide, so that the proof of the funda-mental theorem is completed.
32 CURVES [CH. 1
All curves of given K(s) and r(s) can thus be obtained from each other bya motion of space. The resulting curves are at least three times differ-entiable.
EXAMPLES. (1) Plane curve. Here K(S) may be any function, r = 0. Inparticular, if K is constant, we find from Eq. (8-2) the circle.
(2) Circular helix. K = const, r = const. We see this immediatelyfrom Example 1, Sec. (1-5), since
a = K2
+r2 and b = K2 + r2
uniquely determine the curve
x = a cos s/c, y = a sin s/c, z = bs/c, c = a2 + b2,
and all other curves of the same given K and r must be congruent to thiscurve.
Another way to show this is indicated in Sec. 1-9.(3) Spherical curves (curves lying on a sphere). These are all curves
which satisfy the differential equation in natural coordinates
R2 + (TR')2 = a2, a = any constant. (8-10)
Indeed, when a curve is spherical, its osculating spheres all coincide withthe sphere on which the curve lies, hence Eq. (8-10) holds, where a is theradius of the sphere (according to Eq. (7-7)). Conversely, if Eq. (8-10)holds, then the radius of the osculating sphere is constant. Moreover,according to Eq. (7-6):
c' = t + (- t + R rb) + R'n + (TR')'b - R'n = {Rr+(TR')'}b = 0. (8-11)
Differentiation of Eq. (8-10) shows that for -r5-6 0, R' o 0,
Rr + (TR')' = 0, (8-12)
so that c' = 0, and this means that the center of the osculating sphere re-mains in place (except for r = 0, R' = 0, the circle). Eq. (8-12) is thereforethe differential equation of all spherical curves. The circle fits in for
= 0, R' = 0, provided TR' = 0.From Eq. (8-11) it follows, incidentally, that for a nonspherical curve
(not plane) the tangent to the locus of the centers of the osculating sphereshas the direction of the binormal.
For more information on natural equations see E. Cesaro, Lezioni di geometriaintrinseca, Napoli, 1896; translated by G. Kowalewski under the title: Vorlesun-gen caber natiirliche Geometrie, Leipzig, 1901, 341 pp. See also L. Braude, Lescoordonnees intrinseques, collection "Scientia," Paris, 1914, 100 pp. Other
1-91 HELICES 33
forms of the natural equations of a space curve can be found in G. Scheffers,Anwendung I, pp. 278-287, where the fundamental theorem is proved for(dK/ds)2 = f(K2), r = f(K2).
1-9 Helices. The circular helix is a special case of a larger class ofcurves called (cylindrical) helices or curves of constant slope (German:Boschungslinien). They are defined by the property that the tangent makesa constant angle a with a fixed line 1 in space (the axis). Let a unit vectora be placed in the direction of 1 (Fig. 1-28). Then a helix is defined by
t a = cos a = const.
Hence, using the Frenet formulas:
a is therefore parallel to the rectifying plane of the curve, and can be writtenin the form (Fig. 1-29):
a=tcosa+bsina,which, differentiated, gives
0 = Kn cos a - rn sin a = (K cos a - r sin a)n,or
K/r = tan a, constant.
For curves of constant slope the ratio of curvature to torsion is constant.Conversely, if for a regular curve this condition is satisfied, then we canalways find a constant angle a such that
n(K cos a - r sin a) = 0,
ds (t cos a + b sin a) = 0,
tFIG. 1-28 FIG. 1-29
34
or
Hence:
CURVES [CH. 1
t cos a + b sin a = a, constant unit vector, along the axis.
cos a = a t.
The curve is therefore of constant slope. We can express this result asfollows:
A necessary and sufficient condition that a curve be of constant slope is thatthe ratio of curvature to torsion be constant. (Theorem of Lancret, 1802;first proof by B. de Saint Venant, Journal Ec. Polyt. 30, 1845, p. 26.)
The equation of a helix can be written in the form (line l is here theZ-axis) :
x = x(s), y = y(s), z = s cos a,
which shows that this curve can be considered as a curve on a generalcylinder making a constant angle with the generating lines (cylindricalloxodrome). When K/r = 0 we have a straight line, when K/r = oo, aplane curve.
If we project the helix x(s) on a plane perpendicular to a, the projectionx, has the equation (see Fig. 1-30) :
x,=x -Hence
x; = t - t - a cos a,and its are length is given by
ds; = dx, dx, = sine a ds2.
Since
dx,/ds, = csc at - cot a a,its curvature vector is
d2x,/ ds; = K csc2 an,
Fia. 1-30
and its curvature K, = K csc2 a. In words:
The projection of a helix on a plane perpendicular to its axis has its prin-cipal normal parallel to the corresponding principal normal of the helix, andits corresponding curvature is K, = K csc2 a.
EXAMPLES. (1) Circular helix. If a helix has constant curvature, thenits projection on a plane perpendicular to its axis is a plane curve of con-stant curvature, hence a circle (Section 1-8). The helix lies on a cylinderof revolution and is therefore a circular helix.
7 -9] HELICES 35
(2) Spherical helices. If a helix lies on a sphere of radius r, then Eq.(8-10) holds, which, together with K = T tan a, gives after elimination of r,
r2 = R2[1 + R'2 tan2 a],or
RdR = +ds cot a,r
which, integrated for R, and by suitable choice ofthe additive constant in s, gives
R2 + 82 cot2 a = r2.
The projection of the helix on a plane perpen-dicular to its axis is therefore a plane curve withthe natural equation
FIG. 1-31
R2 + Si cos2 a = r2 sin4 a.
This type of curve is discussed in Sec. 1-8, and since cos2 a < 1, representsan epicycloid (compare Eq. (8-3)):
A spherical helix projects on a plane perpendicular to its axis in an arc ofan epicycloid.
This projection is a closed curvewhen the ratio a/b of Eq. (8-4) isrational. Using the notation of thisformula and of Eq. (8-3), we obtainthat in this case
B _ acosa = A a+2bis rational.From
B _ 4b(a + b = r sine a,a-1-2bA _ 4b(a + b)
a= r sin a tan a,
we findFIG. 1-32. Only three of the six arcs
are drawn.
a = r cos a, b = 2 (1 - cos a) = r sine a/2, (a + 2b = r)
for the radius of the fixed and of the rolling circle. When cos a = * wefind as projection an epicycloid of three cusps (Fig. 1-31); each of thethree arcs represents part of a helix on the sphere (Fig. 1-32).
This case has been investigated by W. Blaschke, Monatshefte fur Mathematikand Physik 19, 1908, pp. 188-204; see also his Diferentialgeometrie, p. 41.
36 CURVES [CH. 1
1-10 General solution of the natural equations. We have seen thatthe natural equation of a plane curve can be solved (that is, the cartesiancoordinates can be found) by means of two quadratures. In the case of aspace curve we can try to solve the third order differential equation in a(s)obtained from Eq. (8-6) by eliminating 0 and y. The solution can, how-ever, be reduced to that of a first order differential equation, a so-calledRiccati equation, which is a thoroughly studied type, but not a type whichcan be solved by quadratures only. This method, due to S. Lie and G. Dar-boux, is based on the remark that
a2 + 02 + y2 = 1 (10-1)
can be decomposed as follows:
(a + i13)(a - iQ) + 'Y)(1 - 'Y).
Let us now introduce the conjugate imaginary functions w and -z-1:
w+i/3_ 1+y. -1_a-i(_ 1+y (10-2)1-y a-i(3' z 1-y a+i13
Then it is possible to express a, 0, y in terms of w and z:
1-wz a_i1+wz y=w+z (10-3)w -z w- z w- z
These expressions are equivalent to the equations of the sphere(Eq. (10-1)) in terms of two parameters w and z. We shall return to thisin Sec. 2-3, Exercise 2.
With the aid of the Eqs. (8-6) we find for w':
dw _ a'+ i/3' a+i13 KO - i,a+try wy' r(i7 -(3w)ds 1-y +(1-y)27 = 1-y +1-y=-iKw+ 1-y
Because of Eq. (10-2) we find for S:
1 + ,Y - w2 + 772R_Z
2w
The elimination of 0 from these two equations gives an equation from whichy also has disappeared:
ds=-2r-iKw+ 2W2. (10-4)
1-10] GENERAL SOLUTION OF THE NATURAL EQUATIONS 37
Performing the same type of elimination for dz/ds, we find that z satis-fies the same equation as w. This equation has the form
ds = A + Bf + Cf, where A (s) 2, B(s) = -iK, C(s) = 2 (10-5)
This is a so-called Riccati equation. It can be shown that its general solu-tion is of the form
c1+f2f Cf3 + f,'
where c is an arbitrary constant and the fi are functions of s.
(10-6)
The fundamental properties of a Riccati equation are:(1) When one particular integral is known, the general integral can be obtained
by two quadratures.(2) When two particular integrals are known, the general integral can be found
by one quadrature.(3) When three particular integrals f1, f2, f3 are known, every other integral f
satisfies the equation
f -f2 f3-f2In words: The cross ratio of four particular integrals is constant.* Eq. (10-6)is a direct consequence of Eq. (10-7).
Let fl, f2, f3, f4 be such functions of s that Eq. (10-6) is the generalintegral of Eq. (10-4). In order to find ai, Ai, -ti, i = 1, 2, 3, we needthree integrals wi and three integrals zi, to be characterized by constantsc1, c2, C3 for the wi, and three constants d1, d2, d3 for the zi:
1 - w1z1 pp 1 + wiz1 W1 + Z1 etc.a1= w1-Z1f Ml=i w1-z1r YI =V)1-Z1f
The nine functions ai, $i, yi must satisfy the orthogonality conditions
ai + Qi + Y+ = 1, i = 1, 2, 3;
alai + 0i0i + 'Y(Yi = 0, i, j = 1, 2, 3, i P1 j.
The first three are automatically satisfied by virtue of Eq. (10-1), and there-fore we have to find the ci, di in such a way that the last three conditions
* Proofs in L. P. Eisenhart, Differential geometry, p. 26, or E. L. Ince, Ordinarydifferential equations (1927), p. 23.
38 CURVES [CH. 1
are also satisfied. One choice of cti, d, is sufficient, since all other choiceswill give curves congruent to the first choice. Now ala2 + 0102 + 'Y1 Y2 = 0can be written as follows:
or
(w1 - w2) (z1 - z2) = - (w1 - z2) (ZI - W2),
2(wlzl + W2Z2) = wlz2 + w2z1 + w1w2 + z1z2,
and if we substitute for w1, w2, zl, Z2 their values (see Eq. (10-6)), with theconstants c1, C2, d1, d2 respectively, we obtain the same relation for theconstants:
2(cid, + c2d2) = c1d2 + CA + C1c2 + d1d2.
Similarly:2(C2d2 + C33) = C2d3 + CA + C2C3 + d2d3,2(c3d3 + cidi) = c3d1 + cld3 + C3C1 + d3d1.
Every solution of these three equations in six unknowns c;, d; will give acoordinate expression for the curve. A simple solution is the following:
Cl = 1, c2 = i, c3 = 00 ; d1 = - 1, d2 =-i, d3 = 0.
To verify this, substitute into the three equations the values ofc1, c2, d1, d2, which gives CA = - 1 as well as c3d3 = + 1 which is compatiblewith c3 = oo, d3 = 0. The c and d form three pairs of numbers, eachpair of which is harmonic with respect to each other pair. This is a directresult of the properties of the general solution of the Riccati equation.Hence w1 = (ff + f2)/(f3 + f4), etc. We thus obtain for the a;:
1 - wlzl _ (1? - fl) - (f2 - f4)WI - zl 214 - J23(JJJ)
= 1-w2,z2= (fl-fl)+(fz-f+)1a2W2 - Z2 L 2(Jlf4 - f213)1 - w3Z3 = f3{4 - fl (10-8)a3 =W3 - Z3 fIf 4 - { I'2{3'
which results in the following theorem.
J J
If the general solution of the Riccati equation
df=-i itds 2
r - ikf + 2 f2
is found in the form (fl, f2, f3, f4 functions of s)
M + J2J __Cf3 + J4,
(c arbitrary constant)
EVOLUTES AND INVOLUTES
then the curve given by the equations
x =f sat ds, y = J sae ds, z =Jsa3 ds,
39
where the a; are given by Eq. (10-8), has K(s) and T(s) as curvature and torsion.
This reduction to a Riccati equation goes in principle back to Sophus Lie(1882, Werke III, p. 531) and was fully established by G. Darboux, Lecons I,Ch. 2. We find Eq. (10-8) in G. Scheffers, Anwendung I, p. 298.
EXAMPLES. Plane curve. When r = 0:
df/f = -iK ds,('
f =ce'°, =J Kds,
fl = e ", f2 = 0, f3 = 0, f4 = 1,
which lead to the Eqs. (8-2) of the plane curve.Cylindrical helix. In this case the Riccati equation (10-4) can be written
in the formw' _ - ari(1 + 2cw - w2). (c constant)
Two integrals can immediately be found by taking w2 - 2cw + 1 = 0.The general solution of this equation can now be found by means of onequadrature. For the details of this problem we may refer to Eisenhart,Differential geometry, p. 28.
1-11 Evolutes and involutes. The tangents to a space curve x(s)generate a surface. The curves on this surface which intersect the gener-ating tangent lines at right angles form the involutes (German: Evolvente;French: developpantes) of the curve. Their equation is of the form(Fig. 1-33):
y = x + fit. (X a function of s)(11-1)
The vector dy/ds is a tangent vectorto the involute. Hence:
t dy/ds = 0,
1 + = 0'Aconst-s=c-s. FIG. 1-33
40 CURVES
The equation of the involutes is therefore
y=x+(c-s)t.
[CH. I
(11-2)
For each value of c there is an involute; they can be obtained by unwindinga thread originally stretched along the curve, keeping the thread taut allthe time.
The converse problem is somewhat more complicated: Find the curveswhich admit a given curve C as involute. Such curves are called evolutes ofC (German: Evolute; French: developpees). Their tangents are normal toC(x) and we can therefore write for the equation of the evolute y (Fig. 1-34) :
Hencey=x+ain+a2b.
dLy
s = t(1 - a1K) + n ( S1 - rat/d
+ b(dd8a2
+ ra)must have the direction of aln + a2b, thetangent to the evolute:
andK = 1/al, R = a1,
ds' - rat ds2 + rala1 a2
which can be written in the form
a2dR-Rda2a2+R2
This expression can be integrated:
tan-1R = fr A + const,
or
a2 = R [cot (Jr ds + const)].
The equation of the evolutes is:
y=x+R [n+cot (frd8+const) b]. (11-3)
FIG. 1-34
FIG. 1-35
C
1-111 EVOLUTES AND INVOLUTES 41
This equation shows that a point of the evolute lies on the polar axis of thecorresponding point of the curve, and the angle under which two differentevolutes are seen from the given curve is constant (Fig. 1--35).
For plane curves:y = x + Rn + XRb. (X a constant)
For X = 0 we obtain the plane evolute. The other evolutes lie on thecylinder erected on this plane evolute as base and with generating linesperpendicular to the plane. They are helices on this cylinder.
The theory of space evolutes is due to Monge (1771, published in 1785, thepaper is reprinted in the Applications). Further studies were published byLancret, Menmires presentees et l'Institut, Paris, 1, 1805, and 2, 1811. Lancretdiscussed the "developpoides" of a curve, which are the curves of which thetangents intersect the curve at constant angle not = 90°.
The locus of the polar axes is a surface called the polar developable(see Sec. 2-4). On this surface lie the oo' evolutes of the curve and alsothe locus of the osculating spheres. This locus cannot be one of theevolutes, since its tangent has the direction of the binormal b (see Eq.(8-11)), while the tangent to the evolute has the direction Rn + alb.
EXERCISES
1. The perpendicular distance d of a point Q(y) to a line passing through P(x)in the direction of the unit vector u is d = I (y - x) x uI (Fig. 1-36). Using thisformula, show that the tangent has a contact of order one with the curve.
d
I
FIG. 1-36 FIG. 1-37
2. The perpendicular distance D of a point Q(y) to a plane passing throughP(x) and perpendicular to the unit vector u is D = I(y - x) uI (Fig. 1-37).Using this formula, show that a plane through the tangent has a contact of orderone, but for the osculating plane which has a contact of order two with the curve.
3. Show (a) that the tangent has a contact of order n with the curve, if x', x",x"', ... x("), but not x("+'), have the direction of the tangent; and (b) that theosculating plane has such a contact, if x', x", x"', ... X("), but not x("+,) lie in theosculating plane.
42 CURVES [CH. 1
4. Show that the distance of a point P(xi) to the plane a x + p = 0 is of theorder of a x, + p, and that of P to the sphere (x - a) (x - a) - r2 = 0 is oforder (x, - a) (x, - a) - r2.
5. Osculating helix. This is the circular helix passing through a point P of thecurve C, having the same tangent, curvature vector, and torsion. Show that itscontact with the curve is of order two. Is it the only circular helix which has acontact of order two with C at P? (T. Olivier, Journal Ecole Potytechn., cah. 24,tome 15, 1835, pp. 61-91, 252-263.) Also show that the axis of the osculating helixis the limiting position of the common perpendicular of two consecutive principalnormals.
6. Starting with the common coordinate equations, find the natural equations of(a) logarithmic spiral:
(b) cycloid:
r = cekB.
x = a(0 - sin 0), y = all - cos 0).
(c) circle involute:
x = a(cos 0 + 0 sin 0), y = a(sin 0 - 0 cos 0).
(d) catenary:y = (a/2) (erl° + e-°i°).
7. Find from Eq. (8-6), in the case of constant K and r, the third order equationfor a, and by integration obtain the circular helix.
8. Prove that when the twisted cubic
x=at, y=bt2, z=t'satisfies the equation 2b2 = 3a, it is a helix on a cylinder with generating line parallelto the XOZ-plane, making with the X-axis an angle of 45°. Determine the equa-tion of the cylinder.
9. The spherical indicatrix of a curve is a circle if and only if the curve is a helix.10. The tangent to the locus C of the centers of the osculating circles of a plane
curve has the direction of the principal normal of the curve; its are length betweentwo of its points is equal to the difference of the radii of curvature of the curve atthese points.
11. Curves of Bertrand. When a curve C, can be brought into a point-to-pointcorrespondence with another curve C so that at corresponding points P1, P the curveshave the same principal normal, then
(a) PIP is constant = a,(b) the tangent to C at P and the tangent to C, at P, make a constant angle a,(c) there exists for C (and similarly for C,) a linear relation between curvature
and torsion
K + r cot a = 1/a, ( take r 0, a 0, a 9-6 2) (A)
These curves were first investigated by J. Bertrand (Journal de Mathem. 15,
1850, pp. 332-350).
1-11] EVOLUTES AND INVOLUTES 43
12. Show that a curve C for which there exists a linear relation between curva-ture and torsion:
PK + Qr = R, (P, Q, R constants Fx- 0)
admits a Bertrand mate, that is, a curve of which the points can be brought into aone-to-one correspondence with the points of C such that at corresponding pointsthe curves have the same principal normal.
13. Special cases of Bertrand curves. Show:(a) When r = 0 every curve has an infinity of Bertrand mates, a = 0.(b) When r 54 0, a = a/2, we have curves of constant curvature. Each of the
curves C and C, is the locus of the centers of curvature of the other curve.(c) When K and r are both constant, hence in the case of a circular helix, there
are an infinite number of Bertrand mates, all circular helices.14. Show that the equation of a Bertrand curve (that is, a curve for which (A),
Exercise 11, holds) can be written in the form
x= a fu dv + a cot a fu x du, (a, a constants)
Here u = u(v) is an arbitrary curve on the unit sphere referred to its arc length(hence u u = 1, u' u' = 1, u' = du/dv). (Darboux, Lecons I, pp. 42-45.)Show that the first term alone on the right-hand side gives curves of constant curva-ture, the second term alone curves of constant torsion.
15. Mannheim's theorem. If P and P, are corresponding points on two Bertrandmates, and C and C, their centers of curvature, then the cross ratio (CC,, PP,) _sec' a = const.
16. Investigate the pairs of curves C, and C which can be brought into a point-to-point correspondence such that at corresponding points (a) the tangents are thesame, (b) the binormals are the same.
See E. Salkowski, Math. Annalen 66, 1909, pp. 517-557, A. Voss, Sitzung8ber.Akad. Miinchen, 1909, 106 pp., where also other pairs of corresponding curves arediscussed.
17. Show that when the curve x = x(s) has constant torsion r, the curve
has constant curvature ± r.
y=-Tn+ Jb ds
18. Show that if in Sec. 1-10 we split a2 + 02 + y2 = 1 into (a + iy) (a - iy) =(1 + #)(1 - 0) and follow the method indicated by Eq. (10-2), we are led to theRiccati equation
LJ' = -$(K - 2r) + (K + ir)w2.19. Loxodromes. These can be defined as curves which intersect a pencil of
planes at constant angle a. Show that their equation can be brought into the form
x = r cos 0, y = r sin 0, z = f (r), where r = Vx2 + y2and
0=tans f 1+(f')2ar.
J r
44 CURVES [CH. 1
20. The tangents to a helix intersect a plane perpendicular to its projectingcylinder in the points of the involute of the base of the cylinder.
21. Find the involutes of a helix.22. Show that the helices on a cone of revolution project on a plane perpendicu-
lar to their axes (the base) as logarithmic spirals and then show that the intrinsicequations of these conical helices are
R = as, T = bs, (a, b constants)
23. Show that the helices on a paraboloid of revolution project on a plane per-pendicular to the axis as circle involutes.
24. A necessary and sufficient condition that a curve be a helix is that
(x "xrrrar1) = -K5)
= 0.Ws- (K
25. Differential equation of space curves. If, instead of using the method of theRiccati equation, we attempt to obtain the x(s) directly as the solution of theFrenet equation for given K and r, we obtain the equation (K, TO 0)
I q 2 'a(iv) -
2K + T arrr + K2 + T2 _ KK - 2(K') + KY) Z +
K2(K - r
) Sr = 0.K T K2 KT K T
26. The parabolas y = x2 and y = N"tr can be obtained from each other by arotation of 90°. They must therefore satisfy the same natural equation. Obtainthis equation in the form
1ds/2=9['482-11.
27. Verify by using Eq. (8-12) that the curve of Exercise 9, Section 1-6:
x = a sin2 u, y = a sin u cos u, z = a cos u,
lies on a sphere.
1-12 Imaginary curves. We have so far admitted only real curves, de-fined by real functions of a real variable. When we admit complexanalytic functions x; of a complex variable u, then we obtain structuresof 002 points, called imaginary curves. The formulas for the are length,tangent, osculating plane, curvature, and torsion retain a formal meaningfor those curves, but now they serve as definition of these concepts. Cer-tain theorems require modification, notably those based on the assumptionthat ds2 > 0. We have to admit, in particular, curves for whicl` ds2 = 0,the isotropic curves. And if we define planes as structures for which alinear relation exists between the x;, or - what amounts to the same - asplane curves, curves for which (x a a) = 0, then we have to admit planessuch as the plane with equation x2 = ixi, for which ds2 can no longer bewritten as the sum of the squares of two differentials, but as the square of
1-12] IMAGINARY CURVES 45
a single differential such as ds2 = dxi in the case of x2 = ix1. We thereforedistinguish between regular planes, of which the equation by proper choiceof the coordinates can be reduced to x3 = 0, and isotropic planes, of whichthe equation can be reduced to x2 = ix1. By planes we mean regularplanes. There are no planes for which ds2 = 0. (Such planes do exist infour-space; e.g., those for which x1 = ix2, x3 = 2x4.)
Isotropic curves in the regular plane x3 = 0 are defined by
ds2 = dxi + dx2 = 0; dx2 = ±i dx1, (12-1)
which equation gives, by integration:
X2 - b = ±i(xi - a). (12-2)
In an isotropic plane we obtain by integration of ds2 = dxi = 0 that x3 = c,x2 = ix1. Hence:
Plane isotropic curves are straight lines. Through every point of a regularplane pass two isotropic lines; through every point of an isotropic plane passesone isotropic line. The isotropic lines in a regular plane form two sets ofparallel lines, those in an isotropic plane form one set of parallel lines.
Isotropic lines were introduced by V. Poncelet in his Traite des propriltesprooectives des figures (1822). They are also called minimal curves or null curves.
Isotropic curves in space x = x(u) satisfy the differential equation
dx- dx = dxl + dx2 + dxi = 0, (12-3)
or
a a = 0, x = dx/du. (12-4)
One solution of this differential equation consists of the isotropic straightlines
X = x + ua, x, a constant vectors, a a = 0, (12-5)
for which, as we see by differentiation, dX dX = a a du2 = 0. Theselines generate, at each point P(x), a quadratic cone
(X - x) (X - x) = 0, (12-6)
the isotropic cone with vertex P. The tangent plane to this cone along thegenerator (12-5) has the equation
(X - (12-7)
This is easily verified if we remember that the tangent plane to the conex2 + y2 + z2 = 0 at (x,, y1, z1) is xx1 + yy1 + zz1 = 0, and that this plane remainsthe same when we multiply x1, y1, z, by a factor X, so that this plane is tangentalong the generator x : y : z = x1 : y1 : z1 = a1 : as : as.
46 CURVES [CH. 1
By selecting x = 0 and taking the XOY-plane through a, the equation ofplane (12-7) becomes x1 = ±ix2, so that (12-7) is an isotropic plane.Eq. (12-7) is, at the same time, the most general form of the equation of anisotropic plane, since a a = 0 for the planes x1 = ±ix2 and therefore alsozero for all positions of a rectangular cartesian coordinate system.
Substitution of the X of Eq. (12-5) into Eq. (12-7) shows that isotropicline (12-5) lies in the plane (12-7). Hence:
The tangent planes to the isotropic cones are isotropic planes, tangent alongisotropic lines, and also, although it seems strange at first:
The normal to an isotropic plane at a point in this plane lies in this planeand is tangent to the isotropic cone at that point.
Isotropic straight lines do not form the only solution of Eq. (12-4).We shall now derive the general solution by a method reminiscent of thatused to pass from Eq. (10-1) to Eq. (10-2). For this purpose we writeEq. (12-3) in the form
dx1 + i dx2 dx3 = u.dx3 dx1 - i dx2
This leads to the equations
dx1 dx2 _ dxj dx2 1
dx3+Z dx3-u, dx3-idx3=-u
Solving these two equations for dxi/dx3 and dx2/dx3, we obtain the solu-tion of Eq. (12-4) in the form
dx1 _ dx2 _ dx3 = F(u) du.u2 - 1 i(U2 + 1) 2u
Writing f"'(u) for F(u), we then obtain by partial integration the equation ofthe isotropic curves in the form:
x1 = (u2 - 1)f" - 2uf' + 2f,x2 = [(u2 + 1)f" - 2uf + 2fli,x3 = 2uf" - 2f, (12-8)
where f(u) is an arbitrary analytic function of a parameter u. This repre-sentation contains, as we see, no integration. It does not give the(straight) isotropic lines.
The tangent line to an isotropic curve at P(x) is given by
y = x + aa,the osculating plane by
(12-9)
y = x + Xi + µ%, or (y - x, x, x) = 0. (12-10)
1-13] OVALS 47
From Eq. (12-4) followsx x = 0. (12-11)
Since
(a X a) . (g X a) = (g j[) (:k g) - (i g)2, (12-12)we see that
(i X :R) (s x it) = 0. (12-13)
Hence, comparing with Eq. (12-7), we find that the osculating plane of anisotropic curve (not a straight line) is tangent to the isotropic cone.
However, all curves whose osculating planes are tangent to the isotropiccone are not all isotropic curves. Such curves are characterized byEq. (12-13). In this case we apply the identity
(a a)(abc)2 = [(a x b) (a x b)][(a x c) (a x c)] - [(a x b) (a x c)]2.(12-14)
Identity (12-14) is obtained by substituting p = a x b, q = a x c into theidentity
(p x q) (p x q) = (p p)(q q) - (p q)2,
and applying to p x q the identity
(u x v) x w = (u w)v - (v w)u,or
(a x b) x (a x c) = - [b (a x c)la = (abc)a.
When in this identity a = g, b = s, c = z, then we obtain for the casethat Eq. (12-13) is satisfi'd,
(s j) (it s 1)2 = [(a X s) (z X g)]2 = 0, (12-15)
since differentiation of Eq. (12-13) gives
0.
Eq. (12-15) expresses the theorem of E. Study:When at every point of a curve the osculating plane is an isotropic plane
the curve is either an isotropic curve or a curve in an isotropic plane.
1-13 Ovals. We shall now present some material which no longerbelongs to local differential geometry, but to differential geometry in the large.Theorems in this field do not deal with the exclusive behavior of curves andsurfaces in the immediate neighborhood of a point, such as the curvature ortorsion "at" a point, but describe characteristics of a finite arc or segment
48 CURVES [CH. 1
of the curve or surface. Such theorems are often obtained by means of theintegral calculus, where local differential geometry is based on the applica-tion of the differential calculus. For this reason we sometimes opposeintegral geometry to differential geometry. Classical integration of arelengths, areas, and volumes can be considered as part of integral geometry.Although there is no reason to attempt a specific distinction between inte-gral geometry and differential geometry in the large, we are inclined to usethe latter term when dealing with such conceptions as curvature and tor-sion, defined in "local" differential geometry.
A section of differential geometry in the large which has been rather wellinvestigated deals with ovals. An oval is a real, plane, closed, twice differen-tiable curve of which the curvature vector always points to the interior. Thenthe tangent turns continuously to the left when we proceed counterclock-wise along the curve. The points can be paired into sets of opposite points,at which the tangents are parallel (Fig. 1-38). If t is the unit tangent vec-tor at P, then t1 = -t is the unit tangent vector at the opposite point P1.Then the curvature K = dp/ds cannot change its sign, since cp is a monotoni-cally increasing (decreasing) function of s.
It will be remembered that in the case of plane curves it is possible to deter-mine the sign of K by postulating that the sense of rotation t -> n is that ofOX 0Y. When, therefore, K maintains the same sign along the curve,the spherical image of the curve is a circle traversed once in the same sense.
We shall assume the curvature K(s) to be differentiable and > 0. Apoint where K has an extreme value is called a vertex. We now prove thefollowing theorem.
Four-vertex theorem. An oval has at least four vertices.To prove this theorem we observe that when the oval is not a circle it
has at least two vertices A and B. Suppose that there are no more, that
K=0
FIG. 1-38
AdK=O
FIG. 1-39
1-13] OVALS 49
dic > 0fn on one side of AB and dK < 0 on the other side (Fig. 1-39). Now
take dR along the curve (R = K-'):
fndR = Rn- fRdn=J Rectds=Jtds= fdx=c
This means that fBn dR has the same value whenA
taken along the arc AB where dR > 0 and along theare AB where dR < 0. If we take the sphericalimage of n (Fig. 1-40), in which A, and B, corre-spond to A and B, then we see that the two in-tegrals give vectors lying in segments of the planein which they cannot add up to zero. Thereforethere are more than two vertices, and since thecurve is closed, their number is even. Four is pos-sible, since an ellipse has four vertices.
0.
FIG. 1-40
This theorem was found by S. Mukhopadhyaya, Bull. Calcutta Math. Soc. I(1909), Coll. geom. papers I, pp. 13-20; here a cyclic point is defined as a pointwhere the circle of curvature passes through four consecutive points, a sextacticpoint where the osculating conic passes through six consecutive points, etc.The theorem was rediscovered by A. Kneser, H. Weber Festschrift 1912, pp.170-180; several proofs have since been given, e.g., one in W. Blaschke, Dif-ferentialgeometrie, p. 31; see also S. B. Jackson, Bull. Amer. Math. Soc. 50,1944, pp. 564-578; P. Scherk, Proc. First Canadian Math. Congress, 1945,pp. 97-102.
A convenient set of formulas for ovals canbe obtained by associating two points P(x)and P,(x1) of the oval with parallel tangentsor opposite points by means of the equations(Fig. 1-41):
x,=x+Xt+un, x=x(s), (13-1)
where µ is the width of the curve at p; thatis, the distance between the two paralleltangents at P. Then, denoting the quantitiesat P, by index 1, we have
ti =-t, n, =-n.
50 CURVES
Differentiation of Eq. (13-1) gives
tI dsi/ds = t + X't + XKn + ,In - ),Kt,
[CH. 1
or, equating the coefficients of t and n on both sides of this equation:
1 + ds,/ds =-X' + µK,0 = XK + µ', (13-2a)
or, in terms of differentials, using the equation K ds = dip:
ds + ds1 = -da + µ d(p, 0 = X dip + dµ. (13-2b)
These equations lead to some simple results in the case of curves of con-stant width, or orbiform curves, characterized by the property that the dis-tance between two parallel tangents is always constant. Then dµ = 0.Then Eqs. (13-2) give X = 0, and integration from rp = 0 to (p = r showsthat the perimeter of the curve P is equal to µr:
P =J -o(ds + dsi) = µ,d(p = AT.
In words:The chord connecting opposite points of a curve of constant width is per-
pendicular to the tangents at these points. AndTheorem of Barbier. All curves of constant width µ have the same per-
imeter µr.The first of these theorems establishes not only a necessary but also a
sufficient condition.A circle is a curve of constant
width, but there are some curvesof constant width which are notcircles. This can be shown by Qfollowing a method indicated byEuler. If we wish to construct acurve of constant width with 2nvertices, we take a closed differ-entiable curve with n cusps havingone tangent in every direction(such curves may be called curvesof zero width). Any involute ofsuch a curve is a curve of constantwidth. Let us, for instance taken = 3 and let us start with a point
FIG. 1-42
R on the tangent at point P (Fig. 1-42). Then, developing the involutefrom P via B, A, C back to P, we obtain a closed curve with two tangents
1-13] OVALS 51
in every direction; the distance between two opposite points Q, R isQR = PQ + PR = arc PB + DB + DB + arc AB - are AC + arc CP =arc BC - F are AB - are AC + 2DB. The curvature vector of the involute,which at Q is along QP, always points to the interior.
Curves of constant width were introduced by L. Euler, De curves triangu-laribus, Acta Acad. Petropol. 1778 (1780) II, pp. 3-30, who called them orbiformcurves, and the curve ABC of Fig. 1-42 a triangular curve. E. Barbier, Journalde mathem. (2) 5 (1860), pp. 272-286, connected the theory of these curves withthat of the needle problem in probability. (See further C. Jordan and R.Fiedler, Contribution a l'etude des courbes convexes fermees (1912).) The Eqs.(13-2) were first established by A. P. Mellish, Annals of Mathem. 32, 1931,pp. 181-190. For more information on differential geometry in the large seeW. Blaschke, Differentialgeometrie and Einfiihrung; also D. J. Struik, BulletinAmer. Mathem. Soc. 37, 1931, pp. 49-62.
EXERCISES
1. The limit of the ratio of arc to chord at a point P of a curve is, for chord -. 0,unity only when the tangent at P is not isotropic. Prove for the case of a planecurve that this limit is
2-Vk-
k+1'when k - 1 is the order of contact of the curve with the isotropic tangent. (E.Kasner, Bull. Am. Math. Soc. 20, 1913-1914, pp. 524-531; Proc. Nat. Ac. Sc. 18,1932, pp. 267-274.)
2. Show that the angle of an isotropic straight line with itself is indeterminate.
3. When a a = 0, and a b = 0, (a) show that b lies in the isotropic planethrough a, (b) that every vector c for which c a = 0 lies in this plane.
4. Isotropic curves are helices on all cylinders passing through them.
5. The equation
s s
(X,-x,)(1 2u -i(Xs-xa)(1 2u +(X3-xa)u=0,
represents an isotropic plane.
6. By means of the expressions g i, (i s a), i x it, (i x a) (g x a) charac-terize: (a) regular straight lines, (b) regular plane curves, (c) isotropic straightlines, (d) isotropic curves, (e) curves in isotropic planes.
(E. Study, Transactions Am. Mathem. Soc. 10, 1909, p. 1.)What is K and r in each case?(Example: For a regular straight line i x # 0, g x it = 0, (a x g) (i x i) = 0;
(iaa)=0.)
52 CURVES [CH. I
7. By using the identity
(jr a g)2 =1.1
x.x... . ... .. ... ...
show that a plane isotropic curve is a straight line (E. Study, loc. cit.).
8. The sum of the radii of curvature at opposite points of a curve of constantwidth is equal to the width.
9. The statements:(a) an oval is of constant width,(b) an oval has the property that PP1 is constant (constant diameter),(c) all normals of an oval are double (i.e., are normals at two points),(d) for an oval the sum of the radii of curvature at opposite points is constant,
are equivalent, in the sense that, whenever one of the statements (a), (b), (c), (d)holds true, all other statements are true (A. P. Mellish).
10. The perimeter of an oval is equal to ir times its mean width, the mean takenwith respect to the angle gyp.
11. Show that for a general oval the width µ satisfies an equation of the form
dµ = f(P),z +
where f(op) = R + R,.
12. The curves with equation
x4 + 2J° = 1,
have eight vertices, situated on the lines x = 0, y = 0, x ± y = 0.
13. Curvature centroid. This is the center of gravity of a curve if its are isloaded with a mass density proportional to its curvature. Show that the curvaturecentroid of an oval and of its evolute are identical.
14. Pedal curve. The locus of the points of intersection of the tangents to a
curve C and the perpendiculars through a point A on these tangents is the pedal
curve of C with respect to A. Find the pedal curve of a circle with respect to a pointon its circumference.
15. If P is a point on a curve C and Q is the corresponding point of the pedalcurve of C with respect to a point A (see Exercise 14), then AQ makes the sameangles with the pedal curve which AP makes with C. Hint: If C is given by x =x (s), write the pedal curve as y = (x n)n.
1-14] MONGE
y=x+lnx
Y1
0
y=x - lnx
Y
0
(a) (b)
FIG. 1-43
53
x
16. The curves y = x - lnx, y = x + lnx are not congruent (Fig. 1-43). Showthat nevertheless both equations have the same natural equation. The reason isthat the two curves can pass into each other by an imaginary motion (G. Scheffers,Anwendung I, p. 17).
1-14 Monge. We have already had an opportunity to mention someof the contributions of this mathematician, who, with Gauss, can be con-sidered the founder of differential geometry of curves and surfaces. Gas-pard Monge (1746-1818) started his career as a professor at the militaryacademy in M6zieres (on the Meuse, in N. France), where he developedour present descriptive geometry. During the Revolution he was anactive Jacobin and occupied leading political and scientific positions; astemporary head of the government on the day of the King's execution heincurred lasting royalist resentment as the chief regicide. After 1795 hebecame the principal organizer of the Polytechnical School in Paris, theprototype of all our technical institutes and even of West Point. Manyleading mathematicians and physicists such as Lagrange and Ampere wereconnected with this School. Monge was a great teacher, and his lessons inalgebraic and differential geometry inspired many younger men; amongthem were V. Poncelet, who established projective geometry, and C. Dupin,who contributed greatly to the geometry of surfaces. Other pupils ofMonge were J. B. Meusnier, E. L. Malus, M. A. Lancret, and 0. Rodrigues,who all have theorems in differential geometry named after them. Monge'smost important papers on the geometry of curves and surfaces have beencollected in his Applications de l'Analyse d la Ggometrie (1807), of which thefifth edition appeared in 1850, with notes of J. Liouville. Monge enjoyedthe confidence of Napoleon, and was discharged as director of the Poly-technical School after the fall of the emperor. He died soon afterwards.The main ideas of Chapter 2 are due to Monge and his pupils. Those ofChapter 1 were in part also developed by these men, and in part by a later
54 CURVES [CH. 1
school of French mathematicians, who associated themselves with theJournal de Mathematiques pures et appliquees, continuously published since1836, when it was founded by Joseph Liouville (1809-1882). These authorsincluded A. J. C. Barre de Saint Venant (1796-1886), who became interestedin the theory of curves through his work in elasticity, and a group ofyounger men, notably F. Frenet (1816-1888), J. A. Serret (1819-1885),V. Puiseux (1820-1883) and J. Bertrand (1822-1900), whose principalpapers in this field were written in the period 1840-1850; J. Liouville gavea comprehensive report on these investigations in Note I to the 1850 editionof Monge's Applications.
CHAPTER 2
ELEMENTARY THEORY OF SURFACES
2-1 Analytical representation. We shall give a surface, in most cases,by expressing its rectangular coordinates x; as functions of two parametersu, v in a certain closed interval:
xi = x;(u, v), u, < u < u2, vl < v < V2. (1-1)
The conditions imposed on these functions are analogous to those im-posed on the conditions for curves in Section 1-1. We consider the func-tions x; to be real functions of the real variables u, v, unless imaginaries areexplicitly introduced. When the functions are differentiable to the ordern - 1, and the nth derivatives exist, we can establish the Taylor formula:
2
xt(u, v) = x'(uo, vo) + hax;au + k avy + 2. h au + k av
x++ .. .0 0 o
rz
).xi(uo+(n11 )!(h-a +kav oxi +n(hau+kav o+Oh, vo+6k),
0<0<1. (1-2)
The parameters u and v must enter independently, which means thatthe matrix
M=ax ay az
au au au
ax ay az
av av av
,orXU yu Zu
(1-3)x yv Zv
11
has rank 2. Points where this rank is 1 or 0 are singular points; when therank at all points is 1 the Eqs. (1-1) represent a curve, as in the case
x = u + v, y = (u + v)2, z = (u + v)3. (1-4)When two determinants of the matrix (1-3) vanish, all three vanish (un-
less one column contains two zeros), but the vanishing of only one determi-nant does not mean that the point is singular. If, for example, the surfaceis given by
x=u+v, y=u+v, z=uv, (1-5)
then xuyv - xvyu = 0, but xuzv - xvzu 0 0, and the surface is a planethrough the Z-axis. Another example is
x = fi(u), y = f2(u), z = v, (1-6)
which represents a cylinder.55
56 ELEMENTARY THEORY OF SURFACES
Singular points may appear becauseof the nature of the surface and alsobecause of the particular choice of thecoordinates. An example of the secondcase is the sphere referred to latitude 0and longitude p:
x = a cos0cosp,
y = a cos 0 sin gyp, (1-7)
z = a sin 0,
M=
FIG. 2-1
a sin 0 cos p -a sin 0 sin p a cos 0-a cos 0 sin (p +a cos 0 cos p 0
(a = const),
[CH. 2
which has a singular point at 0 = 2 (the North Pole if the XOY-plane is
the equator, Fig. 2-1). This is clearly due to the choice of parameters.
For a circular cone, with coordinate representation (Fig. 2-2) :
x = u sin a cos gyp, y = u sin a sin (p, z = u cos a, (1-8)
M=
we find a singular point at u = 0, whichis the vertex, a particular point of thesurface.
When we write the equation of thesurface in vector form:
x = x(u, v) = xiel + x2e2 + x3e3, (1-9)
the condition that the rank of matrixM be 2 can be written in the form
x X X 0 0 (1-10)(xu = ax/au, x,, = ax/8v).
This equation allows a simple geometri-cal interpretation. When we keep vconstant in Eq. (1-1) or Eq. (1-9),the x depends on only one parameter u
sin a cos rp sin a sin p cos a aII
-u sin a sin rp u sin a cos sp 0= const),
FIG. 2-2
2-1] ANALYTICAL REPRESENTATION 57
and thus determines a curve on the surface, a parametric curve, v = con-stant. Similarly, u = constant represents another parametric curve.When the constants vary, the surface is covered with a net of parametriccurves, two of which pass through every point P, forming the family of oo Icurves v = constant and the familyof oo I curves u = constant. At Pthe vector x. is tangent to the curvev = constant, and x is tangent tothe curve u = constant (see Section1-2). Condition (1-10) means thatat P the vectors x, and x do not u = constantvanish and have different directions.(See Fig. 2-3.)
FIG. 2-3
We also call (u, v) the curvilinear coordinates of a point on the surface.As curvilinear coordinates of a point on the sphere we may select latitudeand longitude, a familiar procedure in geography; polar coordinates arean example of curvilinear coordinates in the plane (rectilinear coordinatescan be considered as a special case of curvilinear coordinates). The para-metric curves are also called coordinate curves.
When we pass from one system of curvilinear coordinates to another
u = u(u, v), v = v(u, v), (1-11)
we obtain the equation of the surface in the new form
x = x(u, v).
The tangent vectors to the new parametric lines xu and xs are expressedas follows in terms of xu, x,:
L avxff _ -xuau+xvau'(1-12)
X; _ xu49U
+ xvav
av av,so that
u vxu X x, =
u v(xu X X.). (1-13)
We see that the inequality (1-10) continues to hold for the new system ofcoordinates, provided the functional determinant
au av
(u v _ a(u, v) _ au au0.
v) a(u,v) au av. (1-14)
av av
58 ELEMENTARY THEORY OF SURFACES [CH.2
We shall consider only coordinate transformations of this kind, whichchange systems in which xu and xy have different directions into systems inwhich xfi and xo have different directions, or at any rate restrict our studyof the surface to regions where (1-14) holds. Eq. (1-14) can also be inter-preted by saying that we consider only those transformations by whichindependent parameters pass into independent ones in accordance withformula (1-10).
2-2 First fundamental form. A relation p(u, v) = 0 between the curvi-linear coordinates determines a curve on the surface. Such a curve can
also be given in parametric form:
u = u(t), v = v(t).
The vector dx/dt = x, at a point P of the surface, given by
x = xuu + x v,
(2-1)
(2-2)
is tangent to the curve and therefore to the surface. Eq. (2-2) can bewritten in a form independent of the choice of parameter:
dx = xu du + x dv. (2-3)
When the curve is given by so(u, v) = 0, the du and dv are connected bythe relation
,p du + rp dv = 0. (2-4)
The ratio dv/du = - is sufficient to determine the direction of thetangent to the surface.
The distance of two points P and Q on a curve is found by integrating
ds2 = dxi dx; = dx dxs-i
along the curve; and substituting for dx the values (2-3), we find
(2-5)
ds2 = (xu du + x dv) (xu du + x dv) = E due + 2F du dv + G dv2,
where
(2-6)
E = xu xu, F = xu x,,, G = x,, x,. (2-7)
The E, F, G are functions of u and v. The distance between P and Q onthe curve u = u(t), v = v(t) can now be expressed as follows:
s E (T)2 + 2Fddu dv V=f' t dt + G (dt)2 A
2-21 FIRST FUNDAMENTAL FORM 59
The expression (2-6) for ds2 is called the first fundamental form of thesurface. It is a quadratic differential form; its square root ds can be takenas the length jdxj of the vector differential dx on the surface, and is calledthe element of arc. Since ds is a length,
E due + 2F du dv + G dv2
is always positive (except zero for du = dv = 0), as long as we study realsurfaces; such a form is called positive definite. Since
ds2 = E (E du + F dv)2 + EGE F2dv2
and
we see thatEG-F2> 0.
This also follows from Eq. (1-10), because xu x xv 0, so that
(xu X xv) . (xu X xv) = (xu xu) (xv xv) - (xu X.)2 = EG - F2. (2-8)
This inequality continues to hold under a change of curvilinear coor-dinates, since according to Eq. (1-13) for the new E, F, G the equationholds
EG - F2 = (f4v)2(EG - F2). (2-8a)
With the aid of E, F, G we can express the angle a of two tangent direc-tions to the surface given by du/dv, Su/Sv. Then
and
cos a =
dx = xu du + xv dv, Sx = xu Su + x Sv
dx Sx xu xu du Su + (xu x9) (du Sv + dv Su) + (xv xv) dv SvldxllOxl IdxllSxl
Edudu+F(duOv+dvsu)+GdvOvV'E du2 + 2F du dv + G dv2 E 6u2 + 2F bit Sv + G 3v2
E du Su + F du Sv + dv Su + G dv Svds Ss (ds Ss ds Ss ds Ss
(2-9)
The following cases of Eq. (2-9) are particularly important:1. When a = a/2 we obtain the condition of orthogonality of two direc-
tions on the surface:
E du Su + F(du Sv + dv Su) + G dv Sv = 0. (2-10)
60 ELEMENTARY THEORY OF SURFACES [CH.2
2. The angle 0 of the parametric lines u = constant (hence du = 0,
dv arbitrary) and v = constant (hence Su arbitrary, by = 0), is given by
cos 0 =F dv Su F=Gdv -IE-60 EG
Esin g = EG (2-11)
3. The parametric curves are orthogonal if F = 0.
EXAMPLES. (1) Sphere (coordinates in Eqs. (1-7)). Squaring of the ele-ments of the first row in M gives E = a2, and similarly F = 0, G = a2 cos2 0:
ds2 = a2 d02 + a2 cos' 0 dcp2. (2-12)
Since F = 0, meridians and parallels are shown to be orthogonal.The loxodromes on the sphere (see Exercise 19, Section 1-11) are the
curves which intersect the planes through a diameter, and hence the merid-ians, at a given angle a. Let the meridians be the curves (p = const.Then, in Eq. (2-9), dv = dcp = 0, and therefore
E du Su _ du _ dOcoca=ds-VE-bu2 -ads -ads'
cost a (doe + cost 0 d02) = doe,
dO/cos 0 = ±cot a d(p,
(rp + c) cot a =1n tan 12 + 4)
The ± indicates that the loxodromes can wind around the sphere in a right-handed or a left-handed sense. When for given c and a (and the + sign)
the angle increases from -oo via - c to + oo, tan (2 + increases from 0
via 1 to oo, and 0 from - - via 0 to 2 This shows that the loxodromes
wind in spiral-like fashion around the poles as asymptotic points. Loxo-dromes on the sphere (and in particular on the globe) are also called rhumblines.
(2) Surface of revolution. When the Z-axis is taken as the axis of revolu-tion of the curve z = f(x) in the plane y = 0 (this curve is the profile of thesurface), the resulting surface (Fig. 2-4) can be given by the equations
x = r cos gyp, y = r sin (p, z = f(r). (2-13)
2-21
FIG. 2-4
FIRST FUNDAMENTAL FORM
FIG. 2-5
61
The curves r = constant are the parallels, and the curves (P = constant arethe meridians of the surface. Here
M=and
cos (P sin jp f(r)-rsin,p rcosrp 0
11
ds2 = (1 + f'2) dr2 + r2 dv2. (2-14)When
r = u sin a, f (r) = r cot a = u cos a (a = constant), (2-15)
we have a cone of revolution (see Eq. (1-8)) with the first fundamentalform:
ds2 = csc2 a dr2 + r2 d02 = due + u2 sin2 a dv2. (2-16)
Meridians and parallels are again orthogonal.(3) Right conoid. Such a surface is generated by a straight line moving
parallel to a plane (here the plane z = 0) and intersecting a line perpendicu-lar to this plane (here the Z-axis, Fig. 2-5). Then its equation is
x = r cos (o, y = r sin (p, z = AP). (2-17)
The curves r = constant are the loci of the points at fixed distance r fromthe Z-axis, and the curves (p = constant are straight lines. Here
M=and
cos o sin o 0-rsin,p rcosp f'
ds2 = dr2 + (r2 + f'2) dV2. (2-18)
The coordinate curves r = constant and (p = constant are orthogonal.
62 ELEMENTARY THEORY OF SURFACES [Cu.
When f(go) = app + b (a, b constants) we have the right helicoid (see Section1-3), for which
ds2 = dr2 + (r2 + a2) drp2. (2-19)
The curves r = constant are circular helices of equal pitch. Surfaces ob-tained by the motion of a straight line in space are called ruled surfaces;the right conoids are special cases of ruled surfaces, as are cones andcylinders.
2-3 Normal, tangent plane. All vectors dx/dt through P tangent tothe surface satisfy the Eq. (2-2) and therefore lie in the plane of the vectorsxu and x,,, uniquely determined at all points where xu x xv 0 (comparewith Eq. (1-10)). This plane is the tangent plane at P to the surface. Itsequation is
R=x+Axu+Axv,or
X1-x1 X2 - X2 XT3-x38x1 8x2 8x3
8u 8u 8u 0;
8x1 8x2 8x3
8v 8U 8v (3-1b)
where the derivatives are taken atP (xl, X2, X3).
a, µ parameters, (3-la)
Fio. 2-6
The surface normal, normal for short, is the line at P perpendicular tothe tangent plane. As unit vector in this normal we take (Fig. 2-6) :
N - XU X Xv - XU X Xv
Ixu X Xvi E F2 (3-2) *
We also might have taken -N as unit normal vector, since the sense of Ndepends on the labeling of the coordinate curves.
Since
xu N = 0, xv N = 0, (3-3)
we conclude from Eq. (1-2), using the theory of contact presented in Sec-tion 1-7:
The tangent plane has, of all planes through P, the highest contact with thesurface. This contact is (at least) of order one.
* 'lEG _-F2 will always mean +VEG _-P.
2-31 NORMAL, TANGENT PLANE 63
In the texts on the calculus * it is shown that the area of a region R onthe surface is given by
A =JJ E du dv.
The formula can be made plausible by the consideration that thearea of a small parallelogram bounded by the two vectors dx and 5x isdA = Idx x Sxj = E Idu 5v - Su dvi. Taking for dx and 5x vec-tors tangent to the coordinate lines (Su = 0, dv = 0), and writing dv for Sv (inaccordance with the established custom of integration theory), we obtain
dA = BEG - F2 du dv. (3-4)
EXAMPLES. (1) Circular cone. Here, according to Eq. (1-8) andEq. (2-14) with
f(r) = r cot a, r = u sin a,
we haveN,IEG = u sin a,
and the coordinates of N are
N(NI =-cos a cos (p, N2 =-cos a sin (p, N3 = sin a).
N is independent of r (or u), which means that at all points along a gener-ating line the tangent plane is the same.
(2) Right helicoid. According to Eq. (2-19) :
E= r2+a2,and according to Eq. (2-18) :
Nasinp -acos p r(rra2' r r2rz a
l I
FIG. 2-7
If we call y the angle that N makes with the Z-axis, we have
rcos y = N3, cot y = r.
This shows that when a point P moves along a generating line the tangentplane turns about this line such that its angle with the central axis changesfrom 0° on this axis to 90°, the tangent of this angle changing proportionallyto r (Fig. 2-7). This will be more explicitly discussed in Section 5-5.
* e.g. P. Franklin, loc. cit., Section 1-1.
64 ELEMENTARY THEORY OF SURFACES [CH. 2
(3) Tangential developable. This is the surface gener-ated by the tangent lines to a space curve. If thiscurve C is given by the equation
x = x(s), with unit tangent vector t = t(s),
then the equation of the surface is (Fig. 2-8)
y(s, v) = x(s) + vt(s), (3-5)
where v is the distance from a point P on a generatingline of the surface to its tangent point A on C. Here
y,=t+vKn, yv=t,N = b(or -b).
The tangent plane along a generator coincides with the oscu-lating plane of the point on C through which the generatorpasses. It is therefore the same along a generator.
EXERCISES
FIG. 2-8
1. The following surfaces are given in parametric form.
(a) Ellipsoid:
x = a sin u cos v, y = b sin u sin v, z = ccosu.
(b) Hyperboloid of two sheets:
x = a sinh u cos v, y = b sinh u sin v, z = c cosh u.
(c) Cone:
x = a sinh u sinh v, y = b sinh u cosh v, z = c sinh u.
(d) Elliptic paraboloid:
x= au cos v, y = bu sin v, z=u2.(e) Hyperbolic paraboloid:
x = au cosh v, y = bu sinh v, z = u2.
Find the equations of these surfaces in the form F(x, y, z) = 0. What kind ofcurves are the coordinate curves u = constant, v = constant in each case?
2. Show that (a) the hyperbolic paraboloid can also be given by the equations:
x = a(u + v), y = b(u - v), z = uv,
and (b) the hyperboloid of one sheet is given by
x=au - v, y=b1+uv z=cuv-1u+V u+v' u + v
What are the curves u = constant, v = constant?(c) Also study the surface x = u, y = v, z = us.
2-3] NORMAL, TANGENT PLANE 65
3. Given a surface by the equation z = f(x, y). (a) Find the first fundamentalform and N. (b) Do the same if the equation is F(x, y, z) = 0.
4. Show that the element of area of the surface with equation z = f(x, y) isdA = 1p2 + q2 dx dy, p = az/ax, q = az/3y.
5. Show that the orthogonal trajectories of the family of curves given by
M du + N dv = 0are given by
(EN - FM) du + (FN - GM) dv = 0,and use this formula to find the orthogonal trajectories of the circles r =a cos 0 inthe plane for all values of a (use polar coordinates).
6. Show that the necessary and sufficient condition that the curves
A due + 2B du dv + C dv2 = 0, A, B, C functions of u, v,
form an orthogonal net isEC-2FB+GA=0.
7. Show that the curves dr2 - (r2 + a2) d,p2 = 0 on the right helicoid (see Eq.(2-19)) form an orthogonal net.
8. When the first fundamental form of a surface can be written i : the form
ds2 = due + G(u, v) dv2,
the curves u = constant cut equal segments from all curves v = constant. Sincethe curves u = constant, v = constant are also orthogonal, we call the curvesu = constant parallel.
9. Assume the parametric curves to be orthogonal. Show that the differentialequation of the curves bisecting the angles of the parametric curves is
E du2 - G dv2 = 0.
10. Show that the curves on the cone (1-8) given by
u = c exp((p sin a cot 0),
cut the generating lines at constant angle 0. Show that they project on the XOY-plane as logarithmic spirals.
11. What is the length of a loxodrome on a sphere which starts at the equatorat an angle a with the meridian and ends up by winding around a pole?
12. The locus of the mid-points of the chords of a circular helix is a right helicoid.13. Show that the locus of the projections of the center of an ellipsoid
x2 y2+
22a2+ b2 e2
on its tangent planes has the equation
(x2 + y2 + z2)2 = a2x2 + b2y2 + c2z2.
This is the pedal surface (compare with Section 1-13, Exercise 14) of the ellipsoidwith respect to its center. It is called Fresnel's elasticity surface.
66 ELEMENTARY THEORY OF SURFACES [CH.2
2-4 Developable surfaces. Tangent developables share with conesand cylinders the property that they have a constant tangent plane alonga generating line. Their tangent planes therefore depend on only oneparameter. Such surfaces can be considered as the envelope of a one-parameter family of planes. We shall show that they are the only sur-faces that can be considered as the envelope of such a family of 001 planes.
Such a family can be given by the equation
(4-1)
where the a and p depend on a parameter u. We exclude the case that
a'=da/du=0,which gives a family of parallel planes, as does the case that a and a' arecollinear. The planes determined by the parameters u = u1 andu = u2(u1 < u2) then intersect in a straight line, which also lies in the plane
x (a(u1) - a(u2)} + p(u1) - p(u2) = 0,or, applying Rolle's theorem,
xiai(7I) + xia2(v2) + x3a3(v3) + p'(wl) = 0, u1 1< vi <1 U2, U1 - W1 u2.
When u2 - u1 we find that this line takes a limiting position given by
x a + p = 0, x a' + p' = 0, p' = dp/du. (4-2)
This line is called the characteristic of the plane (4-1).The planes determined by the parameters u = u1j u = u2, and
u = u3(u1 < u2 < u3) intersect in a point, which also lies in the planes
x a'(vi) + p'(w1) = 0, x a'(v2) + p'(w2) = 0, x a"(v3) + p"(w3) = 0;Ui < Vi < u2, U2 < v2 < U3, v1 < v3 < v2, U1 < 4v1 < u2, U2 < w2 < u3, w1 < w3 < w2.
When u3 --> u2 -* u1 we find that this point takes a limiting positiongiven by
+p"=0. (4-3)
This point is called the characteristic point of the plane (4-1). It lies onthe characteristic line. It does not exist when (a a' a") = 0, in whichcase the vector field a(u) is plane. But the vector a is perpendicular to plane(4-1). When the vectors a(u) are parallel to a plane ,r, the planes (4-1)are all parallel to the direction perpendicular to a. In this case the envelopeof the planes (4-1) is therefore a cylinder with generating lines perpendicularto the plane 7r (except in the case that the planes form a pencil and the"envelope" is a straight line).
2-4] DEVELOPABLE SURFACES 67
When the characteristic point is the same for all planes the envelope is acone generated by all characteristic lines.
In the general case there exists a locus of characteristic points, which is acurve C. We shall show that the characteristic line is tangent to C at itscharacteristic point. For this purpose let us consider Eq. (4-3) solved forx, which then becomes a function of u. Differentiation of the first twoequations of (4-3) shows that
0, +p" = 0,
which, when compared with Eq. (4-3), gives
(4-4)
This means that the tangent vector x' to C has the direction of line (4-2).Since it passes through the characteristic point, x' must lie in the character-istic line. Similarly, differentiating the first equation of (4-4), we obtain
x'Comparing this with the second equation of (4-4) we find that x" a = 0,which with x' a = 0 means that the vector x' x x" is parallel to a. Theosculating plane of the curve C at the characteristic point is thereforeidentical with the plane (4-1). We have thus found the following theorem.
A family of oo' planes, which are not all parallel and which do not form apencil, has as its envelope either a cylinder, a cone, or a tangential developable.This envelope is generated by the characteristic lines of the planes, which in thecase of a cone all pass through the one characteristic point and in the case of atangential developable are all tangent to the locus of the characteristic points.
The locus of the characteristic points is called the edge of regression(French: arete de rebroussement; German: Riickkehrkante). It reduces to apoint in the case of a cone. Its name is due to the fact that the intersectionof a tangential developable with the normal plane to this edge of regression atone of its points P has a cusp at that point.
This property expresses the fact that the developable consists of twosheets which are tangent at the edge of regression along a sharp edge. Weshow this property analytically by taking the trihedron (t, n, b) at P as thecoordinate tetrahedron (y,, y2, y3) to which the surface is referred:
y = x + ut =St
+ 32 in + S3(-K2t + K'n + K7-b) + o(s3) (4-5)
r 2
+ u Lt + SKn + 2(-K2t + K'n + KTb) + o(S2) 1
68 ELEMENTARY THEORY OF SURFACES [CH.2
The intersection with the normal plane at P is determined by yI = 0,or by
hence
u=-s - zK3s3+o(s3).
Substituting this value of u into the expressions for y2 and y3i we obtain
y2 = 2 S2 + 0(82) + U{Ks + o(s)) =- 2 s2 + o(s2),
s-6 s3+o(s3)+u[1-2s2+0(S2)]=0,
K Ty3 =
6S3 + 0(s3) + u
2S2 + o(S2) } _ -
3s3 + o(s3).
We thus obtain for the required intersection in first approximation
y3 = -s T2Ryz, (4-6)
which shows that the intersection of the surface with the normal plane tothe curve has a cusp with the principal normal as tangent (Fig. 2-9). Theedge of regression appears as a sharp edge on the surface (Fig. 2-10).For this reason it is sometimes called the cuspidal edge. Comparison of(4-6) with Chapter 1, Eq. (6-4) shows that the cusp of this normal sectionhas a sense opposite to that of the projection of the cuspidal edge on thenormal plane at P.
Cylinders, cones, and tan-gential developables are calleddevelopable surfaces, or simplydevelopables (German, some-times: Torsen). If we let one
b
P n
FIG. 2-9 FIG. 2-10
2-41 DEVELOPABLE SURFACES
FIG. 2-11 FIG. 2-12
69
FIG. 2-13
70 ELEMENTARY THEORY OF SURFACES [CH.2
tangent plane coincide with a plane ,r, we can turn the "next" tangentplane about the characteristic line into ir, then the "next" plane, etc., andin this way develop the surface into the plane. Conversely, we can obtaina developable surface by bending a plane, without stretching or shrinking,so that the metrical properties of the plane are unchanged. This can bedemonstrated with a piece of paper, which can be bent without stretchingaround a cylinder. A piece of paper in the form of a circular sector can bebent around a cone; the radius of the sector becomes the slant height of thecone (Fig. 2-11). A tangent developable of a circular helix can be ob-tained by cutting a circle out of a piece of paper and twisting the remainingpart of the paper along an appropriate circular helix on a cylinder(Fig. 2-12); the tangents to the circle become the generating lines of thedevelopable (Fig. 2-13). We shall later give an exact proof of this propertyof a developable.
The two sheets of the developable are characterized by v > 0 and v < 0in Eq. (3-5); v = 0 gives the edge of regression.EXAMPLES. (1) Developable helicoid (Fig. 2-13). This is the surface gen-erated by the tangents to a circular helix. Since
x, = acosu, x2 = asinu, x3 = bu,
the surface has the equations:
yl = a cos u - av sin u, y2 = a sin u + av cos u, y3 = bu + by.
The parameters are u and v. This surface intersects the XOY-plane in thecurve for which y3 = 0 or v =-u:
yi = a cos u + au sin u,y2 = a sin u - au cos u,
which is a circle involute. This also followsfrom the fact that the projection of the tan-gent to the helix between the point P on thehelix and the intersection with the XOY-plane is bu tan y = au. (Compare withSections 1-2 and 1-11.)
The circle involute intersects itself in aninfinity of points (only one of these points isconstructed in Fig. 2-14). As the tangentsurface participates in the helicoidal motionby which the circular helix is generated, we FIG. 2-14
2-4] DEVELOPABLE SURFACES 71
see that the developable helicoid has an infinity of double lines where the sur-face intersects itself. These double lines are also helices.
(2) Polar developable. This is the surface enveloped by the normalplanes of a space curve, the equations of which have the form
(X - x) t = 0, (4-7)
where x and t are functions of the are length s of the curve, and X is ageneric point, independent of s. The parameter is s. The characteristicsare given by Eq. (4-7) and
-t t + (X - x) Kn = 0, (4-8a)or
(X - 1=R; (4-8)
the characteristic points are given by Eqs. (4-7), (4-8), and
(X - x) (-Kt + rb) = R',
obtained by differentiating either Eq. (4-8) or (4-8a). Hence
(X - x) b = TR'. 4-9)
The edge of regression of the polar developable is the curve
X = x + Rn + TR'b,
which according to Chapter 1, Eq. (7-5) is the locus of the centers of theosculating spheres; the generating lines of the developable are the polaraxes. This polar developable can be taken as the space analog of theevolute of a plane curve, which is the envelope of the normal lines. Thiscan be seen by interpreting Eqs. (4-7) and (4-8) for plane curves. We canalso say that just as two consecutive normals of a plane curve intersectin a point of the evolute, so do three consecutive normal planes of a spacecurve intersect in a point of the locus of the centers of the osculating spheres.
(3) Tangential developable. The envelope of the osculating planes of aspace curve is the tangential developable, since
0,or
=0, or
gives the curve itself as the edge of regression and its tangents as the char-acteristic lines.
72 ELEMENTARY THEORY OF SURFACES [CH. 2
(4) Rectifying developable. The envelope of the rectifying planes of aspace curve is determined by
=0
as the locus of lines in the rectifying plane passing through the point P of
the curve and making an angle tan-1K with the positive tangent direction.r
We shall meet this developable later, where the name will be explained.A space curve is the locus of oo 1 points, a developable the envelope of 00
planes. This leads to a duality between space curves and developableswhich can be indicated as follows:
Curves Developables
2 points determine a line. 2 planes determine a line.3 points determine a plane. 3 planes determine a point.2 consecutive points on a curve de- 2 consecutive planes of the family of
termine a tangent line. 001 planes determine a character-istic line.
3 consecutive points on a curve de- 3 consecutive planes of the family oftermine an osculating plane. 001 planes determine a character-
istic point.The curve is the envelope of o01 tan- The developable is generated by 001
gents. characteristic lines.The curve is the edge of regression of The developable is the envelope of
the developable enveloped by the the osculating planes of the curveosculating planes. generated by the characteristic
points.Plane curve. Cone.
The analogy is not complete. Two points, for instance, always determinea straight line, but two planes may be parallel. To extend the analogy to caseswhich involve elements at infinity and metrical relationships involves a deeperstudy of affine and non-Euclidean geometry in terms of projective geometry.
Developable surfaces are generated by straight lines, but not all surfacesgenerated by straight lines, the so-called ruled surfaces, are developable.An example is the hyperholoid of one sheet, which is a ruled surface, butnot developable, since the tangent plane varies when the point of tangencymoves along a generating line. We discuss this further in Section 5-5.
2-5] SECOND FUNDAMENTAL FORM
EXERCISES
73
1. Find the envelope of the planes
(a) u3 - 3u2x, + 3ux2 - x3 = 0-(b) x, sin u - x2 cos u + x3 tan 0 - au = 0. (0, a are constants)
2. Show that the tangent developable of the cubic parabola
is the surfacex1=u, x2 = u2, x3=u3
4(x2 - x)(x1x3 - x2) - (x1x2 - x3)2 = 0-1
3. The rectifying developable of a curve is the polar developable of all itsinvolutes.
4. When the polar developable of a curve is a cone the curve lies on a sphere.5. When the envelope of the rectifying planes of a curve is a cone the curve
satisfies the condition T/K = as + b, where a and b are constants. Can the en-velope of the osculating planes be a cone?
6. Show that the envelope of the planes which form with the three coordinateplanes a tetrahedron of constant volume, is the surface x y z = constant.
7. The osculating plane of the edge of regression at a point P of a curve cutsthe developable in a generating line (counted twice) and in a curve of which thecurvature at P is s-, K, K being the curvature of the edge of regression at P.
8. The rectifying plane of the edge of regression cuts the developable in a gen-erating line (counted once) and in a curve with a point of inflection at P, the generat-ing line being the inflectional tangent.
9. Envelopes of straight lines in the plane. If a family of oo' lines in the planeis given by a x + p = 0 (Eq. (4-1), Section 2-4), where a = a(s), p = p(s) thenits envelope, if it exists, is given by
Prove that the lines of the family are tangent to the envelope. When is there noenvelope?
10. Find the envelope of a line of constant length a which moves with its endson two perpendicular lines. This curve is a hypocycloid of four cusps.
11. Have the principal normals of a curve an envelope?12. Have the binormals of a curve an envelope?
2-5 Second fundamental form. Meusnier's theorem; The geometryof surfaces depends on two quadratic differential forms. We have alreadyintroduced the first of them, which represents M. The second fundamentalform can be obtained by taking on the surface a curve C passing througha point P, and considering the curvature vector of C at P (see Chapter 1,Eq. (4-4)). When t is the unit tangent vector of C, this curvature vector k
74 ELEMENTARY THEORY OF SURFACES
is equal to dt/ds. We now decompose k into a com-ponent k normal and a component k, tangential tothe surface (Fig. 2-15) :
dt/ds = k = k + k,. (5-1)
The vector k is called the normal curvature vectorand can be expressed in terms of the unit surfacenormal vector N :
kn = (5-2)
[CH. 2
kn k
FIG. 2-15
where K. is the normal curvature. The vector ku is determined by C alone(not by any choice of the sense of t or N), the scalar K. depends for its signon the sense of N. The vector k, is called the tangential curvature vector orgeodesic curvature vector. We shall deal in this chapter with the propertiesof k,,; those of k, belong to the subject matter of Chapter 4.
From the equation N t = 0 we obtain by differentiation along C:
ds =-t _-Nor
dx dx
(5-3)
(5-4)
Let us study first the right-hand side of this equation.Both N and x are surface functions of u and v (which in their turn depend
on C). With the aid of the resulting identities
dN = Nu du + N, dv, dx = x du + x dv, (5-5)
we can write Eq. (5-4) in the form:
KnE due + 2F du dv + G dv2 '
or
__ e due + 2f du dv + g dv2 (5-6)Kn E due + 2F du dv + G dv2
In this equation
e = -xu Nu, 2f = - (xu N + x,, Nu), g = -x N (5-7)
are functions of u and v, which depend on the second derivatives of the xwith respect to u and v, in this respect differing from E, F, G, which depend
2-31 SECOND FUNDAMENTAL. FORM 75
on first derivatives only. We write the denominator and numerator ofEq. (5-6) in the following form:
I =Edu2+2Fdudv+Gdv2 =II = edu2+2fdudv+gdv2
I is the first fundamental form, II the second fundamental form.xu N = 0 and x N = 0, we can also write for e, f, and g:
C =
or, using the expression (3-2) for N:
Similarly,
e _ (Xuuxuxv)
-VEG - Fz
f v
BEG-F2'
Xuu Yuu zuu
xu Y. Zu
xv yv zv
-VEG-P
g = (Xvvxuxv)
E
(5-8)
Since
(5-9,
(5-10a)
(5-10b)
These formulas (5-10) allow ready computation of e, f, g when the equationof the surface is given.
Incidentally, we can derive from Eq. (3-3) that
xu Nv = xv Nu,
so that Eq. (5-7) can be rewritten in the simpler form
e = -xu N,,, f = -xu Nv = -xv Nu, g = -xv N,,. (5-7a)
Returning now to Eq. (5-4) or, what is its equivalent, Eq. (5-6), we seethat the right-hand side depends only on u, v, and dv/du. The coefficients
e, f, g, E, F, G are constants at P, sothat K. is fully determined, at P,by the direction dv/du. All curvesthrough P tangent to the same direc-tion have therefore the same normalcurvature (if the sense of N is thesame for all these curves). Ex-pressed in vector language (Fig. 2-16):
All curves through P tangent to theP same direction have the same normal
FIG. 2-16 curvature vector.
ELEMENTARY THEORY OF SURFACES [CH.2
When we momentarily give to N the sense of kn, and to n the sense of
76
dt/ds, then we can express this theorem in the form obtained from Eq. (5-3) :
KcOSrp = K,,, (5-11)
where p(O <, (p < a/2) is the angle between N and n (Fig. 2-16). Thisequation can be cast into another form for directions t for which K. 5-4 0,hence also K 0. Such directions are called nonasymptotic directions.For curves in such directions we can write R = K ', R = K,,'. Thequantities R and R. are here positive, R. represents the radius of curvatureof a curve with tangent t and co = 0. One such curve is the intersection ofthe surface with the plane at P through t and the surface normal; thiscurve is called the normal section of the surface at P in the direction of C.Eq. (5-11) now takes the form
R cos w = R.
We can thus cast our previous theorem into the form:The center of curvature C1 of a curve C
in a nonasymptotic direction at P is theprojection on the principal normal of thecenter of curvature Co of the normal sectionwhich is tangent to C at P (Fig. 2-17).And in still other words:
If a set of planes be drawn through a Cltangent to a surface in a nonasymptoticdirection, then the osculating circles of theintersections with the surface lie upon asphere.This theorem is known as Meusnier'stheorem.
Co
FIG. 2-17
(5-12)
Jean Baptiste Meusnier (1754-1793), a pupil of Monge at the school in M&-zieres, published the theorem in his Memoire sur la courbure des surfaces, Me-moires des savans strangers 10 (lu 1776), 1785, pp. 477-510, which he wrote afterMonge had shown him Euler's paper (see Section 2-6). In 1783-1784, afternlontgolfier's ascent in a balloon, he (lid fundamental research on " aerostation, "and in this same period collaborated with Lavoisier in his work on the decom-position of water into its elements. He died as a revolutionary general duringthe siege of \Iayence. See G. Darboux' account in Eloges acadenaiques (Paris,1912), pp. 218-262.
2-6] EULER'S THEOREM 77
The last two versions of Meusnier's theorem do not hold for directionsfor which the second fundamental form II is zero, since in those directions,according to Eq. (5-6), Kn = 0. We have seen that there are no (real)directions for which the first fundamental form I is zero. However, it mayhappen that there are real directions for which II is zero:
e due + 2f du dv + g dv2 = 0. (5-13)
This happens, for instance, when there are straight lines on the surface.Directions satisfying Eq. (5-13) are called asymptotic directions. Curveshaving these directions are called asymptotic curves (German: Haupttangen-tenkurven). When Eq. (5-13) is satisfied, Eqs. (5-6) and (5-11) indicatethat a normal section in an asymptotic direction has a point of inflection.
2-6 Euler's theorem. We shall now investigate the behavior of thenormal curvature vector when the tangent direction at P varies. Itsdirection always remains that of the surface normal, but its length may varyfor different directions. When the sense of N has been agreed upon, thesign of K. will express whether kn has the sense of N or the opposite sense.
Since I > 0 the sign of K depends only on II. There are three cases:(1) II maintains the same sign whatever the direction may be. In
this case II is a definite quadratic form; this is expressed by the conditionthat
eg-f2>0.(See the reasoning used for the case of I in Section 2-2.) The centers ofcurvature of the normal sections are all on the same side of the surfacenormal; the normal sections are all concave (or all convex). The point iscalled an elliptic point of the surface; an example is any point on the ellip-soid.
(2) II is a perfect square:
eg-f2=0.
The surface behaves at the point like an elliptic point, except in one direc-tion, where K. = 0; the curves in this direction have a point of inflection.The point is called parabolic; an example is any point on a cylinder.
(3) II does not maintain the same sign for all directions du/dv. In thiscase II is indefinite; the condition is
eg-f2 < 0.
The normal sections are concave when they are cut out by planes in direc-tions lying in one section of the tangent plane, convex when outside this
78
K = 0
ELEMENTARY THEORY OF SURFACES
K=O
[CH. 2
section. The sections are separatedby the directions in which K. = 0, theasymptotic directions (see Eq. (5-13)and Fig. 2-18). The point is calledhyperbolic (saddle point); an exampleis any point on a hyperbolic paraboloid(Fig. 2-19).
FIG. 2-19
The conditions eg - f2 0 do not depend on the choice of curvilinear
coordinates on the surface. This is geometrically evident, since theseformulas express geometrical properties of the surface. It follows analyti-cally from Eq. (1-12) and
Ni N. Bu+N" 8u' No=N"ev+N°8v'
so that according to the definitions (5-7) and the resulting equation
eg - f2 = (xu x xv) (Nu x
for the e, f, g defined in the new coordinate system, the equation holds(compare Eq. (1-13)):
eg - f2 = (U v(eg - f2).
Now we take the equation of the surface in the form (1-2); n = 3 (omitthe index 0 for the derivatives):
x = x0 + hxu + kx + z (h2xuu + 2hkxu, + k2x,,,,)a
+-1- (ha (6-1)3! u
2-6] EULER'S THEOREM 79
Then the distance D of a point Q(x) on the surface near P(xo) to the tan-gent plane is (Fig. 2-20) :
D = (x - xo) N = 2[h2(x,,.,,. N) + 2hk(x,, N) + k2(xv, N)]3
+3!
of which the principal part is (comparewith Eq. (5-9)):
D, = 2(eh2 + 2fhk + gk2) = z II,(h = du, k = dv), (6-2)
where Dp is positive or negative depend-ing on whether Q lies on one or the otherside of the tangent plane. Eq. (6-2)gives a geometrical illustration of II,comparable to the identification of ds2with I:
When the second fundamental formdoes not vanish, it is equal to 2D, where
FIG. 2-20
D, is the principal part of the distance of the point Q(u + du, v + dv) to thttangent plane at P(u, v).
When eg - fl > 0, D, (as well as D) retains its sign; when eg - f2 < 0,it changes sign:
At an elliptic point the surface lies entirely on one side of the tangent plane,at a hyperbolic point it passes through the tangent plane in the asymptoticdirections.
At a parabolic point there is one (asymptotic) direction, in which thecontact is of higher order.
All types of points are illustrated on a torus (a surface obtained byrotating a circle about a line in its plane outside of the circle). The outside
points (obtained by rotating BCA)are elliptic; the inside points (BDA)
B are hyperbolic; the circles obtainedby rotating B and A have parabolicpoints (Fig. 2-21).
D C We can now ask for the directionsin which the normal curvature is a
A maximum or minimum. We shallfrom now on write K instead of ic,,,
FIG. 2-21 all through Chapter 2.
80 ELEMENTARY THEORY OF SURFACES [CH.2
The normal curvature in direction du/dv is given by Eqs. (5-6) :
_ e due + 2f du dv + g dv2 _ e + 2f X + gX26 3)'
=K
E due + 2F du dv + G dv2 E + 2FX + GX2 - "( - )(
where X = dv/du. The extreme values of K can be characterized bydK/dX = 0:
(E + 2FX + GX2)(f+ gX) - (e+2fX+gal)(F+GX)=0.Since
E+2FX+GX2 = (E+Fx)+a(F+GX),e+2fX+gX2 = (e+.f A)+X(f+ga),
we can, in this case, cast Eq. (6-3) into the simpler form:
_ II _ f+gX __ e+fXK (6-4)I F +Ga E+ Fa
Hence K satisfies the equations
(e - KE) du + (f - KF) dv = 0, (f - KF) du + (g - KG) dv = 0. (6-4a)
Elimination of K gives a quadratic equation for X with real roots :
(Fg - Gf)X2 + ( Eg - Ge) X + (Ef - Fe) = 0 (6-5a)
or
dv2
E-du dv
Fdu2G = 0. (6-5b)
e f g
This equation determines two directions dv/du, in which c obtains anextreme value, unless II vanishes or unless II and I are proportional. Onevalue must be a maximum, the other a minimum. These directions arecalled the directions of principal curvature, or curvature directions. Theyare determined either by Eqs. (6-4a), (6-5a), or (6-5b). Since the rootsX,, X2 satisfy the equation (compare with Eq. (6-5a)):
GXla2 + F(Xi + X2) + E1 [G(eF - Ef) - F(eG - Eg) - E(gF - Gf)] = 0,gF - Gf
the curvature directions are orthogonal (according to Eq. (2-10)). This alsoholds for the case that gF - Gf = 0, when, according to Eq. (6-5b), oneof the directions is du = 0. We call the normal curvatures in the curvaturedirections the principal curvatures, and denote them by K, and K2.
Integration of Eq. (6-5) gives us the lines of curvature on the surface,which form two sets of curves intersecting at right angles, or an orthogonal
2-6] EULER'S THEOREM 81
family of curves on the surface. From the existence theorem of ordinarydifferential equations, we can conclude that these curves cover the surfacesimply and without gaps in the neighborhood of every point where thecoefficients of the first and second fundamental forms are continuous, exceptat the points where these coefficients are proportional. Such points arecalled navel points or umbilics, and we exclude them for the moment.Now let us take the lines of curvature as the parametric lines. ThenEqs. (6-5a, b) must be satisfied for du = 0, dv arbitrary, and for dv = 0, duarbitrary. Hence:
gF - Gf = 0, eF - Ef = 0.
In these equations F = 0 because the parametric lines are orthogonal;moreover, neither E nor G can be zero (EG - F2 > 0). We thereforefind that when the parametric lines are lines of curvature:
F=O, f=0. (6-6)
This condition is necessary and, because of Eq. (6-5b), also sufficient.Now Eq. (6-3) takes the form:
e due + g dv2 (du)' dv 2K E due + G dv2 = e ds + g VS (6-7)
This formula can be cast into a simple form. We find by substitutingfirst dv = 0, then du = 0, into Eq. (6-7), that
e gKl = E, K2 = G'
(6-8)
and if we introduce the angle a between the direction dv/du and the curve,ture direction Sv = 0, we find from Eq. (2-9) that
E du Su _.,IEdu sin a = v-G dv (6-9)cos a = - -ds
Eq. (6-7) therefore takes the form:
K = K1 COS2 a + K2 sin2 a. (6-10)
This relation, which expresses the normal curvature in an arbitrarydirection in terms of K1, K2, is known as Euler's theorem. Together withMeusnier's formula it gives full information concerning the curvature ofany curve through P on the surface.
82 ELEMENTARY THEORY OF SURFACES [CH.2
This theorem is one of the contributions to the theory of surfaces due to
Leonard Euler (1707-1783), who enriched mathematics in an endless numberof ways. It was published, with a proof different from ours, in the Recherchessur la courbure des surfaces, A16moires de 1'Academie des Sciences de Berlin 16(1760), published 1767, pp. 119-143, one o: the first papers on the theory ofsurfaces. Our proof follows that given by C. Dupin (see Section 2-7). Euleralso made a start with the theory of developable surfaces in De solidis, quorumsuperficiem in planum explicare licet, Novi Comment. Acad. Petropol., 16, 1771,pp. 3-34. Here Euler introduced x, y, and z as functions of two parameters.This is the first, or one of the first, papers where curvilinear coordinates on asurface are used.
At an umbilic the coefficients of I and II are proportional, so thatEq. (6-8) gives Kl = K2, which means that all normal curvature vectors coin-cide (this certainly holds for real surfaces; see, however, Section 5-6). Atan umbilic the directions of curvature are no longer given by Eq. (6-5),and we have to study the behavior of the surface with the aid of derivativesof higher order than the second. The number of umbilics on a surface is ingeneral finite; on an ellipsoid, for instance, there are four (real) umbilics (seeExercise 12, Section 2-8), provided the axes are unequal. On a sphere allpoints are umbilics (see Section 3-5 for the converse theorem).
The behavior of the linesof curvature near an um-bilic can be studied inDarboux, Surfaces IV,Note VII. There areseveral possibilities. Anexample is offered by thelines of curvature on anellipsoid, where they curvearound an umbilic like con-focal conics (Fig. 2-22).See also A. Gullstrand,Acta mathematica 29,1905, p. 59.
A special case of um-bilic is the umbilic at a
FIG. 2-22
parabolic point, or parabolic umbilic, where the coefficients e, f, and g allvanish. All points of the plane are parabolic umbilics (see Section 3-5 forthe converse theorem). An example of an isolated umbilic of this kind canbe constructed by taking a saddle point (Section 2-6), not with two upwardand two downward slopes on the "height of land," but with three of them.In this case we can descend in directions at an angle of 120°, and ascend inthe bisecting directions. Here every normal section has a point of inflection,
2-71 DUPIN'S INDICATRIX 83
so that K = 0 in all directions. Sucha point has been called a monkeysaddle, since there are two downwardslopes for the legs, leaving one slopefor the tail (Fig. 2-23).
This monkey saddle (Affensattel)is described in D. Hilbert-S. CohnVossen, Anschauliche Geometrie, Ber-lin: Springer, 1932, which in ChapterIV, pp. 152-239, contains a highlyoriginal approach to the study ofcurves and surfaces, stressing thevisual and imaginative side.
2-7 Dupin's indicatrix. Theprincipal curvatures K, and K2 canbe found from Eq. (6-4) by sub-stituting the values for X found by
FIG. 2-23
solving Eq. (6-5a). A short cut is found by observing that these values of Ksatisfy the two equations
(EK - e) + (FK - f)X = 0, (FK-f)+(GK-g)1,=0,
which can be simultaneously satisfied if and only if
EK-e FK-f I=0.FK-f GK-g (7-1)
This quadratic equation in K has KI and K2 as roots. From this equation wederive
FZ)
eQI the mean curvature, (7-2)M (K1 + K2)Eg2(EG- 2f F
and
K= K1K2eg=f2 the Gaussian curvature ksometimes (7-3)= EG - F2' called total curvature).
Eq. (7-3) shows that KI, K2 have the same sign at an elliptic point, anddifferent signs at a hyperbolic point. The total curvature is zero at aparabolic point.
We can now give a simple diagram to illustrate Euler's theorem. Letus take an elliptic point and take both Kl, K2 > 0. Consider the ellipse(Fig. 2-24)
84 ELEMENTARY THEORY
K1x2 + K2y2 = 1, R1 = K11,R2 = K2-')
with principal semi-axes VtG1, V"R2.A line y = x tan a through thecenter intersects the ellipse in thepoints
x= ±Cos a
K1 cos2 a + K2 sin2 Cl'
__/
sinay
_VKl cos2 a + K2 sin2 a
OF SURFACES
FIG. 2-24
The distance OA intercepted by the ellipse on this line is
Hence
OA = /x2 +-y2 =1
_V K1 coS2 a + K2 sin2 a
[CH. 2
OA = V, or OA = /R.
Here R is the radius of (normal) curvature in the direction which makesan angle a with the X-axis.
This ellipse is called the indicatrix of Dupin. If the X-axis representsone of the directions of curvature, then the distance of any point on theellipse to the center is the square root of the radius of curvature in the cor-responding direction on the surface.
When the point is hyperbolic,we can take Kl > 0, K2 < 0. Theindicatrix of Dupin in this case isthe set of conjugate hyperbolas(Fig. 2-25): \ y=xtana
KIX2 + K2y2 = ± 1, R1 = K1 ',R2 = K2 1.
The length OA on a line makingan angle a with the X-axis (corre-sponding to a direction of curvature)is now equal to VIRI. The twoasymptotes of the hyperbolas repre-sent the directions with K = 0, orthe asymptotic directions (hence
FIG. 2-25
the name). They are real for hyperbolic points, imaginary for ellipticpoints.
2-71 DUPIN'S INDICATRIX
The indicatrix of a parabolic point, where
K2 = 0, K = Ki C0S2 1P, Ri = Kl I
85
is a pair of parallel lines in the direction of the single asymptotic line(Fig. 2-26).
We now intersect the surface with a plane, parallel to the tangent planeat an elliptic point, but only a small distance a away from it. We projectthe intersection on the tangent plane. Then the principal part of theintersection, according to Eq. (6-2), will be given by the equation:
E x2 + EG xy +G
y2 = 2e (x = v'E h = E du, y = '/G k = AIG dv).
If we measure x and y in the directions of curvature, this quadraticequation becomes (see Eq. (6-6)), after a similarity transformation xxV'2e, y-+yam:
E x2 + G y2 = 1, KIX2 + K2y2 = 1.
We thus have shown for an elliptic point:The intersection of the surface with a plane close to the tangent plane and
parallel to it is in a first approximation similar to the Dupin indicatrix.In the case of a hyperbolic point we must intersect the surface with two
planes close to the tangent plane, parallel to it, and one on each side. Theprojection of the intersections of these planes with the surface on the tangentplane is again, in a first approximation, similar to the Dupin indicatrix.
In the case of a parabolic point we need only one plane parallel to thetangent plane to get, in a first approximation, the two parallel lines of theindicatrix.
This theory has a meaning only at points where the second fundamentalform does not vanish identically, that is, at points where the tangent planehas ordinary contact with the surface. At a monkey saddle the indicatrix,constructed according to the preceding rule, is given by Fig. 2-27.
FIG. 2-26 FIG. 2-27
86 ELEMENTARY THEORY OF SURFACES [CH.2
The asymptotes of the Dupin indicatrix correspond to the asymptoticdirections on the surface, the axes of the indicatrix to the directions ofcurvature. Hence:
The directions of curvature bisect the asymptotic directions.
Charles Dupin (1784-1873), a pupil of Monge at the Paris Ecole Polytech-nique, wrote the Developpements de geometrie (1813), in which he published anumber of results which carry his name, as well as the treatment of lines of curva-ture and asymptotic lines which now is usually presented. The subtitle of hisbook Avec des applications a la stabilite des vaisseaux, aux deblais et remblais, audefilement, a l'optique, etc. shows how the author never lost contact with engi-neering practice. This is also evident in his second book, Applications degeometrie (1822). Entering the Napoleonic navy as an engineer, Dupin livedto be a promoter of science and industry, a peer of France and a senator underNapoleon III. The notion of mean curvature was introduced by Sophie Ger-main, Crelle's Journal fur Mat hem. 7, 1831, pp. 1-29.
2-8 Some surfaces. 1. Sphere. Here (see Eq. (2-12)) :
ds2 = a2 do2 + a2 cost 9 u = 0, v = (p)x,, (-a cos 0 cos (p,x,v( a sin 9 sin p,
-a cos 0 sin p,-a sin 0 cos rp,
-a sin 0),0 ),
xvv(-a cos 0 cos rp, -a cos 0 sin rp, 0 ),
E = a2 cos 0,
-acos0cosrp -acosBsin sp -a sin 0-asin6cos4p -asin9sinrp acosO
II = a(d82 + cost 0 dp2),
II=+a.
-a cos 9 sin rp +a cos 0 cos rp 0a2 cos B
f=0, g=acos2B,
a3 cos B _a2 cos
(8-1)
This is the case mentioned at the end of Section 2-6, where I and II areproportional. All points on the sphere are umbilics, and since the principaldirections are undetermined at all points all curves on the sphere may betaken as lines of curvature. The + sign of K is due to the choice of
N(-cos 0 cos cp, -cos 0 sin (p, -sin 0),
which means that N is directed toward the center of the sphere. This isalso the sense of KN, the curvature vector of a normal section.
2-81 SOME SURFACES
The asymptotic lines of the sphere satisfy the equation
87
do2 + cost 0 dp2 = 0, (8-2)
and are therefore imaginary curves; since I and II are proportional theyare also isotropic curves. Conversely, all isotropic curves on the sphereare at the same time asymptotic curves. Waiving for a moment our re-striction to real figures only, we find the integral of Eq. (8-2) in the form
otan (2 + 4 = cetlp,
which in the case of the + sign leads to
C209, - 1 2ce1wsin B =
c2e21rp + 1,COS B = c2e14'P + U = e2ty.
Inserting these values into the Eqs. (1-7) for x, y, z, we obtain
x+i a2cu z _ ac2u- 1y= c2u+ c2,u+1
(8-3)
Elimination of u shows that these curves are plane, and since they are
isotropic, they must be straight lines (Section 1-12).We obtain a similar result when in Eq. (8-3) we take the - sign:Through every point of the sphere pass two isotropic straight lines lying on
the sphere. They are the asymptotic lines of the sphere.The Dupin indicatrix at a point of the sphere is a circle.2. Surface of revolution. Here (see Eq. (2-14)):
We findds2 = (1 + f'2) dr2 + r2 d1p2.
rfe +f,2, f =0,
+f'2
We conclude that because of f = F = 0 the coordinate lines are lines ofcurvature (see Eq. (6-6)).
The lines of curvature of a surface of revolution are its meridians and par-allels.
The asymptotic lines are the integral curves of the equation
f" dr2 + rf' drp2 = 0
* The reader will have little trouble to distinguish between the f of the secondfundamental form and the f of f(r).
88 ELEMENTARY THEORY OF SURFACES [CH. 2
and can therefore be obtained by a quadrature. Since
leg
- f2_ rff
+ f'2'
the points of a real surface are elliptic or hyperbolic, depending on whetherf'f" > 0 or < 0 respectively. The circles of parabolic points are given byf' = 0, which gives the points which in the profile have tangents 1 Z-axisand f" = 0, which gives the points of inflection of the profile. Betweenthese circles the asymptotic lines are alternately real or imaginary(when f' = 0 and f" = 0 are not satisfied at the same point of the profile).
When f" = 0 at all points of the surface we have f = ar + b: a circularcone (Eqs. (1-8, 2-16)). All points are parabolic except the vertex r = 0,where e = f = g = 0 (this point is singular, not umbilic).
As soon as f" contains a factor r, the point r = 0 has also e = f = g = 0.This happens, for example, when f = ar'. The profile at the point r = 0has a tangent line with a contact of order 3 in r.
The asymptotic lines are orthogonalwhen (see Exercise 6, Section 2-3) :
E(rf') + G(f")= rf'(1 + f'2)+ r2f" = 0.
When we integrate this equation, weobtain:
r2aa2, or f =a cosh- 1 a + c,
(a, c integration constants),
which means that the profile curve isa catenary. This surface, obtainedby rotating the curve
y = a cosh-1 a
or
FIG. 2-28
a v+ )x=aeosha=2 eaea
about the Y-axis, is called a catenoid (Fig. 2-28). Surfaces on which theasymptotic lines form an orthogonal net are called minimal surfaces. Wehave thus found that the only surfaces of revolution which are also minimalsurfaces are catenoids.
2-81 SOME SURFACES
3. Right conoid. Here (Section 2-2) :
We findds2 = dr2 + (r2 + f'2) d4'2, f = f(4').
-f' ffire = 0, f = Vr2 + g=
r2 + f2.
The asymptotic lines are given by
-2f'drdcp+f"rdcp2=0.
89
Since eg - f 2 < 0, all points are hyperbolic and the asymptotic linesare real. One set is given by d(p = 0, which means that the straight linesare asymptotic lines. The other set is the solution of
- 2f' dr + r f" dp = 0, dr =, cr2 = fr 2f'
and can therefore always be found in finite form. The two sets are orthog-onal when f" = 0 or f = app + b, which characterizes the right helicoid(Eq. (2-19)).
The right helicoid is the only right conoid which is also a minimal surface.In this case the curved asymptotic lines are given by r = constant; theyare circular helices (Fig. 2-29). Bisecting the angle of these curves(Section 2-3, Exercise 9) are the lines of curvature:
-a dr2 + a(r2 + a2) dV2 = 0; + Cl = ±a sinh-1 a
FIG. 2-29 FIG. 2-30
90 ELEMENTARY THEORY OF SURFACES [CH.2
Since r and cp are the polar coordinates of the projection of a point of theconoid on the XOY-plane, the lines of curvature are projected on the XOY-plane in the curves (Fig. 2-30):
r = ±a sinh a ('p + CO.
4. Developable surfaces. For a tangent developable:
y(s, v) = x(s) + vt(s).Here
yx=t+vKn, yv=t,yss = Kn + v(Kn)', ys, = Kn, yvv = 0,
N-vKb --bEE - F2
E = 1 + v2K2, F = 1, G = 1,e=-vKT f=0, g=0.
(8-4)
We conclude that eg - f2 = 0, so that there are only parabolic points.The asymptotic lines are the solution of eds2 = 0, hence they are thestraight lines of the surface. The lines of curvature are the integral curvesof the equation
dv2 -ds dv ds2
E 1 1 = 0,e 0 0
or
ds2 + ds dv = 0,
hence they are the straight lines andthe curves s + v = c, which are theinvolutes of the edge of regression (Sec-tion 1-11).
When, in Eq. (8-4), x(s) reduces toa constant vector a, then Eq. (8-4)represents a cone of which the vertex isthe end point of a:
FIG. 2-31
y = a + vt(s),
where t is now a given unit vector field, determined for instance by a curveon a unit sphere (Fig. 2-31). Here again we find that eg - f2 = 0, andthat the lines of curvature are formed by the generating lines and theirorthogonal trajectories.
2-81 SOME SURFACES
A cylinder is given by the equation
y=x(s)+vawhere a is a constant vector. Here we also find that eg - f2 = 0.
91
We shall now prove that eg - f 2 = 0 is not only a necessary, but also asufficient condition that a surface be developable.
We prove this by means of the identity
eg - ft = (x,. N,.) (x (x,. Nv) (xv N,.) = (x,. x x Nv)
= F2, (8-5)
which shows that eg - f2 = 0 is identical with 0. This canhappen either (a) when N,u or Nv vanishes, or (b) when N is collinear withN (N is perpendicular to N. and N.). In case (a) N depends on only oneparameter and the surface is the envelope of a family of ool planes, andhence (see Section 2-4) a developable. In case (b) we take as one set ofcoordinate curves on the surface the asymptotic curves with equation
e due + 2f du dv + g dv2 = (' du +' dv)2 = 0.
If these curves are taken as the curves v = constant in the new coordinatesystem, then e = f = 0 or x N,. = x N = 0, hence N = 0, whichbrings us back to case (a). The theorem can also be stated as follows(see Eq. (7-3)) :
A necessary and sufficient condition that a surface be developable is that theGaussian curvature vanish.
EXERCISES
1. What is the second fundamental form when the surface is given by theequation z = f(x, y)?
2. The differential equation of the developable surfaces z = f(x, y) is rt - s2 =0.(Here and in the following text we often use the notation
p = az/ax, q = az/ay, r=a2z/8x2, s = a2z/axay, t = a2z/ay2.)
3. The differential equation of all minimal surfaces z = f(x, y) is
t(1 + p2) - 2pqs + r(1 + q2) = 0.
4. What is the geometrical meaning of the general integral of the partial dif-ferential equations
(a) xp + yq = 0, (b) yp - xq = 0?
5. Find the equation of the lines of curvature in the case that the equation ofthe surface is z = f(x, y).
92 ELEMENTARY THEORY OF SURFACES [CH. 2
6. Show that the surfaceN
z =n=2
where the coefficients of x and y are all constants, is developable.7. The sum of the normal curvatures at a point of a surface in any pair of or-
thogonal directions is constant.8. Find the radii of principal curvature at a point of a surface of revolution
x = u cos v, y = u sin v, z = f(u), and interpret the answer geometrically.9. Find the radii of principal curvature at a point of a developable surface.
10. The asymptotic lines on the surface
x4 y4z=a4 b4
are the curves in which the surface is met by the two families of cylindersx2 y2
=x2 y2
a2+
b2constant, a2 - b2 = constant.
11. Asymptotic lines on surfaces of revolution. Show that the surface of revo-lution with the profile z = f(r), given in Table A, has asymptotic lines of whichthe projection on the XOY-plane has the equation r = r((p), given in Table B,where r, rp are polar coordinates.
Table A Table B
(a) z = (a) p +c=JVIn r(b) z = 6r-1 (b) p + c = V In r(c) z=6lnr (c) rp+c=Inr(d) z = (r - a)2 (d) r = a cos2(,p/2 + c)
l(e) z = Cr a.2 - r2 - a2 cos'- c =cos-1(e) P +r
aJ a
Plaster models of these surfaces and of some others (twelve in all) were con-structed in 1885 at the University of Munich under the supervision of A. Brill andcan be inspected in many cabinets of mathematical models.
12. Find the umbilics of the ellipsoid and prove that the tangent planes at thesepoints are parallel to the circular sections of the ellipsoid (that is, to those planeswhich intersect the ellipsoid in circles).
13. Find the parabolic curves on a torus (Fig. 2-21) by computation.14. Show that the asymptotic directions of a surface, given by F(x, y, z) = 0,
are given by
0, Fxdx+F,dy+F,dz = 0.15. Show that the directions of curvature in the case of Exercise 14 are given by
F. dFx dxF dF dy = 0, F5dx+F5dy+F.dz = 0.F. dF, dz
2-91 A GEOMETRICAL INTERPRETATION 93
16. Show that a surface F(x, y, z) = 0 isdevelopable if
F.. Fz,y Fz. Fx
Fvz Fvv Fv. Fv
Fox Foy F F.Fz F Fz 0
= 0, F y = 8F/i3x, etc.
17. Cylindroid. The locus of the endpoints of the curvature vectors at a pointP of a surface belonging to all curves pass-ing through P is a right conoid with theequation
z(x2 + y2) = K,x2 + K2y2.
This surface is called conoid of Pliicker orcylindroid; the surface normal is its doubleline (Fig. 2-32).
2-9 A geometrical interpretation of asymptotic and curvature lines.The lines of curvature can be characterized by the following property, dueto Monge.
A necessary and sufficient condition that a curve on a surface be a line of cur-vature is that the surface normals along this curve form a developable surface.
To prove this theorem, let the curve be given by x = x(s). Then
(dx/ds) N = t N = 0.
by
A point on the ruled surface formed by the surface normals can be given
y = x(s) + uN(s),
where u is the distance of point y to point x. Then
(9-1)
y. = t + uN., Yu = N, You = No, Yuu = 0,
hence g = 0, and the condition of developability eg - f2 = 0 requires thatf = 0 or
(t N N,) = 0.
N is perpendicular to t and N so that the only ways in which this conditioncan be satisfied is (a) No = 0, (b) t is collinear with No. In the case (a) the
94 ELEMENTARY THEORY OF SURFACES [CH.2
surface normals form a cylinder or a plane, which are developable surfaces.Case (b) can be expressed as follows:
dN = X dx (along the curve); (a) X = 0, (b) X 0. (9-2)
Eq. (9-2) is equivalent to
N. du + N, dv = X(x. du + x dv),
which vector equation is equivalent to the two following scalar equations,obtained by scalar multiplication with x. and x respectively:
(e+XE)du+ (f +XF)dv = 0,(f + XF) du + (g + XG) dv = 0,
which in turn are equivalent to Eq. (6-4a), with X = -ic; conversely,Eq. (6-4a) is equivalent to Eq. (9-2). The curves are therefore lines ofcurvature and only these; moreover, we have found a new equation tocharacterize these lines:
dN+Kdx=0, (9-3)
where K is the normal curvature in the direction dx of the line of curvature.This is the formula of Rodrigues.
Olinde Rodrigues (1794-1851), another of Monge's pupils, published his re-sults in the Correspondance sur l'Ecole Polytechnique 3 (1815) and in the Bull.Soc. Philomatique 2 (1815). His name is also attached to a theorem in Legendrefunctions. He became a follower of St. Simon, and was an editor of the col-lected works of this reformer.
Eq. (9-2) with K = 0 represents a cylinder or a plane, and with Kconstant a cone. This can be verified by differentiating Eq. (9-1):
dy = dx + u(-K dx) +Ndu = dx(l - Ku) +Ndu,
which is zero when u = K 1, du = 0, and only in this case. The surfacenormals along a line of curvature, in all other cases, form a tangentialdevelopable, so that they are tangent to a space curve, the edge of regres-sion. Each surface normal is tangent to two edges of regression; the pointsof tangency are at distances Kl 1 and K2-' from P; they are the centers ofprincipal curvature. The two families of lines of curvature on the surface,forming an orthogonal net, thus determine two sets of developable surfacesintersecting at right angles along the surface normals. The center surfacesare the loci of the centers of principal curvature, of the surface. A simple
2-91 A GEOMETRICAL INTERPRETATION
FIG. 2-33
95
example is offered by the meridians and parallels of a surface of revolution,which are lines of curvature because along the meridians the surface normalsform planes, and along the parallels form cones. The cones and planes aredevelopable surfaces. They are orthogonal to each other; one of the centersurfaces here degenerates into the axis. The general case is illustrated byFig. 2-33, where CI and C2 are the lines of curvature through P, and Q andR are the points on the center surfaces. The edges of regression of thedevelopable surfaces lie on the center surfaces.
Monge was led to the lines of curvature of a surface in his MWmoire sur latheorie des deblais et des remblais, Memoires de I'Acad. des Sciences, 1781, pp. 666-704, which was based on the following engineering problem: To decompose
96 ELEMENTARY THEORY OF SURFACES [CH. 2
two given equivalent volumes into infinitely small particles which correspondto each other in such a way that the sum of the products of the paths describedby the transportation of each particle of the first volume (the "deblai") to itscorresponding particle of the second volume (the "remblai ") and the volumeof the particle is a minimum. The orbits were supposed to be straight lines.This led to rectilinear congruences; that is, to families of straight lines withthe property that through one point of a region of space one and only one linepasses. These lines can always be arranged in two sets of developable surfaces;when these surfaces are normal, the lines of the congruence are normal to a setof surfaces on which the developable surfaces cut out the lines of curvature. Seethe monograph of P. Appell, Le probleme geometrique des deblais et des remblais,Memorial des sciences mathematiques, 29, 1928, 34 pp.
The asymptotic lines are characterized by the equation II = 0 or,according to Eq. (5-8) :
0. (9-4)
This equation can be written in the form t dN = 0, and since t N = 0,also in the form N dt = 0, or
K = 0 by N N. n = 0. This leads to the fol-lowing characterization of asymptotic lines:
All straight lines on a surface are asymptotic lines. Along a curved asymp-totic line the osculating plane coincides with the tangent plane.
The converse is also true, and when we consider that any plane through astraight line may be considered as an osculating plane of the line, we cansummarize our results as follows:
A necessary and sufficient condition that a curve on a surface be asymptoticis that the tangent plane of the surface coincide with the osculating plane of thecurve.
As an example we can take the straight lines of a ruled surface, whichhere form one family of asymptotic lines (see our example of the developablesurfaces and of the right conoids, Section 2-8). The hyperboloid of onesheet has two sets of straight lines, which are its asymptotic lines. Theisotropic lines on the sphere are its asymptotic lines (Section 2-8).
2-10 Conjugate directions. In his study of the indicatrix which carrieshis name, Dupin pointed out that the directions on the surface correspond-ing to conjugate diameters in the indicatrix lead to curves with interestingproperties. He called these directions conjugate; they determine conjugatefamilies of curves.
2-101 CONJUGATE DIRECTIONS
The diameters y = mix and y = m2x in the ellipse (Fig. 2-34)X2 1 2
a+b2
97
are called conjugate if one diameter d1 is parallel to the tangents at the end pointsB1, B2 of the other diameter. Since the tangent at Bi(xi, m2x1) has the equation
xxi m2yxi = 1a2 + b2
we havem1 = -b2/a2m2,
or
m,m2 = - b2/a2.
The relation between two conjugatediameters is reciprocal: the tangents atthe end points A,, A2 of d1 are parallelto d2. For the hyperbolas
2 y2a2 b2 ± 1
the condition is m1m2 = b2/a2.
FIG. 2-34
Conjugate directions with respect to the indicatrix, both in the ellipticand in the hyperbolic case, are related by the equation
K1m1m2 =- -K2
Since m = tan p = 4E du (see Eq. (6-9)) and K2 = E g (see Eq. (6-8)),
we find for the equation determining conjugate directions:
,4E -du
GSvGeEg'
or
e du Su + g dv Sv = 0. (10-1)
This equation for conjugate directions is obtained for the special case
that f = 0. To obtain expressions valid for the general case, in whichf 0, we substitute for e, f, g, their values (5-7a). Eq. (10 1) then canbe written (adding f Su dv and f du Sv, which are here zero) :
x. - N.
du + x Nv 0,
98 ELEMENTARY THEORY OF SURFACES [CH.2
or
(10-2a)
This equation may also be written
Sx dN = 0. (10-2b)
In an arbitrary curvilinear coordinate system this equation takes the form
edu5u+f(duSv+dvSu)+gdv5v= (10-3)
Since asymptotic lines are defined by dx dN = 0 we see that asymptoticlines are self-conjugate. This agrees with the fact that the asymptotes ofthe indicatrix are self-conjugate diameters. One family of curves may beselected arbitrarily; it determines the conjugate family.
Eqs. (10-2a, b) allow a simple geometrical interpretation. The tangentplanes along a curve C(x(s)) on the surface form a developable surface.Let the equation
X=x+upindicate the generators of this surface. Along the generator the unit sur-face normal vector is N, so that from N dX = 0, where
dX = dx + udp + pdu
it follows that 0 = N. dx + uN dp +N p du. Since N dx = 0 andN p = 0, we conclude that
which leads to
If we denote the direction of p by ax, we find that
ax dN = 0,
which means that the directions of dx and ax are conjugate. In other
words:The generating lines of the developable surface enveloped by the tangent
planes to a surface along a curve C on the surface have along C a direction
conjugate to the direction of C. In simpler, but less exact, words:The line of intersection of two tangent planes at consecutive points in a direc-
tion on a surface has the conjugate direction. This relation is reciprocal.
2-11 TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 99
The lines of curvature form a conjugate set. This follows from theproperties of the indicatrix (its axes are conjugate). It also follows fromthe orthogonality relation dx Sx = 0, if we take dx and Sx to be in thedirections of curvature. In this case, we obtain from Rodrigues' formulaSx = - Ki 1 SN that dx SN = 0.
When the parametric lines are conjugate, Eq. (10-3) is satisfied fordu = 0, Sv = 0, which means that
f = 0.
Conversely, when f = 0, the parametric lines are conjugate.Since the lines of curvature are both orthogonal and conjugate, we have
returned to Eq. (6-6), expressing that the necessary and sufficient conditionthat the parametric lines are lines of curvature is:
F=0, f=0.EXAMPLE. Let us find the curves conjugate to the parallels 0 = u =
constant on the sphere (Section 2-2). They satisfy the equation
g dv Sv = g dip Sip = 0, or Sip = 0,
which means that the meridians are conjugate to the parallels. Indeed,the planes tangent along the parallels envelop cones whose generating linesare tangent to the meridians; the planes tangent along the meridiansenvelop cylinders whose generating lines are tangent to the parallels.
It is convenient to combine in a table the conditions satisfied by E, F, G,e, f, g when the parametric curves belong to one or another of the specifictypes of curves which we have discussed. These curves are
orthogonal, when F = 0,lines of curvature, when F = 0, f = 0,conjugate, when f = 0,isotropic, when E = 0, G = 0 (since in this case I = F du dv),asymptotic, when e = 0, g = 0 (since in this case II = f du dv).
2-11 Triply orthogonal systems of surfaces. A point on a surface or inthe plane is determined by two parameters, which form a curvilinear sys-tem of coordinates; sometimes this system may be rectilinear as in the caseof cartesian coordinates in the plane. We can in a similar way determine apoint in space by means of three parameters, or curvilinear coordinates.For this purpose we transform the cartesian coordinates (x, y, z) of a pointP by means of the equations
x = x(u, v, w), y = y(u, v) w), z = z(u, v, w), or x = x(u, v, w) for short.(11-1)
100 ELEMENTARY THEORY OF SURFACES [CH. 2
The surfaces u = constant, v = constant, w = constant are the coordinatesurfaces; they intersect in the coordinate lines. Examples are the trans-formations to cylindrical or spherical coordinates, which can be written asfollows :(a) x = u cos v, y = u sin v, z = w,(b) x = u cos v cos w, y = u cos v sin w, z = u sin v. (11-2)
Since
dx = x du + x, dv + xw dw,
the square of its length takes the form
dudv xw xw dw2. (11-3)
Of particular interest are those curvilinear coordinates for which
xu. x,, = x,, - xw = Xw - xu = 0. (11-4)
The coordinate lines, and therefore also the coordinate surfaces, are per-pendicular. This orthogonality can be proved in the same way as in thetwo-dimensional case (Eq. (2-10)). Examples of orthogonal systems ofcurvilinear coordinates are (a) the cartesian, (b) the cylindrical, and (c) thespherical system of coordinates, where the respective coordinate surfacesare (a) planes, (b) planes and cylinders, and (c) planes, spheres, and cones.The coordinate surfaces, in this case, form a triply orthogonal system ofsurfaces. Every transformation (11-1), subject to the conditions (11-4)determines such a system. There exists no simple way of finding suchsystems, since it requires the solution of three partial differential equations(11-4) in three dependent and three independent variables. We shalltherefore content ourselves with a few examples, of which the simplesthave already been presented. Another example is the system expressedby the equation
a2x2
X+ b2 y2 X + c2
z2
X= 1, a2 < b2 < c2, (11-5)
which is a system of quadric surfaces with the origin as center. WhenX < a2 it represents ellipsoids, when a2 < X < b2 it represents hyperbo-loids of one sheet, and when b2 < X < c2, hyperholoids of two sheets.When X > c2 the surfaces are imaginary. For given (x, y, z) Eq. (11-5)represents a cubic equation in X, which, as can he shown (see Exercise 7,Section 2-11), has three real roots, one < a2, one between a2 and b2, andanother between b2 and c2. Through every point of space passes one ellipsoid,
2-I1] TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 101
one hyperboloid of one sheet, and one hyperboloid of two sheets. When z = 0they cut out of the XOY-plane a set of conics with the same foci, sinceb2 - X - (a2 - X) = b2 - a2 is independent of it. Such a system of conicsis called a confocal system. For this reason we call Eq. (11-5) a system ofconfocal quadrics, although there are no points which can be called commonfoci.
The tangent plane at a point P(x1, y1, z1) to a quadric with the origin asits center has the equation
xxi + yyi + zzi = 1.a2-it b2-i c2-iWhen two quadrics passing through P are given by XI, it2, XI s it2, thenthey form an angle a of which the cosine is given by a fraction with thenumerator:
I yi zi(a2 - it1) (a2 - X2)
+(b2 - it1) (b2 - it2)
+ (e2 - 1) (e2 - it2)
But x1y1z1 satisfy also Eq. (11-5) for it = itI and it = X2, hence we find that
FIG. 2-35
102 ELEMENTARY THEORY OF SURFACES [CH.2
xl [a2 1 1 - a2 1X2J
yl [b2 1 b2 1 2J z1 [C2X1 c2 1 2J - 0,
which shows that a = 90°. In \v'ords:The quadrics of a confocal system form a triply orthogonal system (Fig. 2-
35).When we express x, y, z in terms of the X1, X2, X3 determining the three
orthogonal quadrics passing through the point (x, y, z) :
x = X(XI, X2, X3), (11-6)
we obtain a system of elliptic coordinates in space.We thus have established the existence of other triply orthogonal sys-
tems than the trivial ones determined by Eqs. (11-2). This lends interestto the following theorem, also due to Dupin:
The surfaces of a triply orthogonal system intersect in the lines of curvature.To prove it let us define the system by means of Eq. (11-1) and Eq.
(11-4). From Eq. (11-4) we find by differentiation
hence
xuw ' Xv = -Xu ' Xvw = +Xw ' Xuv = -Xv ' Xuw,
Xvw ' XU = Xwu ' Xv = Xuv ' Xw = 0. (11-7)
Let us single out for consideration the surface w = 0. Its surfacenormal is in the direction of xw; the vectors xu and x are tangent vectors,functions of the curvilinear coordinates (u, v) on the surface.
The equation xu xv = 0 means that the F on this surface vanishes; theequation xw xu = 0 means, according to Eq. (5-9) that f vanishes.Coordinate systems for which f = F = 0, are formed by lines of curvature.Instead of w = constant we could take any surface of the set and reason ina similar way. The lines of intersection are therefore all lines of curvature.
This theorem does not give us much valuable information for the cases(11-2), but it gives us a simple way of finding the lines of curvature on anellipsoid, a hyperboloid of one sheet, or a hyperboloid of two sheets. If, forinstance, we wish to find the lines of curvature on the ellipsoid
x2 2 z2a+ b2+C2 = 1, (11-8)
we "imbed" this ellipsoid in the triply orthogonal system (11-5). Then wesee immediately:
The lines of curvature on the ellipsoid (11-8) are the curves in which theellipsoid is intersected by the two sets of hyperboloids in the triply orthogonalsystem (11-5) to which the ellipsoid belongs.
2-11] TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 103
The lines of curvature on the ellipsoid are therefore space curves of thefourth degree.
We have been able to see immediately that an ellipsoid can be "im-bedded" in a triply orthogonal system, and so can the system of ellipsoids(Eq. (11-5), X < a2) to which it belongs. It can be shown, however, thatnot every single infinity of surfaces can thus be made part of a triply orthog-onal system. For instance, a system of quadrics (11-8), in which a2, b2, c2depend on a parameter u, can be part of a triply orthogonal system only if
a3(b2 - c2)a' + b3(c2 - a2)b' + c3(a2 - b2)c' = 0,* a' = da/du, etc.This is not satisfied, for instance, for a family of ellipsoids which are similarwith respect to their common center. It can be shown that a single in-finity of surfaces must satisfy a partial differential equation of the third orderto be "imbeddable" in a triply orthogonal system.
Several textbooks (Forsyth, Eisenhart, et al.) have chapters on triply or-thogonal surfaces. A full treatment is given in G. Darboux, Legons sur lessystemes orthogonaux et les coordonnees curvilignes (Paris, 1910). Gaston Darboux(1842-1917), to whose works we must often refer, in 1880 succeeded Chasles asprofessor of higher geometry at the Paris Sorbonne and in 1900 succeededBertrand as secretaire perpetuel of the Academie des Sciences. In his manypapers and books he combined geometrical intuition with a mastery of algebraand analysis. His Legons are not only a great source of information on surfacetheory, but also belong to the best written mathematical books of the nineteenthcentury.
EXERCISES
1. Show that the coordinate lines of the surfaces
y = x,(u) + x2(v),where x1(u) and x2(v) are arbitrary vector functions, are conjugate lines. Thesesurfaces are called translation surfaces.
2. Find the family of curves on a right conoid conjugate to the orthogonal
trajectories of the generating lines.3. If two surfaces intersect each other along a curve at a constant angle, and
if the curve is a line of curvature on one surface, then it is also a line of curvatureon the other. Also prove the converse theorem (0. Bonnet, Journ. Ec. Polyt. 35,1853, a special case by Joachimsthal, Journal fur Mathem. 30, 1846, pp. 347-350).
4. Third Fundamental form. Apart from I = dx dx and II = -dx dN athird fundamental form III = dN dN is occasionally introduced. Prove thatif K is the total curvature and M the mean curvature
KI-2M II+III=0.
* See e.g. A. R. Forsyth, Differential Geometry, p. 453.
104 ELEMENTARY THEORY OF SURFACES [CH.2
5. Theorem of Beltrami-Enneper. The torsion of the asymptotic lines througha point of the surface is (E. Beltrami, 1866, Opere matem. I, p. 301).Hint: Use the result of Exercise 4.
6. Spherical image. When through a point unit vectors are drawn parallelto the surface normals along a curve C on a surface, the end points describe thespherical image of C on the unit sphere. Show that III (Exercise 4) is the squareof the element of are of the spherical image. Also show that at correspondingpoints the spherical image (a) of the lines of curvature is parallel to the lines ofcurvature, (b) of the asymptotic lines is perpendicular to the asymptotic lines.
7. Show that the spherical image of a family of curves of a surface S is orthogonalto its conjugate family on S.
8. Show that the systems of surfaces
(a) x2 + y2 + z2 = u, y = vx, x2 + y2 = wz,(b) x2 + y2 + z2 = ux, x2 + y2 + z2 = vy, x2 + y2 + z2 = wz,
form a triply orthogonal set.9. Prove that the conics of the confocal system in the plane
a2
x2+ b2 y2 = 1 (X variable)
form an orthogonal set (Fig. 2-36): (a) by following a method similar to that ofSection (2-11), (b) by considering the transformation w = sin z of one complexplane on the other.
10. Prove that Eq. (11-5) in X, with fixed x, y, z, has three real roots. Hint:Show that the roots of Eq. (11-5) are separated by the roots of the equation of
Exercise 9.
FIG. 2-36
CHAPTER 3
THE FUNDAMENTAL EQUATIONS
3-1 Gauss. The material of Chapters 1 and 2 bears the strong imprintof the methods of Monge. In this chapter and in the next we discussanother approach, connected with the name of the great German mathe-matician, Gauss.
For both men, theory and practice were intimately related. Monge,as a craftsman and engineer, saw in a surface primarily the boundary of asolid body, and consequently stressed the properties of a surface in relationto the surrounding space. He selected his topics, in the main, by theirvisual or engineering appeal, often finding his theoretical guidance in theirpossible relationship to the theory of partial differential equations. Gaussapproached the theory of surfaces primarily as a result of his work on tri-angulation, where the emphasis is on measurements between points on thesurface of the earth. Consequently he saw in a surface not so much theboundary of a solid body, as a fleece or film, a two-dimensional entity notnecessarily attached to a three-dimensional body. A piece of such a sur-face can be bent and we can ask for the properties of the fleece which donot change under bending. A two-dimensional being, living on this sur-face and unaware of any outside space - like the beings of Abbott'sFlatland,* which live in the plane unaware of any space of which theplane may be a part - would not be able to find out what asymptoticlines or lines of curvature are. But he would be able to find the road ofshortest distance between two points measured along the surface, or theangle of two directions on the surface, that is, the intrinsic properties ofthe surface. Thus, with his characteristic understanding of theory andpractice alike, Gauss drew from his work as a surveyor the inspiration forhis profound reappraisal of the general theory of surfaces.
Carl Friedrich Gauss (1777-1855) was director of the astronomicalobservatory at Gottingen from 1807 to his death. One of the greatestscientists of all times, his work combined the fertility and particularoriginality of the eighteenth century mathematicians with the criticalspirit of modern times.
The fields of his activity ranged from the theory of numbers to complexvariable theory, and from celestial mechanics to the electric telegraph, of
* E. A. Abbott, Flatland - a romance of many dimensions - by a square (1884).Several editions; among others, Boston, Little, Brown & Co., 1941, xiii + 155 pp.
105
106 THE FUNDAMENTAL EQUATIONS [CH. 3
which he is one of the inventors. His fundamental work on surface theory,the Disquisitiones generales circa superficies curvas of 1827 (Werke 4,pp. 217-258) was, like much of Gauss' work, written in Latin, but thereexists an English translation, General investigations on curved surfaces,Princeton, 1902, 127 pages. Gauss published other works on questionspertaining to geometry, such as his paper on conformal representation(1822), but he kept some of his boldest ideas to himself, notably his ideason non-Euclidean geometry (see Section 4-7).
3-2 The equations of Gauss-Weingarten. We can penetrate intoGauss' mode of thinking by asking whether there exist any relations be-tween the coefficients of the first and of the second fundamental form. Itis clear that such relations cannot be purely algebraic, since E, F, G dependon only x and x,,, whereas e, f, g depend also on xuu, xuv, and xvv (algebraicrelations can, of course, exist in special cases to characterize special sur-faces or the nature of the surface at certain points, such as umbilics). Thegeneral relations between E, F, G, e, f, g must be differential relations.We shall now show how these relations can be found.
This demonstration can be accomplished by means of certain formulaswhich were used by Gauss himself in his paper of 1827. We shall derivethem by introducing, at a point P of a surface, a moving trihedron, now com-posed, not of three mutually orthogonal unit vectors, as in the case of spacecurves, but of the three linearly independent vectors xu, x,,, N, of which xuand x lie in the tangent plane normal to N (Fig. 3-1). These vectorssatisfy the equations for scalar multiplication:
XU x N
FIG. 3-1
xu
xv
N
E
F
0
F
G
0
(2-1)
This moving trihedron depends on two parameters (u, v), whereas in thecase of space curves it depends on only one parameter.
Every vector can be linearly expressed in the three basic vectors of themoving trihedron. When we do this for xuu, xuv, and xvv, we obtain equa-tions of the form
xuu = alxu + a2xv + a3N,xuv = R1xu + a2Xv + 03N, (2-2)xvv = 'lXu + 1'2xv + 73N,
3-2] THE EQUATIONS OF GAUSS-WEINGARTEN 107
where the coefficients a1 ... 73 have to be determined. We immediatelyfind, by using Table 2-1 and Section 2-5, that
(2-3)
Moreover, using again Table 2-1 and introducing the notation
[11, 1] = x,.,, x,,, [11, 2] = x,,,, x,, (2-4)we find
[11, 1] = Eal + Fat, [11, 2] = Fat + Gal,
or, solving these equations for al and a2,
_ G[11, 1] - F[11, 2] E[11, 2] - F[11, 1]a1 EG - F2
' a2 = - EG - F2 (2-5)
The bracket symbols [11, 1] and [11, 2] can be expressed in terms of deriva-tives of E, F, G, because
Hence:2xu x,,,, = E,,, 2x x,,,, = E,,, x,. xn + x,,,, x,, = F,,.
[11, 1] = zEu, [11, 2] = Fu - zE'ro,
and, if we introduce these expressions into Eq. (2-5), we see that the coef-ficients a1 and a2 can be expressed in terms of E, F, G and their first deriva-tives. A similar reasoning gives us /31, /32 and y1, 112-
Changing the notation, we denote a1 by rill and a2 by ri1. If we changethe R; and y;(i = 1, 2) similarly, we obtain the formulas
&,,, = 111x+, + r2,lx, + eN,x,,,, = r12x + r2 12 X, + fN,xroro = r22xu + ra2xv + gN,
where the rk (i, j, k = 1, 2) are defined as follows:
T,2 2EF - EE,, - FE11 2(EG - F2)
EG,, - FE,r12 2(EG F2)' r12 2(EG - F2)'
FGvGu=
2GFr rzEGv-2FF,+FG
at2(EU
F2)2 2(EG - F2)
(2-6)
(2-7)
The r),, are called the Christoffel symbols. It is convenient to introduce notonly rat but also rt1 by the definition
rile = rk1 (hence rat = rig, rs1 = r?2) (2-8)
108 THE FUNDAMENTAL EQUATIONS [CH. 3
The [i j, k] are often called Christoffel symbols of the first kind; in that casewe call the rk Christoffel symbols of the second kind. They depend exclusivelyon the coefficients of the first fundamental form and their first derivatives.
Eqs. (2-6) are the Gauss equations which we intend to derive. They havealso been called the partial differential equations of surface theory.
The Christoffel symbols are called after Erwin Bruno Christoffel (1829-1901),who taught first at Zurich and after the Franco-Prussian war of 1870-1871 atStrassbourg. Christoffel introduced his symbols in a paper on differential formsin n variables, published in Crelle's Journal fur Mathem. 70, 1869 (Gesamm.Abhandlungen I, 1910, p. 352), denoting our r;k by { ik 1. The change to ourpresent notation has been made under the influence of tensor theory.
It will be found useful to complement the Gauss Eqs. (2-6) by the twoequations which express the derivatives N. and N. in terms of the movingtrihedron. Since N. and N, lie in the tangent plane, the expressions will beof the form
N. = p,xu + p2x,, N. = g1xu + q2x,.
With the aid of Table 2-1 and Section 2-5, we obtain
-e = p1E + p2F, -f = q1E + q2F,-f = p1F + p2G, -g = qiF + q2G,
or
Nu =fF - eG zu + e F
X"EG - F2 EG - F2gF - fG fF - gEN. EG - fin xu + EG - F2 xv. (2-9)
It is customary to call these equations after J. Weingarten (Crelle's Journalfur Mathem. 59, 1861).
An interesting application of the Gauss equations is obtained by referringthe surface to a conjugate set of curvilinear coordinates. Then f = 0, andthe second of the Eqs. (2-6) becomes
xuv - r;2xu - r12x,, = 0. (2-10)
This is a set of three differential equations for the rectangular coordinatesx; of the surface. Conversely, when the second of the Eqs. (2-6) takes theform (2-10), the curvilinear coordinates (u, v) on the surface are conjugate.The r;2, r12 may be any functions of u and v, since their expressions interms of the E, F, G and their derivatives of the corresponding surface aredetermined by Eq. (2-10) itself. We thus have found the theorem:
3-2] THE EQUATIONS OF GAUSS-WEINGARTEN 109
Three functionally independent solutions of the partial differential equa-tion
,p,,, - Acp. - Bcpro = 0, A = A(u, v), B = B(u, v) (2-11)
determine a surface on which the curves u = constant, v = constant are aconjugate set of curves.
When the curves v = constant are asymptotic, e = 0, and we find in asimilar way:
Three functionally independent solutions of the partial differential equation
,puu - Apu - Bpn = Of A = A(u, v), B = B(u, v) (2-12)
determine a surface on which the curves v = constant are asymptotic lines.These conditions are necessary and sufficient.
The importance of these theorems lies in the fact that the more generallinear partial differential equation of the second order,
Bpuo + Cp. + Dvu + EIpro = 0,
where A, ... E are functions of u and v, can always be transformed intoeither form (2-11) or form (2-12) by a real or imaginary substitution of theindependent variables. For further information we refer to Darboux'Surfaces If pp. 102-145, where many applications are made connecting thetheory of partial differential equations, and their theory of characteristics,with that of surfaces. We confine ourselves to a very simple special case,obtained by taking A = Of B = 0 in Eq. (2-11). Then we obtain
ccuv = 0, (2-13)
of which the general solution is p = fl(u) + f2(v), where fi and f2 are arbi-trary functions. Hence:
The surfaces
x; = U;(u) + V;(v), i = 1, 2, 3, (2-14)
where the U; and V; are arbitrary func-tions of u and v respectively, have a con-jugate set of parametric lines. Thesesurfaces are called translation surfaces(see Exercise 1, Section 2-11), sincethey can be obtained by moving thespace curve x = U(u) parallel to itselfin such a way that one point of thecurve moves along the space curvex = V(v). The curves U(u) and V(v) FIG. 3-2
110 THE FUNDAMENTAL EQUATIONS [CH. 3
may be interchanged and still yield the same surface. An example is acylinder obtained by moving a straight line 1 parallel to itself along acircle c (Fig. 3-2); it can also be obtained by moving the circle c parallel toitself along the line 1 (that is, moving the plane of circle c with c in it parallelto itself such that c continues to intersect 1). Straight lines and circlesform a conjugate set of curves.
3-3 The theorem of Gauss and the equations of Codazzi. The equa-tions of Gauss define the coordinates x; of a surface as functions of u and vby means of a set of differential equations. These equations are not inde-pendent, but certain compatibility conditions are satisfied. They areexpressed by the equations:
(xuu)ro = (xuv)u; (xuv)v = (xroro)u-
We have therefore for any surface:
12X- +fN),av (rilxu + ri,xv + eN) = au (ri2xu + 1,2
(3-1)
av (r12xu + r12xv + fN) = u (r22xu + 122x1 + gN).
The terms in xuu, xu,,, and xvv can again be expressed in terms of x,,, x,,, andN, by means of the Gauss equations, while the terms in N. and Nv can heexpressed in terms of xu and xv by means of the Weingarten equation.In this way we obtain two identities between the vectors x., xv, N, whichcan be satisfied only if the coefficients of xu, x, and N are identically zero.In this way we obtain six scalar equations. We shall study in detail thethree resulting from the first equation of (3-1) :
(a) coeff. of xu :49V
r + r,,,112 + r;, r212 + e EG - F'2a I I I 2 I fF eG
(h) coeff. of xv: av rI + r121r22 + e EG - F2 (3-2)
= aU 112 + r12r11 + r, 2r22 + f EG F
(c) coeff. of N: gri, + av = er12 +
Three other equations can be derived from the second equation in (3-1).The Eqs. (3-2) can be written in the form:
rl, + 1,122(a) F EG - F2 = -49
r112 -49V
3-3] THE THEOREM OF GAUSS 111
49(b) -E EG - F2 = au r, - av r,l + r12r1 - riir12 + r12r12 - ri1r22
(e) a - of= er12 + f(r12 - r'h) - grit. (3-3)
av au
The Eqs. (3-3a) and (3-3b) have in common that they contain thecoefficients of the second fundamental form only in the combinationeg - P. Moreover, this combination appears as numerator of the fraction(eg - f2)/(EG - F2), which is (see Section 2-7) the Gaussian curvature ofthe surface. Eqs. (3-3a) and (3-3b) both express the property that thisGaussian curvature depends only on E, F, G and their first and secondderivatives. We call an expression which depends only on E, F, G andits derivatives a bending invariant, and thus have found Gauss' theorem:
The Gaussian curvature of a surface is a bending invariant. (This is a" Theorema egregium," " a most excellent theorem," wrote Gauss.)
This same result is expressed, in different analytical form, by the firsttwo equations obtained from the second of Eqs. (3-1). The third equationgives us a result which has in common with the Eq. (3-3c) that it containsderivatives of coefficients of the second fundamental form. We write thisnew third equation together with (3-3c) :
ae aJ49V au = er12 + f(ri2 - r,,,,) - grit,
Of - = 6ra2 +f(r22 - r12) - grit.(3-4)
These two equations are known as the equations of Codazzi or of Mainardi-Codazzi.
These formulas are found in the paper by D. Codazzi (1824-1875) in his answerto a "contours" of the Paris Academy (1860, printed in the Mem. presentes aI'Academie 27, 1880); also in the Annali di Matem. 2, 1868-1869, pp. 101-119.They were, at that time, already published by G. Mainardi, Giornale IstitutoLombardo 9, 1856, pp. 385-398. Gauss, in his Disquisitiones has these formulasin principle, though not explicitly. Their fundamental importance was firstfully recognized by 0. Bonnet, Journal de l'Ecole Polytechnique 42 (1867), pp.31-151.
There also exist compatibility conditions for the Weingarten equations(2-9). However, these equations, obtained by expressing that Nvand by using the Gauss equations (2-6), can be shown to be equivalent tothe Codazzi equations (see Exercise 21, Section 4-2).
112 THE FUNDAMENTAL EQUATIONS [CH. 3
When we call the Gaussian curvature a bending invariant, we mean thatit is unchanged by such deformations of the surface (we always think of alimited region) which do not involve stretching, shrinking, or tearing.This bending leaves the distance between two points on the surface, meas-ured along a curve on the surface, unchanged, and also the angle of twotangent directions at a point. We can easily obtain an idea of this bendingby deforming a piece of paper without changing its elastic properties (weshould not wet it); a curve drawn on this piece of paper conserves its lengtheven if its shape is changed as a result of the bending of the paper.
When the curvilinear coordinate lines retain their position on the surfaceduring the bending and the measurement of the coordinates remains thesame, then the coefficients of the first fundamental form and all their deriva-tives with respect to the curvilinear coordinates also remain the same. Afunction containing E, F, G and their derivatives is therefore a bending invari-ant; such an invariant can also contain arbitrary functions P(u, v), ¢(u, v), .. .or the derivatives dv/du, d2v/due, .... Properties of surfaces expressibleby bending invariants are called intrinsic properties.
Both the expression of Gauss' theorem and Codazzi's formulas can bewritten in many forms. The formulation (3-3) of Gauss' theorem doesnot give an expression for the Gaussian curvature in which E, F, G takeequivalent positions. F. Brioschi, in 1852 (Opere I, p. 1) gave an expres-sion which is more satisfactory from this point of view. To obtain it weuse the determinant expressions for e, f, g (Section 2-5):
K EG - F2 (EG 1 F2)2 [(xuuxuxv)(xvuxuxv) - (xu.xuxv)2l,
X Y_ x - xI
(EG-F2)2
uu v uu u1
xu.xvv xu'xu
{Il xv'xuro xv'x,u
xuv'xuv xuv'xu(3-5)
xv'xuv xv'xu xv'xv
In the terms of the determinants we recognize E, F, G and the symbols[ij,k] of Eq. (2-4). We still need expressions for and X.,- xuv, butsince each of these expressions occurs with the same factor EG - F2, weneed only their difference. We can verify, by the same method as used forEq. (2-4), that
xuu xvv - xuv ' xuv = - I Evv + F. -
We thus obtain for K the expression
E - G FF - 2FE 0 1E 2G1 ( j
K
2 vv+ uv uu ivuu22i E FF - G
, u2
E F2E (3-6)172 2P0 v Z u v2-
( ) II 2Gv F G 2Gu F G
xuu'xv
xu'xv
xv'xv
3-3] THE THEOREM OF GAUSS 113
This takes a much simpler form in the case that the parametric lines areorthogonal. Then F = 0 and we can verify without much difficulty thatin this case
K EG Lau \ Eaau
u) + ( __ )J(3-7)=- -1 Fat a
Eqs. (3-5) through (3-7) are also called Gauss' equations. They allexpress the "Theorema egregium."
The Codazzi equations take a simple form when we do not only takeF = 0, but also f = 0, which means that we take the lines of curvature ascoordinate lines. Then (Sec. 2-6) we have for the principal normal curva-tures KI and K2:
KI = e/E, K2 = g/G,
and the Codazzi equations, now reduced to
e = 2Ev (E + G), gu = 21 2 G. (E + G),
take the form
(3-7a)
OKI 1 Eu aK2 1 Gu
49V 2 E (K2 - KI), au 2 G (KI - K2). (3-8)
EXERCISES
1. Derive the second of the Codazzi Eq. (3-4) from the second Eq. (3-1).
2. Derive Rodrigues' formula for the lines of curvature from the Weingartenequations.
3. Compute the Christoffel symbols for polar coordinates in the plane.
4. Show that when the surface is given by z = f(x, y) (see Exercise 2, Sec. 2-8),
r1 = pr riz = Ps rIz = pt ,+p2+q2 ] +p2+q2 +p'+q2r2 qr r2 - -- 4s r2 - qt K = rt - s2
II = 1+p2+q21 12+p2+92'
22 1+p2+q2' (1+p2+g2)25. The locus of the centers of the chords of a space curve C is a translation
surface on which C is an asymptotic curve.
6. Show that the paraboloid z = axe + by' is a translation surface. What arethe curves u = constant, v = constant (Eq. 2-14) in this case?
7. When D2 = EG - F2, show that
au In D = r,1, + riz, a In D = r;2 + rz2
114 THE FUNDAMENTAL EQUATIONS [CH. 3
8. When w is the angle between the coordinate curves, show that
aw D 2 D I aw D D I
au E rI1 - r12, TV = - E 12 -G
r22.
9. Verify the simplified equations of Gauss and of Codazzi (3-7) and (3-8).
10. Verify the following expressions for the Gaussian curvature:
K 1
E F G _ 1 1 a aFFF
GGv
2./EG-F2 l au 1/EG--F2 av 11EG-F2
(G. Frobenius, see Blaschke I, p. 117.)
1 a D 2 a D 2 l a D, a D l 1K = - ( rl) - - r12) _ - - 22) - (-r12)JD av E au E au G av
(J. Liouville, Journal de MathEm. 16 (1851), p. 130.)
11. Show that the direction cosines Ni of the unit surface normal vector satisfyequations of the form
<p,,,, + A,p + &p + C,p = 0,cp,,,, + A,p + B1',o + CI<p = 0, A ... C, functions of u and v.
12. Show that K is invariant under a change of curvilinear coordinates on thesurface.
13. Tensor notation. We introduce the notation
9n =E, 912=F, 922=G,= G = - F E
911 EG - F2' 912 EG - F2' 94 EG - F2Show that
9i;g'k = Sj, i, j, k = 1, 2.
Here we sum on the index which is repeated (here i). The Kronecker symbol Sj is 1when k = j, and zero when k 7-1 j; for example S; = 1, 62 = 0.
In the next exercises we use the notation of Exercise 13; the indices i, j, k, .. .run over 1 and 2. We call u = u1, v = U2-
14. Show that
[ij, k] = 1 (a9,k + 09,k a &),2 8u; au; auk
a [jk,11,[jk, 11 = 9,,rjk.
15. Show that Gauss' equations (2-6) can be written
x;; = r jak + h;,N,
x, = x,,, x_ = x,., etc., and e = hu, ,f = h2, = h12, 9 = h22.
3-41 CURVILINEAR COORDINATES IN SPACE 115
16. Show that Weingarten's equations (2-9) can be written:
Ni = - (gklhi,)xk.
(Here the summing is both on 1 and on k.)17. Show that Codazzi's equations can be written:
aui - rihk, - - r,hki-18. By introducing the symbol of Riemann:
IN, ij } = Ruk = au rtki3u;
rjk r r,1y - rjkrm (sum on m),
show that the Eqs. (3-3) can be written as
gn2(eg -f2) = R121 = {ln, 12}.19. By introducing the symbol
(kl, ij) = Ri;kt = Rtjkg.,,
show that the Gaussian curvature K can be written:
R1212 _ (12, 12).K EG-F2 EG-F2
20. Show that when we substitute E, F, G fore, f, g into the Codazzi equations,these equations are identically satisfied.
3-4 Curvilinear coordinates in space. The theory developed in the twoprevious sections can be considered as an aspect of a more general theory,dealing with families of surfaces in space. We shall give an outline of thistheory for the special case that our given surface belongs to a triply or-thogonal system of surfaces, which for one surface does not involve a re-striction of generality. In this case we shall take the parametric linesalong the curves of intersection of the surfaces. This means not only thatthe parametric lines are orthogonal, but also, according to Dupin's theorem,that they are lines of curvature. Using the notation of Section 2-11, weintroduce the curvilinear coordinates (u, v, w) by means of the equation
x = x(u, v, w),
which establishes three families of oo I surfaces
u(x, y) z) = 01, v(x, y, z) = C2, w(x, y, z) = C3,
or for short:ui = C1, i = 1, 2, 3. (4-1)
We also know (Section 2-11) that the relations exist:
XU xv = x x,, = x, xu = 0, (4-2a)xuv X. = xvw xu = xwu xv = 0, (4-2b)
116 THE FUNDAMENTAL EQUATIONS [CH. 3
of which the last three express Dupin's theorem. We can consider the sur-face of the previous sections as one of the surfaces w = u3 = constant.For these surfaces f = 0, F = 0.
At each point we have a set of three linearly independent orthogonalvectors x,,, x,,, x, , and also the set of the gradient vectors vu, vv, vw, whichhave the same directions as the x,,, x,,, x,,, respectively.
The gradient vector field of a function of position f(x, y, z) is defined as
Vf = grad f = e, f + ezfy + e3f:.
Since df = dx of this vector field is perpendicular to the surface f = constant.The length of of is df/dn, where do is the element of length in the directionnormal to f = constant.
The ds2 of space, expressed in the curvilinear coordinates u, v, w, takesthe form (Section 2-11):
ds2 = (xu dug + (xv . dv2 + (x. x.) dw2,
for which it is customary to write either one of the following notations:
ds2 = gii due + 922 dv2 + g33 dw2 = Hi du2 + H2 dv2 + H3 dw2. (4-3)
The g-notation is called the tensor notation (see Exercise 13, Section 3-3),the H-notation is called Larne's notation for this ds2. We take the Hipositive, H, means H;Hi = (Hi)2.
If we now introduce three unit vectors ul, u2, u3 in the direction of x,,, x,x,, respectively (in the sense of increasing u, v, w), then since
IVul = du/dn = Hl-', 1vui4 = Hs', i = 1, 2, 3,
the following relations exist at each point P of space:
ui = hixu = Hi vu, u2 = h2x = H2 vv, u3 = h3x. = H3 Vw, (4-4)
where the hi are defined byhi=Ht', i=1,2,3.
The trihedrons (x,,, x,,, xw), (ul, u2, u3), and (Vu, Vv, Vw) depend on thethree parameters (u, v, w), and their motion in space can be used to explorethe geometry of space very much in the way we have used the moving tri-hedron depending on one parameter for the investigation of space curves.This motion can be expressed in terms of the derivatives of either one ofthese trihedrons. Following the method of Section 3-2, we write in analogyto the Gauss equations:
x,, = r;,x ri,x + r xw (and 2 similar equations),xuv = r,2xu + ri2xv + r12xw (and 2 similar equations). (4 0)
CURVILINEAR COORDINATES IN SPACE 117
Eqs. (4-2b) show that all rjr (i, j, k = 1, 2, 3, all different) vanish. Forthe other r we find by a method similar to that used in Section 3-2:
l, = h, au H, H, auh, = dH, (= Eu/2E for w = c),
r12 = h, H, h,H2 dH, (= Ev/2E for w = c), (4-6)
r =-H,h2 avH, =-H,h2 d 11 for w = c),
and in general:
t dH; t dH; dH;9
, ri, = - Hhs dsru =ds: ,
vi= h'H' ds ,
where ds; = H; du; represents the ds in the direction normal to u; = con-stant. Hence
xuu =dH, xu - h2H, dH, x - h3H, dH, x., etc.ds, ds2 ds3
xuv = h,H2 dHl xu + h2H1 dH2 x,,, etc.(4-7)
The corresponding equations for the unit vectors are obtained from equa-tions such as auu, = au(h,xu). We obtain:
dH, dH,auu, = - ds2 U2 - ds u3,s
avu, = dH2 u2,
8.u, = 3 u3, and 2 other sets of 3 equations.ds,
(4-8)
Since u; is the unit normal vector of surface u; = const, the equationsfor a;u,, i p j, express Rodrigues' theorem. The equation for h, auu, ex-presses the decomposition of the curvature vector of the lines of curvaturein its tangential and normal component.
When two arbitrary functions E(u, v), G(u, v) of two variables are given,then it is always possible to find surfaces for which ds2 = E due + G dv2 isthe first fundamental form (see Section 5-3). This is not the case whenthree arbitrary functions v, w), g22(u, v, w), g33(u, v, w) are given, andit is required that they be used as coefficients of a ds2 of space of the form
* Contrary to Exercises 13-19, Section 3-3, we do not here sum on indices whichoccur more than once.
118 THE FUNDAMENTAL EQUATIONS [CH. 3
(4-3). It can be shown that the g;; must satisfy six differential relations,the compatibility equations. We can obtain them, in a way similar to thatused in Section 3-3, by evaluating equations of the type xuuv = xuvu Weobtain them in a simpler way by performing a similar operation on the unitvectors in Eq. (4-8) :
We obtainau av auu1, etc.
au L d 2u2]
=a° [dH` u2 dHl u31,
from which, by the use of Eq. (4-8), we obtain three scalar equations ascoefficients of u1, u2, u3 respectively. The coefficient of ul is identicallyzero. The other coefficients are
a dH2 a dHl dHl dH2+ (4-9)
au dsl av ds2 ds3 ds3a dH, dH, dH2
(4-10)av ds3 ds2 ds3
There are three equations of the form (4-9), and six of the form (4-10).As we said before, it can be shown that six of them are independent, but weshall not prove this here (see Exercise 6, this section). The Eq. (4-10) canalso be written in the form
a2H, d a d 49H2 a H H HH a) + (h (4-10a)+ h)(a )(o
- - ( )av aw9, , 1 l 2 .3 v 3 W2
ds3
s3 ft WT
The equations of the type (4-9) and (4-10) are equivalent with theGauss-Codazzi equations for the surfaces u; = constant. To show it forthe surfaces w = constant we notice that by comparison of Eqs. (4-5) and(2-6) we can express e and g in terms of r :
H 3 Hl
e = 3r =- , dH 2 ds3' (4-11)
g = H3r23 2 = - H2 drs32 2I d922d
We can write these equations also in the form e = --- dE/ds3 andg = -I dG/ds3, provided we understand that the E and G are here func-tions of three variables, in which w = constant is substituted after differen-tiation. They are interesting expressions for the coefficients of the secondfundamental form in terms of those of the first.
Eq. (4-9) becomes, after substituting for dHl/ds3 and dH2/ds3 theirexpressions (4-11) in terms of e and g,
3-41 CURVILINEAR COORDINATES IN SPACE 119
au (hi2ri2) -a49
v(ha12r,1) = hih2eg,
which is equivalent with the Gauss equation (3-3b), in which f = F = 0.Similarly, Eq. (4-10) becomes
(-hie) _ (h2Hir12)(-h2g),49V
which is equivalent with the first of the Codazzi equations (3-4).
The equations of this section were derived by Gabriel Lame (1795-1870), whoin and after 1837 introduced curvilinear coordinates for the solution of the dif-ferential equations of heat and elasticity. He collected his results in the Legonssur les coordonnees curvilignes (1859). He introduced the concepts of isothermicsystems (see Section 5-2) and differential parameters (see Section 4-8, Exer-cise 11).
His work was used by G. Darboux for his investigations of triply orthogonalsystems (Section 2-11); he generalized it to systems which intersect in conjugatesystems (Legons sur les systemes orthogonaux, p. 361). Formulas for generalsystems of curvilinear coordinates in space can be found in papers by the AbbeAoust (e.g. Annali di matem. 6, 1864, pp. 65-87) who generalized Lame's work.This work has since 1887 been superseded by that of G. Ricci-Curbastro, whocreated in the tensor calculus a new instrument to deal with questions pertain-ing to general curvilinear coordinates, not only for the geometry of three, butalso for n > 3 dimensions.
EXERCISES
1. The sets of mutually orthogonal vectors a;, b;, (i = 1, 2, 3), are called re-ciprocal if a; b, = 0, i s j; ai bi = a2 b2 = as ba = 1. Show that whenx = x(u, v, w) introduces a set of curvilinear coordinates (not necessarily or-thogonal), the sets xv, s,,, xw and Vu, Vv, Vw form a reciprocal set.
2. Show that, in the notation of Section 3-4,
Hiui + H2uy Haw = 1, Iliuzu + H2v.v + H3w.w = 0, cycl.hix2 - Q'xfl + h2x2w = 1, h?x y h3xwyw = 0, cycl.
3. Show that h,x. = h2ha(y zw - cycl.,Hiu. = H2!I3(vVw. - v:wv), cycl.
4. Find the rotation vector p which expresses the motion of the system of unitvectors (u,, u2, US) of Section 3-4 in the direction of ui.
5. Show that Eqs. (4-7) and (4-8) are equivalent with
au Vu = - rl, VU - r12 Vv - r13 Vw, and 2 other equations,
av Vu = - r12 Vu - r12 Vv, and 5 other equations.
120 THE FUNDAMENTAL EQUATIONS [CH.3
6. Show that Eqs. (4-9) and (4-10) can be written (see Exercise 18, Section 3-3) :
Rijk = 0, where i, j, k, 1 = 1, 2, 3.
(It can be proved that 6 of the Ruk are independent, see L. P. Eisenhart, Riemanniangeometry, Princeton, 1926, p. 21.)
3-5 Some applications of the Gauss and the Codazzi equations.(1) An interesting illustration of Gauss' theorem is presented by the cate-noid,
x = u cos v, y = u sin v, z = c cosh-' - = f(u), (5-1)
and the right helicoid,
x = ul cos v1, y = ul sin vi, z = av1 = fl(v). (5-2)
The first fundamental form of the catenoid is (Section 2-8):
z
ds2 =U2
ac2
du2 + u2 dv2,
that of the right helicoid,
dsi = du? + (ui + a2) ME
If we now write
a = c, v = vl, u = u- -+a2 or u, = N1u2 -- a2 (5-3)then
dsi = ds2,
and we see that for 0 < v < 27r and -a <, u < + a we have established aone-to-one correspondence between the points on both surfaces such that atcorresponding points the first fundamental forms are equal.The total curvature of the catenoid is (Section 2-8) :
ff=
-ezK = u(1 + f,2)2 u , since
c
-%/u_ C2
The total curvature of the right helicoid is:
- (f 1) 27
a2K1 __[U11 + (f )212 (ul + a2)2'
and we can readily verify that the substitution (5-3) also leads to K = K1.This shows that a correspondence can be established between the points of acatenoid and of a right helicoid such that at corresponding points the E, F, Gand therefore the Gaussian curvature are the same.
3-5] APPLICATIONS OF GAUSS AND CODAZZI EQUATIONS 121
(a)
(d)
(b)
(e)
FIG. 3-3
(c)
(f)
This is a one-to-one correspondence so long as -a <, u < a, and0 <, v < 27r, hence for one full turn of the helix.
It can be shown (Section 5-4) that one surface can actually pass into theother by a continuous bending. In Fig. 3-3 we show six different stages inthis deformation. This can be demonstrated with a flexible piece of brassapplied to a plaster model of a catenoid and bent so as to be applied to amodel of a right helicoid (Fig. 3-4, models by Brill, Darmstadt). Eqs.(5-2) and (5-3) show that the circles on the catenoid pass into the helicesof the helicoid, and the catenaries of the catenoid into the straight lines of
122 THE FUNDAMENTAL EQUATIONS
FIG. 3-4
[CH. 3
the helicoid (that is, for the parts of the surfaces for which -a < u <+a,0 <, v < 27r, hence for one full turn of the helix).
(2) As an illustration of the Codazzi equations we prove the theorem:The sphere is the only surface all points of which are umbilics.The normal curvatures at an umbilic are all equal, and we can write
K = KI = K2 (Section 2-7). The Codazzi Eqs. (3-8) then show that aK/au = 0and aK/av = 0, hence that K is a constant. We also find from Eq. (2-9) thatNu = - Kxu, N = - Kx,,; here we use the fact that e/E = f/F = g/G = K.If we now associate with a point (x) on the surface a point
y = x + RN, R= K ', constant,
then yu = yv = 0, and all vectors x + RN going out from the points on thesurface meet in the fixed point (y). Hence we conclude that
(x - y) (x - y) = R2,which is the equation of a sphere. When K = 0 we find that N is a constantvector, and dx N = 0 can be integrated into x N = constant, the equa-tion of a plane (which can be considered as a sphere of infinite radius).
(3) This result is related to the following theorem due to D. Hilbert:In a region R of a surface of constant positive Gaussian curvature without
umbilics the principal curvatures take their extreme values at the boundary.
3-51 APPLICATIONS OF GAUSS AND CODAZZI EQUATIONS 123
Since the surface is not a sphere K, K2, so that we take K, = K,/K2.Let K, assume a maximum at a point P inside the region R. Then K2 as-sumes a minimum at P. Such a point is not an umbilic and is a regularpoint of the surface, so that the region near P is covered simply and withoutgaps by lines of curvature, which we can take as coordinate lines. Henceat P, aK,/ av = 0 and aK2/au = 0, and hence, according to the Codazzi
, 0;Eqs. (3-8), E, = G = 0. Moreover a2K,/av2 < 0 and a2K2/au2 >
hence, differentiating Eqs. (3-8), we obtain, since K, > K2, that Ev, >, 0,G,,,, >, 0. But K, at this point P, can be written, according to Eq. (3-6):
K 2EG (E"" + G,.,.),
which has a right-hand member < 0 and a left-hand member > 0 (E > 0,G > 0). This is impossible, so that nowhere in the interior of the regioncan we have aK,/av = aK2/au = 0. The normal curvatures must mono-tonically increase or decrease along the lines of curvature.
The proof requires the existence of a sufficient number of continuousderivatives, since the only criterion used for an extremum of K, and K2 is thevanishing of their first derivatives.
This result leads to theTheorem of Liebmann. The only closed surface of constant positive curva-
ture (without singularities) is the sphere.Indeed, when a surface of constant curvature is not closed and is not a
sphere, the maximum of all larger principal curvatures lies on the boundary.But a closed surface has no boundary, so that the larger principal curva-tures must be equal at all points of the surface. This, however, means thatboth K, and K2 must be constant on the surface, which can only happen(Eq. (3-8)) if all points are umbilics. The only possibility is therefore thesphere.
This proof again requires that at all points on the surface a sufficientnumber of continuous derivatives of x with respect to u and v exist. In-deed, when we take two congruent spherical caps and place them together ina symmetrical position along their boundary circle, we obtain a closed sur-face of constant positive curvature which is not a sphere, but which has aline of singular points.
This is our first example of surface theory in the large. We can express thetheorem of Liebmann by saying that a sphere cannot be bent. A spherewith a hole in it can be bent, and we can also "crack" a sphere.
This theorem was suggested by F. Minding, Crelle's Journal fur Mathem.18, 1838, p. 368; the first proof dates from H. Liebmann, Gottinger Nachrichten,1899, pp. 44-55. The present proof follows W. Blaschke, Differentialgeometrie
124 THE FUNDAMENTAL EQUATIONS [CH. 3
I, sect. 91. Hilbert's proofs are in Trans. Amer. Math. Soc. 2, 1901, pp. 97-99,also Grundlagen der Geometrie, Anhang V.
This theorem of Liebmann is a special case of the general theorem that aclosed and convex surface cannot be bent, which theorem seems to go back toLagrange (1812). See A. Cauchy, Comptes Rendus 21, 1845, p. 564. A proof ofthis theorem was also given by Liebmann. It can also be proved that convexpolyhedrons cannot be bent.
3-6 The fundamental theorem of surface theory. We have seen thatthe coefficients of the two quadratic forms
I = E due + 2F du dv + G dv2, EG - F2 9,1 0, (6-1)II = e due + 2f du dv + g dv2, (6-2)
satisfy the Gauss-Codazzi equations. We owe to 0. Bonnet the proof ofthe converse theorem, called the
FUNDAMENTAL THEOREM. If E, F, G and e, f, g are given as functions ofu and v, sufficiently differentiable, which satisfy the Gauss-Codazzi equa-tions (3-4) and (3-6), while EG - F2 0 0, then there exists a surfacewhich admits as its first and second fundamental forms I = E due +2F du dv + G dv2 and II = e due + 2f du dv + g dv2 respectively. Thissurface is uniquely determined except for its position in space. For realsurfaces with real curvilinear coordinates (u, v) we need EG - F2 > 0,E> 0,G> 0.
The demonstration consists in showing that the Gauss-Weingartenequations (equivalent to fifteen scalar equations):
xuu = riix,. + ri1xo + eN, (a)xuo = rl2xlu + ri2xv + fN, (b)X" = r22xu + rs2xti + gN, (c) (6-3)
(EG - F2)Nu = (fF - eG)xu + (eF - (d)(EG - P)N. = (gF - fG)xu + (fF - gE)xo, (e)
under the additional conditions (validity at one point suffices)
NN=1, xuN=0, xoN=0, xxu=E, xuxo=F,xoxo = G, xuuN = e, xuoN =f, x,N = g. (6-4)
determine the twelve scalar functions of u and v given by x, xu, x,, Nwith just such a number of integration constants that the surface x = x(u, v)is determined but for its position in space. Instead of a general demonstra-tion of this theorem, which requires some knowledge of the integrationtheory of mixed systems of partial differential equations, we shall illustrate
3-6] FUNDAMENTAL THEOREM OF SURFACE THEORY 125
the nature of this demonstration by means of a simple example as follows:Given the differential forms
I = due + cos2 u dv2,II = due + cos' u dv2,
find the surface of which I and II are the first and second fundamental forms.Since E = 1, F = 0, G = cos' u; e = 1, f = 0, g = cos' u, we find for
the Christoffel symbols
r1, = r22 = r12 = r22 = 0, r12 = - tan u, r22 = sin u cos u,
which satisfy the Gauss-Codazzi equations (3-3), (3-4), as direct si.bstitu-tion shows.
The Gauss-Weingarten equations (6-3) are in this case
x1515 = N, (a)x,,, =-tan ux,j (b)X,, = sin u cos ux15 + cos2 uN, (c) (6-5)N. = x15, (d)N. =-x,,. (e)
We have to solve these 15 equations for the x;. Eqs. (6-5a) and (6-5d)give by elimination of N :
or
x,51515 + x, = 0,
x = a(v) cos u + b(v) sin u + c(v).
Then Eq. (6-5b) gives
b' cos u = -b' sine u sec u - c' tan u, (b' = db/dv, c' = do/dv)or
b' = c' = 0.
Hence b and c are constant vectors, and
x = a(v) cos u + b sin u + c.
From this equation and from Eqs. (6-5c), (6-5b), and (6-5e), we obtain:
a"' cos u = sin u cos ux.. + cos2 uN, =-a' cos u,or
a,,, + a' = 0 ,
a = p cos v + q sin v + r, p, q, r constant vectors.
The solution of Eq. (6-3) is therefore [r = 0, according to Eq. 6-5(c)]
x=p cosvcosu+gsinvcosu+bsinu+C.
126 THE FUNDAMENTAL EQUATIONS [CH. 3
We must now select p, q, b, c so as to satisfy the additional conditions(6-4). Since
sinvcosu+gcosvcosu,
we find that for all u and v:
(p - p) sin2vcos2u-2(p q) sinvcosvcos2u+ (q q) cost v cost u,
or
Similarly we derive from x,, x = 0:
Then we obtain from x. x = 1:
b form a system of mutually orthogonal unit vectors.The other equations (6-4) are now automatically satisfied. Denoting thevectors p, q, b by e1, e2, e3 respectively, we obtain as the general solutionof our set of equations:
x = e1cosvcosu + e2 sinvcosu+ e3 sin u + c,
which is the equation of the unit sphere. By the choice of el, e2, e3, c wecan place this sphere in any position of space, selecting any orthogonalsystem of meridians and parallels for our u- and v-coordinates.
The first to give a proof of the fundamental theorem was 0. Bonnet, JournalEcole Polytechnique, cah. 42 (1867); there are also proofs in Bianchi's Lezioni,section 68, and in Eisenhart's Differential geometry, pp. 158-159. The reader maycompare this fundamental theorem with that for space curves, in which we con-sider a system of ordinary differential equations, which does not need condi-tions of the Gauss-Codazzi type for complete integrability. In the case ofcurves, we have given an actual method to carry out the integration (Section1-10), which leads to a Riccati equation. The analogous problem in the caseof surfaces can again be reduced to a Riccati equation. A complete discussionof this case is in G. Scheffers, Anwendung II, pp. 393-414. The general theoryof mixed systems of partial differential equations on which the proof of thefundamental theorem is based can be studied in T. Levi Civita, The absolutedifferential calculus, Chap. 2, Sec. 8.
CHAPTER 4
GEOMETRY ON A SURFACE
4-1 Geodesic (tangential) curvature. We are now able to continue ourstudy of the intrinsic properties of the surface. As in Section 2-5 we startwith the curvature vector of a curve C on the surface. This curvaturevector
dt/ds = Kn = k
of the curve at a point P with the tangent Ik,% AKn=k
direction t lies in the plane through Pperpendicular to t. This plane also con-tains the surface normal, in which the unitvector N is laid (Fig. 4-1). The projectionof the curvature vector k on the surfacenormal is, according to Meusnier's theorem(Section 2-5), the curvature vector of thenormal section in direction t; we havecalled this vector ku. We shall now studythe projection of k on the tangent plane,which is called the vector of tangential
FIG. 4-1
curvature and which we have denoted by k,. Hence we have the relation,already expressed in Eq. (5-1), Chapter 2:
k = (1-1)
in words:The curvature vector is the sum of the normal and the tangential curvature
vectors.The tangential curvature vector is also called the geodesic curvature
vector. To find an expression for k, we introduce a unit vector u per-pendicular to t in the tangent plane in such a way that the sense t --+ u isthe same as that of xu -+ x,,. We then introduce the tangential curvature orgeodesic curvature K, by means of the equation
k, = K,u. (1-2)
Where Meusnier's theorem gives us IK cos'pj for the magnitude K. ofthe normal curvature vector ku, we find JK,I = IK sin (pi as the magnitude ofthe geodesic curvature vector. We shall now show that this vector, con-
127
128 GEOMETRY ON A SURFACE [CH. 4
trary to k,,, depends only on E, F, G and their derivatives (as well as on thefunction ik(u,v) = 0 defining the curve C), and is therefore a bendinginvariant.
From Eq. (1-1) follows, since u k = u 0:
Kg = u (dt/ds) = u V.
Hence, since t -* u --> N has the positive (right-handed) sense:
Kp = (N X t) t' = (tt'N). (1-3)
The unit vector t satisfies the equation
t = xuu' + u' = du/ds, v' = dv/ds;hence
so thatt' = Xuu(u')2 + xvv(v')2 + Xuu" +
x xuu)(u')3 + (2xu x xu + Xv x xuu)u'2v'
+ (xu X x,,,, + 2x x x x,»,)(v')3] N+ (xu x x.n) N(u'v" - u"v'). (1-5)
The coefficients of (u')3, (U')2V', etc. are all functions of E, F, G and theirfirst derivatives. Indeed:
(xu x xuu) N = (x,, x xuu) (xu xE
(xu xu) (xuu xro) - (xu xuu) (xu x9)
-VEG _-F2_ E[l1, 2] - F[11, 1]
Er11 E.Similarly:
(xu x xuv) N = r12 E, etc.(xu x x,,) N = VE .
Therefore, according to Eqs. (1-3) and (1-5):
K = [r 21(u')3 + (2r12 - rl,)(u')2v' + (r22 - 2r,2)u'(v')2 - r22(y')3+ u'v" - u"v']V'EG - F2, (1-6)
where the square root is taken positive, and this equation shows that K,depends on only E, F, G, their first derivatives, and on u', u", v', v":
The tangential curvature of a curve is a bending invariant.Since kg, lies in the tangent plane and is perpendicular to t, it is also a
bending invariant.
4-1] GEODESIC CURVATURE 129
When we apply Eq. (1-6) to a plane curve, taking u = x, v = y, E = 1,F = 0, G = 1, then we obtain
Kg = - u'v" - u"v', (1-7)
which means that Kg = dcp/ds. Our choice of the sign of Kp therefore makesKp the ordinary curvature of a plane curve (p. 15). The tangential curva-ture vector itself is unaffected by the choice of sign of K0. It depends onlyon the geometrical structure of the curve. Formula (1-1) can now bewritten (comp. Chapter 2, Eq. 5-2):
k= Kpu. (1-8)
When we construct the cylinderwhich projects the curve C on thetangent plane at P, then the generat-ing lines of this cylinder have thedirection of N, and the normal to thecylinder at P has the direction of u(Fig. 4-2). The geodesic curvature
FIG. 4-2
vector is therefore the normal curvature vector of C as curve on the pro-jecting cylinder, hence the curvature vector of the normal section of thecylinder in the direction of t. This normal section is the projection C' ofC on the tangent plane. This means:
The tangential curvature vector of a curve at P is the (ordinary) curvature
vector of the projection on the tangent plane of the surface at P.
EXAMPLE: The great circles on a sphere have geodesic curvature zero, sincethey project on the tangent plane at each of their points as straight lines.
FIG. 4-3
The small circle of radius r on a spherewith radius a projects on the tangentplane at one of its points as an ellipse ofsemi-axes A = r and B = rV a2- r2/a. Thegeodesic curvature of such a small circle istherefore constant and equal to the curva-ture of this ellipse at the vertex of theminor axis, which is equal to B/A2 =
a2 - r2/ar (Fig. 4-3). This is equal tothe curvature of the circle obtained bydeveloping the small circle on the plane asa curve on the cone which is tangent tothe sphere along the small circle (seeExercise 2, Section 4-8).
130 GEOMETRY ON A SURFACE [CH. 4
The geodesic curvature was already known to Gauss. The first publicationon this subject was by F. Minding, Crelle's Journal fur Mathem. 5, 1830, p. 297,who in the next issue (Vol. 6, 1830, p. 159) showed its character as a bendinginvariant. The name geodesic curvature is due to 0. Bonnet, Journal EcolePolytechn. 19, 1848, p. 43. Ferdinand Minding (1806-1885), whose name weshall repeatedly meet, was a professor at the German university of Dorpat (nowTartu, Estonia).
We present here for further use the expressions for the geodesic curvatureof the parametric lines:
BEG - Fz(KG)-const = r21(u')3 E = r2 EKE
since u'
V'EG - F2a iB G - F" _ -r-r( ) (v) Eu-const = 22 zKo GVG(1-9)
In particular, when the parametric lines are orthogonal, we haver2 r22 = - 2TU/E; and s1 and s2 are taken along the respective11
parametric lines,
(K0)1 = (Ka)v=covet = -1 Ev = - 1 a In-\/E _ - d
In,/E-,2 EvG vG av dsz (1-10)
(Kg) 2 = (Kg)u_conet =GG-VE- 1 8lna _+ Sl1n G.
For polar coordinates in the plane, where E = 1, G = r2 = u2, we find forthe circles r = const, Kp = +r 1, which illustrates again the choice of thesign.
Let us now introduce the unit vectors it and i2 along the orthogonalparametric lines. Then, according to Eq. (1-8):
dil(K,)1 = i2 - )2
A= -11 '(K'
(1-11),ds1' O
ds2
where si and 82 are taken along the respective parametric lines. We nowconsider a curve C through P, x = x(s), making an angle 0 with the curvev = constant. Then its unit tangent vector t satisfies the equation:
t = it cos 0 + i2 sin 0.
Differentiating t along C and using the formula
dLx = aia du aia A dL dLds au ds + 49v ds
_dsl
cos 0 + ds2 sin 6 (a = 1, 2),
we obtain for the curvature vector of C:
1-2] GEODESICS 131
dt _ di, di, di2 di2 dods ds1 cost
B +d82
Cos B sin B +dsl-cos B sin B +
d82sine B + u ds'
where, according to the definition (comp. Eq. (1-2)),
u = -ii sin B + i2 cos B. (1-12)
Hence, by virtue of Eqs. (1-2) and (1-12) we obtain for the geodesiccurvature K° of C:
Kds ds, d82 ds,
+12 d$2
Cos2 0 sin 0,
or, because of Eq. (1-11):
Kpds
+ (K°)1 Cos 0 sin2 0 + (K°)2 sin3 0 + (K°)1 cos3 0 + (K°)2 COS2 0 sin 0,
or finally
K° =ds
+ (K°)1 COS 0 + (K°)2 Sin 0. (1-13)
This formula is known as Liouville's formula for the geodesic curvature.It can be found in a note to Liouville's edition of Monge's Applications(1850).
4-2 Geodesics. Geodesic lines are sometimes defined as lines of short-est distance between points on a surface. This is not always a satisfactorydefinition, and we shall therefore define geodesics in a different way, post-poning the discussion of the minimum property. Our definition of geodesicsis that they are curves of zero geodesic curvature.
This means that straight lines on the surface are geodesics, since in thiscase the curvature vector k vanishes. For all curved geodesics our defini-tion means that the curvature vector at each point coincides with thenormal curvature vector. In other words, the osculating planes of a curvedgeodesic contain the surface normal. We can therefore state the followingproperty :
All straight lines on a surface are geodesics. Along all curved geodesics theprincipal normal coincides with the surface normal.
Along (curved) asymptotic lines osculating planes and tangent planescoincide, along (curved) geodesics they are normal. Through a point of anondevelopable surface pass two asymptotic lines (real or imaginary).Through a point of a surface passes a geodesic in every tangent direction.
132 GEOMETRY ON A SURFACE [CH.4
This last property becomes plausible when we realize that we can findat a point on the surface in every direction a curve of which the osculatingplane passes through the surface normal. It can be shown analytically byconsidering the equation of the geodesic lines. We obtain this equation bysimply writing K, = 0 in Eq. (1-6) and obtain
u'v" - u"v' rll(u')3 - (2r12 - ril)(u')2v' + (2P12 - r22)u'(v')2+ r22(v')3. (2-1)
This equation is derived from Eq. (1-3), so that the accents indicate dif-ferentiation with respect to the arc length. However, since
du d2v dv d2u du d2v dv d2u dt 3ds ds2 - ds ds2 - [dt dt2 - dt ate ds '
this equation (2-1) is still correct when the accents denote differentiation withrespect to any parameter.
It is often convenient to express the geodesics in another way. This canbe accomplished by noting that along these lines N has the direction of ± n.Hence,
or (differentiation is again with respect to s) :
which equations (compare with Eq. (1-4)) are equivalent to
(xuu x.) (u')2 + 2(xu, xu)u,v' + (xvro xu) (v')2 + Eu" + Fv" = 0,(xuu x,) (u')2 + 2(xu, ' x,)u'v' + (xo xti) (v')2 + Fu" + Gv" = 0.
Eliminating v" from these equations and also u", substituting xuu xu =[11, 1], and then introducing the Christoffel symbols from Section 3-2, weobtain the equation of the geodesics in the form:
dlu du AIVds2
+ rii (ds)2 + 2r12 ds ds + r22 Cds)2 = 0,
d2V
ds2+ rI
(d )2 + 2ri2 ds ds + r22 (as)2 = 0.
(2-2)
The history of geodesic lines begins with John Bernoulli's solution of theproblem of the shortest distance between two points on a convex surface (1697-1698). His answer was that the osculating plane ("the plane passing throughthree points `quolibet proxima "') must always be perpendicular to the tangentplane. For the further history see P. Stackel, Bemerkungen zur Geschichte dergeodatischen Linien, Berichte sachs. Akad. Wiss., Leipzig, 45, 1893, pp. 444-467.
4-2] GEODESICS 133
The name "geodesic line" in its present meaning is, according to Stackel, dueto J. Liouville, Journal de mat hem. 9, 1844, p. 401; the equation of the geodesicswas first obtained by Euler in his article De linea brevissima in superficie qua-cumque duo quaelibet puncta jungente, Comment. Acad. Petropol. 3 (ad annum1728), 1732; Euler's equation refers to a surface given by F(x, y, z) = 0 (seeExercise 22, this section).
It seems at first strange that Eqs. (2-2) give two conditions, whereasKp = 0 is only one condition. However, Eqs. (2-2) also express only onecondition, since they are related by the relation ds2 = E due + 2F du dv +G dv2. When we eliminate ds from (2-2) we obtain again one equation forthe geodesics, now in the form
dU22v = r22 (du)3 + (2r12 - raz)(du)2 + (ri, - 2ra2) du - r21. (2-3a)d
This equation can immediately be found from Eq. (2-1), taking u as pa-rameter. It is of the form:
v" = A(v')3 + B(v')2 + Cv' + D, (2-3b)
where A, B, C, D are functions of u and v.From Eqs. (2-3a, b) we see that when at a point P(u, v) a direction
dv/du is given, d2v/due is determined, that is, the way in which the curveof given direction is continued. More precisely, using the existencetheorem for the solutions of ordinary differential equations of the secondorder:
Through every point of the surface passes a geodesic in every direction.A geodesic is uniquely determined by an initial point and tangent at that
point.
The existence theorem states that a differential equation
d&v/due = f(u, v, dv/du),
where f is a single-valued and continuous function of its independent variablesinside a certain given interval (with a Lipschitz condition satisfied in our case),has inside this interval a unique continuous solution for which dv/du takes agiven value (dv/du)o at a point P(uo, vo).*
EXAMPLES. 1. Plane. The straight lines in the plane satisfy Eqs. (2-2)or (2-3); they are the geodesics in the plane. Through every point passesa line in every direction.
* Compare P. Franklin, Treatise on advanced calculus, Wiley, N. Y., 1940, pp.518-522.
134 GEOMETRY ON A SURFACE [CH. 4
2. Sphere. The osculating planes of the geodesics all pass through thecenter, and the geodesics are therefore all curves in planes through the
center (see Exercise 3, Section 1-6). The great circles of a sphere are its
geodesics.
3. Surface of revolution. Here (Section 2-2) we have,
henceE = 1 + f'2,
ffl,rll = 1 +(f')2,
F = 0, G = u2, f = f (u),
r12= r22=-1+f'2' r11 = r12 = r22 = 0.
The equations of the geodesics (2-2) are here:
d2u f'f" du 2 u dv 2 rl2v 2 du dvds2+1+(f')2 d) -1+f'2\ds) =o, ds2+udsds=o.
One of these equations suffices; we take the second:
dv/ds2
+ 2duuds = 0, u2
(dv\ = c, c = constant.
This value substituted into the equation for ds2 gives
u4 dv2 = c2(1 + f'2) du2 + c2u2 dv2,
f 1 + (f')2uV u2 - C2f
The geodesics of a surface of revolution can be found by quadratures.The value c = 0 gives v = constant; the meridians. The parallels
u = constant are geodesics when f' = ac, which means that along suchparallels the tangent planes envelop a cylinder with generating lines parallelto the axis.
EXERCISES
1. Find the geodesics of the plane by integrating Eq. (2-2) in polar coordinates.2. Find the geodesics of the sphere by integrating Eq. (2-2) in spherical co-
ordinates.3. Find the geodesics on a right circular cone.4. Show that the geodesics of the right helicoid can be found by means of
elliptic integrals.5. Find the equation of the geodesics when z = f(x, y).6. The geodesics on cylinders are helices.7. Theorem of Clairaut. When a geodesic line on a surface of revolution makes
an angle a with the meridian, then along the geodesic u sin a = constant, whereu is the radius of the parallel. (A. C. Clairaut, Mem. Acad. Paris pour 1733, 1735,p. 86.)
4-2] GEODESICS 135
8. Show that the evolutes of a curve are geodesics on the polar developable.9. When a curve is (a) asymptotic, (b) geodesic, (c) straight, show that
(a) and (b) involve (c). Do (a) and (c) involve (b)? Do (b) and (c) involve (a)?10. When a curve is (a) geodesic, (h) a line of curvature, (c) plane, show that
(a) and (b) involve (c). Do (a) and (c) involve (b)? Do (b) and (c) involve (a)?11. Prove that for a curve x = x(s) with tangent direction given by the unit
vector t, DK, = u (t x,)av
(t x,.). (D = 11EG
12. Use the formula of Exercise 11 to prove Liouville's formula.13. Show that when the parametric lines are orthogonal
h EG[a au (K2V'G)J
K _ dK, - dK2 -K,2 - K2.
dsl ds2
(J. Liouville, Journal de mathem. 16, 1851, p. 103.)14. Bonnet's formula. When a curve on a surface is given by So(u, v) = constant,
show that1 a 1"V, I a Fop,. - E(p,
K0 - 2 all F2 - 2P 2T
IV 2 - , - 2EG - F pn p p, + G pu Epn Gpu
(0. Bonnet, Paris Comptes Rendus 42, 1856, p. 1137.)15. The curvature of a line of curvature is, but for the sign, equal to its normal
curvature times the curvature of its spherical representation.16. Show that the geodesics on a torus x = r cos cp, y = r sin (p, z = a sin 0
are given by (c is an arbitrary constant, r = p + a cos 0) :
d,p =ca dr
rv r2 - c2 a
17. Show that the finite equation of the geodesics on a paraboloid of revolutioncan be written with the aid of elementary functions.
18. Surfaces of Liouville. There are surfaces of which the line element can bereduced to the form
ds2 = (U + V) (due + dv2),
where U is a function of u alone and V of v alone. Show that the geodesics can beobtained by a quadrature. (J. Liouville, Journal de Mathem. 11, 1846, p. 345.)
19. Show that surfaces of revolution are special cases of Liouville surfaces andthat Clairaut's theorem (Exercise 7) may be extended to Liouville surfaces in theform
U sine co - V cos' w = constant,
where co is the angle at which the geodesic cuts the curve v = constant.20. A surface for which the radii of principal curvature are constants is a sphere.21. Prove that the compatibility conditions for the Weingarten equations
(2-9), Chapter 3, lead to the Codazzi equations.
136 GEOMETRY ON A SURFACE [CH.4
22. Show that the equation of a geodesic on a surface given by F(x, y, z) = 0 is
F. dx d2x
Fl, dy d2y = 0, (FT dx + F,, dy + F. dz = 0).F. dz d2z
4-3 Geodesic coordinates. Let us consider an arbitrary curve Co on asurface and the geodesics intersecting this curve at right angles (Fig. 4-4).This is quite general, since through a point on any curve Co passes one andonly one geodesic in a direction perpendicular to the tangent; we consideronly that part of the surface where the geodesics do not intersect. Wetake the geodesics as the parameter curves v = constant, and the orthogonaltrajectories of these geodesics as the curves u = constant. The curve Cocan be selected as u = 0. The ds2 of the surface has no F:
ds2 = E due + G dv2,
and ds2 is further specialized by thecondition that the geodesic curvature
vanishes. This, according toEq. (1-10), means that E = 0: C.
ds2 = E(u) du2 + G(u, v) dv2.
If we introduce a new parameter
u1 =fuV du,
ds2 = du2 + G(u, v) dv2 (3-1)
where we have replaced u1 again by u; the parameter u now measures thearc length along the geodesic lines v = constant, starting with u = 0 on Co.Eq. (3-1) is called the geodesic form of the line element and (u, v) form a setof geodesic coordinates; or, for short, geodesic set. On every surface we canfind an infinite number of geodesic sets, depending on an arbitrary curve Co,along which the curves v = constant can still be spaced in an arbitrary way.
Another way to introduce geodesic coordinates is to consider at anarbitrary point 0 on the surface (Fig. 4-5) the geodesics starting from thispoint in all directions (v = constant), as well as their orthogonal directions(u = constant). We find again for ds2 the form (3-1), in which u is now
4-31 GEODESIC COORDINATES 137
the arc length along the geodesics, measured from 0. In this case wespeak of geodesic polar coordinates. They also form a geodesic set.
For the arc length of the curve v = c, between its intersections with thecurves u = ui, u = u2 we find, according to Eq. (3-1):
LZfs= du=u2-u1j (3-2)
which means that the segments on all the geodesics v = constant included be-tween any two orthogonal trajectories are equal. And conversely:
If geodesics be drawn orthogonal to a curve C, and segments of equal lengthbe measured upon them from C, then the locus of their end points is an or-thogonal trajectory of the geodesics.
1This is the reason that the orthogonal trajectories of a system of 00geodesics are called geodesic parallels.
Geodesic coordinates were introduced and extensively used by Gauss in hisDisquisitiones of 1827. Parallel curves in the plane were already studied byLeibniz in 1692, who called the involutes of a curve "parallel."
We have fixed only the parameter u (but for a constant), and can stillintroduce any function of v as a new parameter v, (which means that we canstill space the geodesics in any desired way). In the first case, illustratedby Fig. 4-4, we take as the parameter v1 the arc length along Co(u = 0),beginning with an arbitrary point on Co (for which we can take the inter-section with v = 0) :
vl = J 9 G dv.0
Hence in the new parameter G(0, vi) = 1. We still have Co itself to dis-pose of; we select it as a geodesic. Then 0, or, according toEq. (1-10):
d
[au vG(u, v)]u=o = 0.
In the second case, that of geodesicpolar coordinates, we have as a firstcondition that at 0: G(0, v) = 0. Wenow select as new parameter vi the
,I=rnnctn»t
angle which the geodesic v = constant n /u=constantmakes with the geodesic v = 0. Thisangle can be found most easily byrealizing that the curves u = constant FIG. 4-5
esteGeo
138 GEOMETRY ON A SURFACE [CH.4
for decreasing u (Fig. 4-5) approach ordinary circles, since for very smallu the curve u = uo = constant differs little from the circle of radius uo andcenter 0 in the tangent plane. This leads us to the equation (u being theare length along the curves v = constant) :
91
G dvllim ° = 1,U-0 uvl
or, applying 1'H6pital's rule,
' I,, G ldvl=vl or Ia G ]=1.o au
We
Lauu-_oI
We have thus found the following result:We can always reduce the ds2 of a surface to geodesic coordinates
ds2 = du2 + G(u, v) dv2,
by a method (Fig. 4-4) which allows us to write
G = 1, [au G]u_o = 0 (3-3)
and also by a method (Fig. 4-5, geodesic polar coordinates) which allows us towrite
V'G(o, v) = 0, [au G]u=o = 1. (3-4)
Since for geodesic coordinates
rll =.L =r12=0,1
r12 = 2 G", r22 = -2
Gu, r22 =2
G°,
we find for the geodesic curvature of the orthogonal trajectories of the
geodesics
K2 = (KG)u=coast = ,au G_ / 2G'
and for the Gaussian curvature of the surface
1 a2/G (Cu)2 - 2GGu.Kau2 4G2
(3-5)
(3-6)
4-31 GEODESIC COORDINATES 139
This simple expression for K (rational in G and its derivatives) allowsus to deduce some interesting interpretations of the Gaussian curvature.In the plane the ordinary system of polar coordinates
ds2 = dr2 + r2 42 = dug + u2 dv2
is an example of geodesic polar coordinates. Here the conditions (3-4) arefulfilled. We now write, in accordance with this, for a system of geodesicpolar coordinates on an arbitrary surface:
G = G + u[au \/G(u, v)]uao + 2[49"U2 v/G(u, v)]umo + o(u2)
or
''G = u + a1u2 + a2u3 + o(u3),
where al and a2 are functions of v, which can be expressed in terms of theGaussian curvature Ko at 0(0, v) by means of Eq. (3-6) and its derivatives:
a2U Ko a3Koa
N/G/3K\
au2 au3 au au Jo
Hence
2a1 =-Ko(u + ...)o, 6a2 = -Ko - aKl o(u + ...)oor
ai = 0,
and we obtain for VIG- the formula
a2 = -jKo,
= u - *Kou3 + Ri(u, v), (3-7)
where R,(u, v) is of the order n, n > 3, in u. Hence
G = u2 - *Kou4 + R2(u, v), (3-8)
where R2(u, v) is of the order n, n > 4, in u.
Since the circumference C of a curve u = uo = constant and the area Ait encloses are given by
/'2rVG-dv, G = G(uo, v),C
=
J0
A =fLJ '/ du dv, G = G(u, v),0 0
140 GEOMETRY ON A SURFACE
we arrive at the following two expressions for Ko:
Ko = 3 lim 27ru - C7ru- .o U3 '2-A
Ko = 12 lim7ru
7r u-.0 u4
[CH. 4
(3-9)
The closed curve u = uo is the locus of all points at constant geodesic dis-tance from a point (here 0), that is, at constant distance measured along ageodesic. Such curves are called geodesic circles. The formulas (3-9) givea geometric illustration of the Gaussian curvature K as a bending invariantby comparing circumference and area of a small geodesic circle with thecorresponding quantities for a plane circle of equal radius.
4-4 Geodesics as extremals of a variational problem. A popular (andthe oldest) way to describe geodesics is to call them the curves of shortestdistance between two points on a surface. This is certainly the case forthe plane, where the geodesics are the straight lines, which are the shortestdistances between any two of their points. But we must qualify thisstatement even for so simple a case as that of the sphere. For example, itis true that the shortest distance between two points P and Q on a sphere isalong a geodesic, which on the sphere is a great circle. But there are twoarcs of a great circle between two of their points, and only one of them is thecurve of shortest distance, except when P and Q are at the end points of adiameter, when both arcs have the same length. This example of thesphere also shows that it is not always true that through two points onlyone geodesic passes: when P and Q are the end points of a diameter anygreat circle through P and Q is a geodesic and a solution of the problem offinding the shortest distance between two points.
Another simple example is offered on a cylinder, where two points on thesame generator can be connected not only by means of the generator (theshortest distance), but also by an infinite number of helices of varyingpitch, which wind around the cylinder and are all geodesics.
In order to find the shortest distance between P and Q, if such a oneexists, we have to find the function v = v(u) for which s obtains a minimumvalue. In order to find this function, let us suppose that the curve C solvesthe variational problem, so that the distance PQ, measured along C, isshorter than the distance PQ measured along any other curve C' betweenP and Q, when C' is taken close to C. Take C as the curve v = 0 of anorthogonal coordinate system
ds2 = E due + G dv2,
4-4] GEODESICS AS EXTREMALS OF A VARIATIONAL PROBLEM 141
and let the curve C' be determined by a small variation an = Sv alongthe curves u = constant, measured from C. If Chas the equation v = v(u),then we take v so small that it can be identified with bv. When C' passesthrough P(u = uo) and Q(u = u,), then v(uo) = v(ui) = 0. The distancesPQ along C and along C' are now
fus E ++ (v')2G du,s + Ss = fou,
respectively; here E = E(u, 0), El = E(u, Sv), G = G(u, Sv).Since
E(u, av) = E(u, 0) + &E, + , E,, = [8E(u,
we can write for the first variation as of the are length, neglecting all termsof order higher than Sv (including (v')2):
Ss= [-IE-rl+SvE"13-N/ Eldu
= f Z Sv - du =fui2SvE
du
u' E,,
w 2EV an ds. Us = VE du measured along C)
Hence, according to Eq. (1-10)
as =- wK, an ds,U'
(4-1)
where K, is the geodesic curvature along C(v = 0).A necessary condition that s be a minimum is that as = 0 for all an.
This condition is fulfilled if the geodesic curvature K, vanishes along thecurve C between P and Q. A necessary condition for the existence of ashortest distance between P and Q is that it be measured along a geodesicjoining P and Q. In other words:
If there exists a curve of shortest distance between two points on a surface,then it is a geodesic.
We can also find this result by means of the general rules of the calculus ofvariations. They state that the curves which solve the variational problem
o f(x, y, y') d, = 0, y' = dy/dx,
142 GEOMETRY ON A SURFACE [CH. 4
are solutions of the Euler-Lagrange equations,
af d of =0.*ay dx ay'
Such curves are called the extremals of the variational problem. In the case ofthe geodesics, where
f(u, v, v') = N/E
the Euler-Lagrange equations are
Ev + 2Fv' + G (v')2 -
22Fv' + Gv'2, v' = dv/du,
d 2(F + Gv') = 0.E+2Fv'+Gv'2 du '. E+2Fv'+G(v')2
The left side of this equation can also be obtained if we substitute into Bonnet'sformula for the geodesic curvature (Exercise 14, Section 4-2) the expressions
pp(u, v) = v - v(u), v', v = 1, d = a + v' a .du au av
This shows that the Euler-Lagrange equations of our problem can be writtenin the form Kg = 0, so that the geodesics are the extremals of the variationalproblem as = 0.
This result does not inform us when a given geodesic actually is a curveof shortest distance. A full discussion of this problem would lead us toofar. A partial answer can be obtained if we return to our curve C(v = 0)of Fig. 4-6, now a geodesic, and postulate that this curve is a part of afield of geodesics, at any rate in a region R close to C and including P and Q.Such a field is a one-parameter family (or congruence) of geodesics such that
vi 0there passes through every point of R one
r and only one geodesic of this family. We
FIG. 4-6
now take these geodesics as the curvesu = u, v = constant of our region and their or-
thogonal trajectories as the curves u = con-stant. Then, varying again the curve v = 0from C into CI between P(uo) and Q(u,), wehave now E = El = 1 and
as = 1 -+ GO du - du > 0,uofu,
since G= EG - F2> 0. The arc length ofC between P and Q is therefore smaller than any other curve C' betweenthe same points, and we have found the theorem:
* See e.g. F. S. Woods, Advanced calculus, Ginn & Co., N. Y., 1934, Chap. 14.
4-4] GEODESICS AS EXTREMALS OF A VARIATIONAL PROBLEM 143
FIG. 4-7
If an arc of a geodesic can be im-bedded in a field of geodesics, then itoffers the shortest distance between twoof its points as compared to all othercurves in the region for which the fieldis defined.
In other words, if there is but onegeodesic are between two points ina certain region, then that arc givesthe shortest path in that region be-tween these points.
This proposition can be illustratedon the sphere (Fig. 4-7). If we takeon a great circle two points A and Bwhich are not diametrically opposite,then we can imbed in a field of greatcircles that part of the great circlebetween A and B which is less thanhalf the circumference, but the otherhalf cannot be imbedded in this way.Indeed, any geodesic close to thisother half passes through two dia-metrically opposite points of thesphere, which cannot happen in afield.
If the point Q moves on the geo-desic C away from P, then it mayhappen that it will reach a certainposition where PQ will no longer bethe shortest distance between P andQ. Suppose that this happens atpoint R. Then, if the geodesicsthrough P near C have an envelopeE, it can be shown that R is thepoint where C is tangent to E forthe first time. This point R is calledconjugate to P on the geodesic C.An example of such geodesics isoffered by the ellipsoid (see Fig. 4-8,where we see the projection on theplane of symmetry).
FIG. 4-8
144 GEOMETRY ON A SURFACE [CH. 4
Further discussion of this case is difficult without full use of the calculusof variations. However, we can show by simple reasoning that conjugatepoints do not exist on regions of a surface where K is negative. For thispurpose let us consider a geodesic C, and now consider it as the curve u = 0of a geodesic coordinate system
ds2 = due + G dv2.
The distance between two points P(v = vi), Q(v = V2) on C, measuredalong C, and measured along another curve u = u(v) passing through Pand Q is
s = f"-\IG- dv, s + as =J u'2 + G dv, u' = du/dv,
respectively; we take u small and u' finite. By a reasoning similar to theone used in the case of geodesic polar coordinates we find for V a seriesexpansion of the form
so that
= 1 - .-Ku2 + O(U2), K = K(0, v),
s + Ss (1 - Ku2 + o(u2)) dv = f[i + (u'2 - Ku2) + e] dv,
where a is small compared to u2. Hence
aas = .. (u'2 - Ku2 + e) dv.
For K < 0 we can always keep as > 0 provided we stay in a sufficientlysmall neighborhood of C.
The question whether a geodesic is the shortest distance between two pointswas first raised by C. G. J. Jacobi in his Vorlesungen uber Dynamik (1842/43,see his Ges. Werke). The theory is discussed in G. Darboux's Legons III, Ch.V, and in books on the calculus of variations. Fig. 4-8 is taken from A. vonBraunmuhl, Mathem. Annalen 14 (1879), p. 557.
4-5 Surfaces of constant curvature. We have seen that surfaces ofzero Gaussian curvature are identical with developable surfaces (Sec-tion 2-8). The simplest example of a surface of constant non-zero Gaussiancurvature (or surface of constant curvature) is a sphere, where K = a-2 (athe radius) and is therefore positive. It can easily be seen that thesphere is not the only surface with this property. We have only to take
4-51 SURFACES OF CONSTANT CURVATURE 145
a cap of a sphere (as usual we discuss here properties holding for certainsegments of a surface, not necessarily for surfaces as a whole) and think ofit as consisting of some inelastic material, perhaps thin brass. Then it ispossible to give this piece of brass all kinds of shapes, and all surfaces thusformed have the same constant curvature as the spherical cap.
This by itself does not prove that all surfaces of the same constantcurvature (or at any rate certain regions on it) can be obtained from eachother by bending. But we can prove here a theorem giving at least anapproach to this subject. For this purpose we define an isometric corre-spondence between two surfaces as a one-to-one point correspondence such thatat corresponding points the ds2 are equal. We say that the surfaces are iso-metrically mapped upon each other. Bending is a way of obtaining iso-metric mapping, but an isometric correspondence need not be the result ofbending. We can now prove Minding's theorem:
All surfaces of the same constant curvature are isometric.We distinguish among the cases K > 0, K = 0, K < 0.I. Let K = 0. Take a system of geodesic polar coordinates
ds2 = due + G dv2.
Since K satisfies Eq. (3-6), the differential relation
1 82yK au'
(5-1)
(5-2)
we find that is of the form
= ucl(v) + c2(v). (5-3)
We can impose on G the conditions (compare with Section 4-3)
(V'G)u_o = 0, (av/au)u_o = 1. (5-4)
Hence V = u andds2 = du' + U2 dv2.
This expression for ds' can be obtained for all surfaces with K = 0 by takingon it a geodesic polar coordinate system. This can be done in 003 ways(oo' possible points for origin, ool choices of v = 0). All surfaces of zerocurvature therefore are isometric. Taking x = u cos v, y = u sin v, weobtain ds2 in the form
ds' = dx2 + dye, (5-5)
which shows that all developable surfaces can be isometrically mapped ona plane, the curvilinear coordinates corresponding to the rectangular
cartesian coordinates in the plane.
146 GEOMETRY ON A SURFACE [CH. 4
This isometrical mapping can actually be accomplished by bending.To show this, let us take a tangential developable with the equation(Section 2-3)
and
y=x(s)+vt(s), y.=t+Kvn, y=tds2 = (1 + K2v2) ds2 + 2 dv ds + dv2. (5A
The shape of the curve x = x(s) is fully determined by its K(s) and T(S).Let us now consider a family of curves given by K = K(s) and Ti = µT(s),where µ takes all values from µ = 1 to µ = 0. For µ = 0 we obtain aplane curve. When µ passes to all values from µ = 1 to µ = 0, the corre-sponding tangential developable maintains its ds2 in the form (5-6), sinceK(s) does not change. Thus the surface passes through a set of continuousisometric transformations from (5-6) to the case T = 0, which is a plane.Then the tangents to the space curve have become the tangents to theplane curve with the same (first) curvature. In other words (see theremark made in Section 2-4 (Fig. 2-12)), when a piece of paper with a curveC on it is bent, and C becomes the edge of regression C1 of a developable sur-face, then C1 passes into a space curve with the same (first) curvature as C.
II. Let K > _0. Again take a system (5-1) of geodesic polar coordi-nates. Now V'G is a solution of
a2J/8u2 + K/ = 0. (5-7)
This is a linear differential equation with constant coefficients; hence
= cl sin V (u + c2), (5-8)
where c1 and c2 are functions of v. If we impose upon v the conditions(5-4) we obtain
= sin uv/K-_
(5-9)VK
and
ds2 = due +
K
(sine 'f1 u) dv2. (5-10)
Reasoning as in the case K = 0, we see that we can find, on every sur-face with the same K, and in ao9 ways, a coordinate system in which ds2takes the form (5-10), and the isometric mapping is accomplished. Inother words:
Every surface of constant positive curvature K can be isometrically mappedon a sphere of radius K-1. This theorem only holds, of course, for suchregions of the surface where we can introduce a system (5-1); those regionscan then be isometrically mapped on corresponding regions of a sphere.
4-61 ROTATION SURFACES OF CONSTANT CURVATURE 147
III. Let K < 0. Reasoning in the same way as before, we obtain theequation
a2//au2 - L-,I-G = 0, L =-K, (5-11)
which, under the conditions (5-4), has the solution
v _ sinh uVL,N/L
so that
ds2 = due + (sinh dv2, (5-12)
which shows that here again all surfaces of this type can be isometricallymapped on each other, and in o03 ways.
This theory is due to Minding, Journal fur Mathem. 19, 1839, and was furtherelaborated by many other mathematicians, notably by E. Beltrami. Excellentexpositions of the theory of surfaces of constant curvature can be found inDarboux's Legons or Bianchi's Lezioni. Hilbert, loc. cit., Section 3-5, alsoproved that there does not exist a surface of constant negative curvature whichis without singularities and everywhere analytical.
4-6 Rotation surfaces of constant curvature. There is a great varietyof surfaces of constant curvature, which can be gathered from the fact thatall the many varieties of developable surfaces are surfaces of zero curvature.The best known types of surfaces of constant curvature K 0 0 are the rota-tion surfaces of constant curvature. We shall here give an outline of theirtheory.
Such surfaces can be given by the parametric representation (Sec-tion 2-2),
x = r cos gyp, y = r sin gyp, z = f (r), (6-1)
with corresponding first fundamental form
ds2 = (1 + f'2) dr2 + r2 dlp2.
By means of the transformation
du = V11 _+P dr, (6-3)
Eq. (6-2) can be brought into the Ageodesic form
ds2 = due + G doe,G = G(u) = r2. (6-4)
B
FIG. 4-9
(6-2)
1.48 GEOMETRY ON A SURFACE [CH. 4
FIG. 4-10
The function G has the special propertythat it depends only on u, and it canbe readily shown (see Eq. (6-7)) thatthis property is characteristic of all sur-faces isometric with rotation surfaces.In this case the Eq. (5-7) takes theform
d2VG-/du2 + KV = 0(with straight d).
The solution for K = a-' > 0 isgiven by
VIG = c, cos u/a + c2 sin u/a,c,, c2 constants, (6-5)
and for K < 0, K = - b-2 it is given by
= cle°ib + c2e-u/b,C1, c2 constants. (6-6)
The c, and c2 can always be selectedin accordance with conditions (5-4),but sometimes it is more convenientto select them otherwise. For the
moment we leave them undetermined. Since according to (6-2), (6-3),and (6-4) :
r z = f (r) = J 1 - (du)2 du, (6-7)
we find for the profile of the rotation surfaces of constant curvature: (a) forK = a-2 > 0: r = cl cos u/a + c2 sin u/a,
z =f- a-(- c, sin u/a + c2 cos u/a)2 du,
(b) for K =-b-2 < 0: r = cl exp. (u/b) + c2 exp. (-u/b),
z f-\11-- b-2[c exp. (u/b) - Cl exp. (- u/b)]2 du,
where the choice of c, and Cl may result in different types of surfaces.
"I ROTATION SURFACES OF CONSTANT CURVATURE
FIG. 4-11
149
These equations show that, in general, the determination of rotation sur-faces of constant curvature leads to elliptic integrals. However, there arespecial cases in which these integrals may be evaluated in terms of ele-mentary functions. One case is that in which K = a 2, cl = a, c2 = 0.Then ds2 = due + a2(cos2 u/a) dv2,
r = a cos u/a, z = a sin u/a + constant,
and we obtain the sphere (also obtained for c2 = a, cl = 0). Another caseis that in which K = - b-2, cl = b, c2 = 0. Then ds2 = due + b2e2u /I dv2,and
'l - exp. (2u/b) du.r = b exp. (u/b), z = fv/ (6-8)
Instead of evaluating the integral, we observe that here
dr r (6-9)dz b2 - r2,
which shows that the curve r = f (z) has the property that the segment ofthe tangent between the point of tangency A and the point of intersectionwith the Z-axis is constant and = b (both ± b - r2 give the same curve,Fig. 4-9). This curve is the tractrix, and the corresponding rotation sur-face is called the pseudosphere, with p8eudoradius b. We have thus foundthe following result:
Every surface of constant negative curvature K = - b2 can be isometricallymapped on a pseudosphere of pseudoradius b.
150 GEOMETRY ON A SURFACE [Cii. 4
FIG. 4-12
The pseudosphere has a singular central circle and thins out asymptoti-cally on both ends (Fig. 4-10).
We show in Fig. 4-11 some rotation surfaces of constant positive curva-ture, and in Fig. 4-12 some rotation surfaces of constant negative curvature.Their profiles can be expressed with the aid of elliptic functions.
4-7 Non-Euclidean geometry. The geometry of surfaces of constantcurvature receives additional interest through its relation with the so-called non-Euclidean geometry. This geometry was the outcome of thecentury-long quest for a proof of the statement, introduced in Euclid'sElements (c. 300 B.c.) as the fifth postulate:
"If a straight line falling on two straight lines in the plane makes theinterior angles on the same side less than two right angles, then the twostraight lines, if produced indefinitely, will meet on that side on which arethe angles less than two right angles."
This postulate is equivalent to another one, known as the parallelaxiom.* This asserts that through a point outside of a line one and only
* The precise difference between "postulates" and "axioms" in Euclid is notclear.
4-71 NON-EUCLIDEAN GEOMETRY 151
one parallel to that line can be drawn. From it we can deduce that thesum of the angles of a triangle is equal to two right angles. Both fifthpostulate and parallel axiom are far from self-evident, so that already inAntiquity attempts were made to prove them, that is, to obtain them bylogical deduction from other postulates or axioms considered more imme-diately evident (such as the axiom that two points determine a line).Though in Antiquity and all through modern times until c. 1830 severalresults were obtained which enriched our understanding of the axiomaticstructure of geometry, a proof of the parallel axiom was never found.Many so-called demonstrations smuggled in some unavowed assumptions.Some mathematicians - the more penetrating thinkers - derived fromthe denial of the parallel axioms some startling conclusions which they con-sidered absurd, but what seems "absurd" to one generation may still belogical and acceptable to another. Per-haps the most valuable contribution tothis type of literature was made by theItalian Jesuit G. Saccheri ("Euclid freedfrom every birthmark," 1733, Latin).Saccheri considered a quadrangle ofwhich two opposite sides AB, CD areequal and perpendicular to the third sideBC (Fig. 4-13). Then he consideredthree possibilities:
(a) The angles at A and D are obtuse("obtuse angle hypothesis").
(b) These angles are right angles("right angle hypothesis").
D
A
FIG. 4-13
C
I
B
(c) These angles are acute ("acute angle hypothesis").He then proved that when one hypothesis is accepted for one quadrangle,it must be accepted for all quadrangles. From this follows again that thesum of the angles of a triangle is greater than, equal to, or smaller thantwo right angles provided hypothesis (a), (b), or (c), respectively, holds.Saccheri's work remained little known, but several other mathematicianscame to related results, notably Legendre. Saccheri and Legendre re-jected the hypothesis of the obtuse angle because it leads to an absurdityif we assume that a line has infinite length; they also had objections to theacute angle hypothesis.
Gauss seems to have been the first to believe in the independence of theparallel axiom, so that he accepted the logical possibility of a geometry inwhich the parallel axiom is replaced by another one, such as the hypothesisof the acute angle. Gauss believed that the geometry of space can befound by means of experiments, hereby opposing a favorite tenet of the
152 GEOMETRY ON A SURFACE [CH. 4
Kantian philosophy prevalent in his days. The first mathematician tostate publicly (1826) that a geometry can be construed independently ofthe parallel axiom was N. I. Lobacevskii, professor at Kazan, whose firstwork on this subject was published in 1829. He was followed independ-ently by the young Hungarian officer John Bolyai (Bolyai Janos), whoseessay was published as an Appendix to his father's big tome on geometry(1832). Gauss commended this paper, writing that he could not praise it,as this would mean praising himself, since he had come to the same con-clusions -a kind of praise eminently fitted to discourage the ambitiousyoung officer.
Both Lobacevskii's and Bolyai's geometry are based on the hypothesisof the acute angle. In this case the lines through a point P in a plane canbe divided into two classes, those that intersect a given line 1 not passingthrough P, and those that do not intersect it. These classes are separatedby two lines which can be called parallel to l through P. In this geometryas well as in that of Euclid a straight line is an open line of infinite length.If we postulate that straight lines are closed and have a finite length, wecan construct a geometry based on the hypothesis of the obtuse angle; thisgeometry was sketched by B. Riemann (1854) inside the framework of themore general so-called Riemannian geometries. Under the obtuse anglehypothesis all lines passing through a point P outside a line 1 in a planeintersect 1. Thus all three assumptions of Saccheri could now be introducedas the bases of possible geometries, which became known as Euclidean andnon-Euclidean geometries.
However, most mathematicians continued to believe that although therewere as yet no logical inconsistencies discovered in the non-Euclideangeometries, it might very well be possible to find such inconsistencies aftera more penetrating study of the properties of these geometries. Conse-quently, little attention was paid to them despite the authority of Riemann.This situation was radically changed through a paper by the Italian mathe-matician E. Beltrami, An attempt to interpret the non-Euclidean geometry(1868). Beltrami proved that when we take the geodesics of a surface ofconstant negative curvature and interpret them as "lines," interpretingangles and lengths according to the ordinary methods of differentialgeometry, then we obtain a geometry in which the hypothesis of the acuteangle holds. The whole geometry of Lobacevskii-Bolyai could thus beinterpreted on a surface of constant negative curvature, parallel lines be-coming asymptotic geodesics. Thus Beltrami proved that the consistencyof Euclidean geometry implied the consistency of Lobacevskil-Bolyai geom-etry, since an inconsistency in the latter could be interpreted as an incon-sistency in the theory of surfaces of constant negative curvature, whichitself is based on Euclidean postulates. Every inconsistency in one
4s THE GAUSS-BONNET THEOREM 153
geometry implied an inconsistency in the other. Beltrami's mapping ofthe one geometry on the other was followed by several other such repre-sentations, not only for the case of the acute angle hypothesis, but also forthe hypothesis of the obtuse angle. We now know that Riemann's geom-etry can be interpreted on a sphere by taking the great circles as "straightlines," provided we identify two diametrically opposite points (otherwisetwo points would not always determine one and only one line). Thelogical equivalence of the three geometries has thus been fully established.
More details can be found in books on non-Euclidean geometry, notably inJ. L. Coolidge, The elements of non-Euclidean geometry, Oxford, 1909, or R.Bonola, Non-Euclidean geometry, Chicago, 2d ed., 1938. A neat summary isfound in J. L. Coolidge, A history of geometrical methods, Oxford, 1940, pp. 68-97.English translations of Loba6evski1 and Bolyai's papers were published byG. B. Halsted: Geometrical researches on the theory of parallels, Chicago, London,1914, 50 pp.; New principles of geometry, Austin, Tex., 1897, 27 pp. (both byLoba6evski1); The science absolute of space, Chicago, London, 1914, xxx, + 71 pp.(by Bolyai). Beltrami's Saggio di interpretazione della geometria non-euclideais found in Opere I, pp. 374-405. There exists a French translation: AnnalesEcole Normale 6, 1869, pp. 251-288. See also H. S. M. Coxeter, Non-Euclideangeometry, Toronto, 2d ed., 1947.
It should be stressed that Beltrami's method only allows us to map apart of the Lobacevskil plane on a part of a surface of negative curvature.Hilbert has shown (see Section 3-5) that it is impossible to continue ananalytical surface of constant negative curvature indefinitely without meet-ing singular lines (at any rate when this surface lies in ordinary Euclideanspace, see also Section 4-5).
4-8 The Gauss-Bonnet theorem. There exists a classical theorem,first published by Bonnet in 1848, but which probably was already knownto Gauss, which we shall prove as an example of the differential geometryof surfaces in the large. It is an application of Green's theorem, knownfrom the theory of line integrals and surface integrals in the plane, to theintegral of the geodesic curvature.
Green's theorem asserts that if P and Q are any two functions of x and y inthe plane for which the partial derivatives 8P/8y and 8Q/8x are continuousthroughout a certain region R with area A bounded by a closed piecewise smoothcurve C, then (Fig. 4-14), the sense of C being counterclockwise,
154 GEOMETRY ON A SURFACE
JP dx + Q dy= ff( - a )dxdy.
The region R may be the sum of a finitenumber of areas, each of which is suchthat any straight line through any interiorpoint of the region cuts the boundary inexactly two points. The theorem holdsequally on a surface if x and y are re-placed by the curvilinear coordinates uand v, the parametric lines covering R in
FIG. 4-14
[CH. 4
such a way that through every point inside R passes exactly one curve u = con-stant and one curve v = constant.
If P(u, v) and Q(u, v) are two functions of u and v on a surface, then,according to Green's theorem and the expression in Chapter 2, Eq. (3-4)for the element of area:
1 P du + Q dv = fJ\aQ aP\ 1 dA,
.J ). . EG-F2
where dA is the element of area of the region R enclosed by the curve C.With the aid of this theorem we shall evaluate
JK,ds,
where Kp is the geodesic curvature of the curve C. If C at a point P makesthe angle 0 with the coordinate curve v = constant and if the coordinatecurves are orthogonal, then, according to Liouville's formula (1-13):
Kp ds = dO + KI(COS 0) ds + K2(sin 0) ds.
Here KI and K2 are the geodesic curvatures of the curves v = constant andu = constant respectively. Since
cos e ds = V du. sin 9 ds = ' dv,
we find by application of Green's theorem:
fK. ds = fd0 +J f(c
(K2/) - v (KIV )) du A.
The Gaussian curvature can be written, according to Chapter 3, Eq.(3-7),
4-8] THE GAUSS-BONNET THEOREM
K 1 r a G a Eu112 EG Lau E + av EGJ
155
E a av (K1\ /E)],
so that we obtain as a result the formula
fKQdS =fd0- f rKdA. (8-1)C J JA
The integral ffA K d A is known as the total or integral curvature, or
curvature integra, of the region R, the name by which Gauss introduced it.Let us first take a smooth curve C, as the boundary of region R on which,
therefore, there are no points where the slope has discontinuities. Then
we can contract C continuously without changing (do, since this is an0
integral multiple of 2a. Let A be a simply connected region, that is, aregion which by continuous contraction of C can be reduced to a point.
is reduced to approach a point
then (do = 27r, and we have found thec
theorem that
fKo ds + f f. dA = 2a. (8-2)A
When C consists of k arcs of smoothcurves (Fig. 4-15) making exterior angles01, 02, .., 0k at the vertices, A1, A2, ...,Ak, where the arcs meet, then we must
FIG. 4-15
keep in mind that d0 in Eq. (8-1) measures only the change of 0 along
the smooth arcs, where we measure (K, ds, and not the jumps at the
vertices. The total change in 0 along C is still 2a, but only part of it isdue to the change of 0 along the arcs. The remainder is due to the angles01, 02, ... 0k. For the sum of the line and area integral we therefore get2a - 01 - 02 - - O. This result is expressed in the Gauss-Bonnettheorem:
If the Gaussian curvature K of a surface is continuous in a simply con-nected region R bounded by a closed curve C composed of k smooth arcs making
156 GEOMETRY ON A SURFACE
at the vertices exterior angles 01, 02, ... Ok, then:
jxds +fJ K dA = 2a - B,A
where K, represents the geodesic curvature of the arcs.
[CH. 4
i= 1,2,...k
This theorem was first published by 0. Bonnet in the previously mentionedpaper in the Journ. Ecole Polytechnique 19 (1848), pp. 1-146, as a generalizationof Gauss' theorem on a geodesic triangle, see application II of this section.Bonnet proved it first for geodesic triangles, using the method of our Exercise 6,this section. The relation of this theorem to Green's theorem was clarified byG. Darboux, Legons III, p. 122, etc.
The integral curvature JJ K dA has a simple geometrical interpreta-
tion, by means of which Gauss originally introduced it. When we drawthrough a point 0 lines parallel to the surface normals at the points of aregion R on a surface S bounded by an are C, then they intersect the unitsphere in a region R1 bounded by an arc C1 which is the spherical imageof C (see Exercise 6, Section 2-11). We can introduce coordinates (u, v)on the unit sphere such that a point and its spherical image have the sameu and v. The equation of C1 is N = N (s), where s is the arc length of C.The element of are do of C1 is given by doe = dN dN, and the element ofarea dA. of R1, according to Chapter 2, Eq. (3-4), by
dA. _ (N (N, N,) - (N N,)2 du dv= (N X N,) (Nu X N,) du dv.
According to the Weingarten formulas, Chapter 3, Eq. (2-9):
N.XN, _ (Ff - Ge) (Ff - Eg) - (Fe-Ef)(Fg-Gf)x,.XXV(EG - F2)2
- eg f2 x X x, = Kx, x x,.EG - F2Hence
dA. = I K I (x, x x,) (x x x,) du dv = j K I v'EG --F2 du dv = I K I dA,
so that the integral (absolute) curvature of a region on the surface is equal to thearea of its spherical image. This was the property stressed by Gauss.
This property of the integral curvature was already known to the Frenchschool of Monge before Gauss pointed out its significance for the intrinsic geom-etry of a surface, see 0. Rodrigues, loc. cit. Section 2-9.
4-8] THE GAUSS-BONNET THEOREM 157
When the area becomes smaller and smaller, we obtain the property that,with the appropriate sign,
KGAI-o
£AA.4
The Gaussian curvature is the limit of the ratio of the small areas on thespherical image and on the surface which correspond to each other, when theareas become smaller and smaller.
Here we must take into consideration that for a hyperbolic point thecontour of AA, is traversed in a direction opposite to that of AA.
This definition of curvature corresponds to that given by K = dip/ds forplane curves.
We shall finally give some applications of the theorem of Gauss-Bonnet.I. When we apply this theorem to a triangle formed by an arc of a curve
between two neighboring points P and Q and the geodesics tangent to C at
P and Q (Fig. 4-16) we find that ffK dA is of order (A(p)3, so that but
for quantities of order (A p)2 and higher (where A p is the angle between thetangent geodesics at P and Q):
fp
Q
K,ds=2r- (-Op)-r-a=Ap,
where Op is the angle between the geodesics. When Q approaches P wefind
K. = dcp/ds, (8-3)
which formula generalizes for surfaces the well-known formula for the ordinarycurvature of a plane curve and gives us at last a simple definition of thegeodesic curvature in terms of quantities on the surface.
H. When the boundary C consists of k geodesic lines, intersecting atinterior angles al, a2, ..., ak, a; = r - 8;, i = 1, . . . k, we have a geodesicpolygon. The integrals of the geodesic curvature vanish and we obtainfor the integral curvature:
FIG. 4-16 FIG. 4-17
158 GEOMETRY ON A SURFACE [CH. 4
ffKdA = 2r- k7r + jai _ a; - (k - 2)7r.
When k = 3 we find the theorem (Fig. 4-17) :The integral curvature of a geodesic triangle is equal to the excess of the sum
of its angles over 7r radians. Gauss wrote: "this theorem, if we mistake not,ought to be counted among the most elegant in the theory of curved sur-faces."
This theorem is of particular interest for a surface of constant curvature.In this case the integral curvature is equal to KA, where A is the area of thegeodesic polygon. We express this result in the theorem:
The area of a geodesic triangle on a surface of constant curvature is equal tothe quotient of (ai + a2 + a3 - 7r) and K, where the a are the angles of thetriangle.
This result generalizes the well-known theorem of Legendre for a sphere,where ai + a2 + a3 - 7r is the spherical excess:
The area of a spherical triangle is equal to the product of its spherical excessinto the square of the radius.
It also shows that the sum of the angles of a geodesic triangle is greaterthan 7r for a surface of constant positive curvature and less than 7r for asurface of constant negative curvature. For developable surfaces thesum is 7r.
III. A surface which can be brought into one-to-one continuous corre-spondence with the sphere is called topologically equivalent to the sphere.Such surfaces can be obtained by continuous deformation of a sphere with-out breaking it and without bringing separated parts of its surface together.Such a surface can be separated into two simply connected regions I and IIby a closed smooth curve C without double points (Fig. 4-18). If we applythe Gauss-Bonnet theorem to such a curve, first as boundary of region I,and then of region II, we obtain, if we traverse C both times in such direc-tion that the enclosed region stays on the left side:
J Kp ds +f rK dA = 27r,
J'K JfJI
d s + K dA = 27r.
Add ing these two equations, we ob-tain the result
FiG. 4-18ffKdA = 47r.
4-8] THE GAUSS-BONNET THEOREM
FIG. 4-19
159
In words:The integral curvature of a surface which is topologically equivalent to the
sphere is 4r.This theorem characterizes the integral curvature as a topological invari-
ant. If we apply a similar reasoning to a surface which is topologicallyequivalent to a torus, then we see that it can be changed into a simplyconnected region R by two cuts (Fig. 4-19). These cuts can be so appliedthat this region R is bounded by four smooth curves intersecting at 7r/2 ra-dians. When we apply the Gauss-Bonnet theorem to this boundary, weobtain
fK11 ds +JfK dA = 27- 4. = 0.
The integral of the geodesic curvature vanishes, and
ffKdA = 0.
The integral curvature of a surface which is topologically equivalent to a torusis zero.
EXERCISES
1. Integrate the differential equation (6-9) of the tractrix. Write r = b sin gyp.2. The geodesic curvature of a curve C on a surface S is equal to the ordinary
curvature of the plane curve into which C is deformed when the developable sur-face enveloped by the tangent planes to S along C is rolled out on a plane.
3. A geodesic C on S is also a geodesic on the developable surface envelopedby the tangent planes to S along C.
4. When a geodesic makes the angle 0 with the curves v = constant of a geo-desic coordinate system for which ds2 = due + G dv2, then
dO + (a.'/au) dv = 0.
160 GEOMETRY ON A SURFACE [CH. 4
5. If two families of geodesics intersect at constant angle, the surface is de-velopable.
6. Prove Gauss' theorem on geodesic triangles by taking two sides as the co-ordinate curves v = 0 and v = vo of a geodesic coordinate system. This was Gauss'proof of his theorem. Hint: Use the result of Exercise 4.
7. Show that for a surface of negative curvature the hypothesis of the acuteangle holds for quadrangles formed by geodesics (see Section 3-7).
8. The integral curvature of a closed surface of genus p (or connectivity 2p + 1),that is, a surface which can be made into a simply connected region by 2p simpleclosed cuts (e.g., a sphere with p handles) is 4ir(1 - p).
9. Prove Euler's theorem for convex polyhedrons F - E + V = 2 (F = numberof faces, E = number of edges, V = number of vertices) by applying the Gauss-Bonnet theorem (W. Blaschke, Differentialgeometrie I, p. 166).
10. By using the Gauss-Bonnet theorem, show that two geodesics on a surfaceof negative curvature cannot meet in two points and enclose a simply connectedarea. (Compare the theorem on p. 144.)
11. Differential parameters. E. Beltrami, in 1864-1865, introduced the follow-ing expressions, in which u and v are curvilinear coordinates on a surface:
Ego - F(,pu,Pv + pv4,u) + Gpu4GuEG - F2
a Fcp 8 Ew,, - F,p/EG-F2 auvEG-F2+avv/EG-F2I'
which he called the first and second differential parameter of the functions ¢ and pof u, v, respectively. Show
(a) that the derivative dco/ds of a function rp(u, v) on the surface in the directionds is maximum when (d(p/ds)2 = V (,p, p),
(b) that for the plane and the notation of vector analysis
V (,p, 4,) = (grad rp) (grad vi), Ap div grad p = Lap cc,
(c) that ± K, = V + VI P,v/o(sc, 'P)
(Beltrami, Opere mat. I, pp. 107-198. Lame, in the Lecons sur les coordonneescurvilignes, 1859, already had used differential parameters for the plane and forspace.)
12. Show that when two independent systems of geodesic parallels are selectedas parametric curves, the element of are can be written
ds2 = csc2 w(du2 + 2 cos w du dv + dv2),
where w is the angle between the parametric curves.
13. Geodesic ellipses and hyperbolas. The locus of a point for which u + v =constant in the coordinate system of Exercise 12 is called a geodesic ellipse. Simi-
THE GAUSS-BONNET THEOREM 161
larly, we call the locus for which u - v = constant a geodesic hyperbola. Showthat ds2 can be written
ds2 = (csc2 w/2) due + (sect co/2) dv2,
where the coordinate lines are geodesic ellipses and hyperbolas. Also show theconverse theorem, that if ds2 can be written in this form, the coordinate lines aregeodesic ellipses and hyperbolas.
14. Derive Gauss' form of the differential equation of the geodesics:
E dO = FdIn -G,.dv) -where 0 is the angle of the geodesic with the curve v = constant. Compare thisformula with Liouville's formula (1-13) and with Exercise 4.
15. Rectifying developable. Every space curve is a geodesic on its rectifyingdevelopable. This is the explanation of the term rectifying. See p. 72.
CHAPTER 5
SOME SPECIAL SUBJECTS
5-1 Envelopes. The method used in Section 2-4 to determine develop-able surfaces as envelopes of a single infinity of planes can be generalized tosingly infinite families of more general surfaces. The method in the case ofplanes can be summarized (see Chapter 2, Eqs. (4-2) and (4-3)) by statingthat if the family of planes is given by
F(x) y, z, u) = 0, (1-1)
where u is the parameter, and we add to Eq. (1-1) the equations
F. = a F(x, y, z, u) = 0, (1-2)Fu = y, z, u) = 0, (1-3)
then, except for special cases, which can be exactly enumerated in this caseof a family of planes:
Eqs. (1-1) and (1-2) together for fixed u give the characteristic line of asurface (1-1),
Eqs. (1-1), (1-2), (1-3) together for fixed u give the characteristic pointof the surface (1-1).
Elimination of u from Eqs. (1-1) and (1-2) gives the envelope of thefamily (1-1) as the locus of characteristic lines.
Solving of x, y, z in terms of u from Eqs. (1-1), (1-2), and (1-3) givesthe edge of regression of the envelope as the locus of characteristic point.
If we now apply this method to the case that Eq. (1-1) represents ageneral family of surfaces, then we can say in general that provided thisfamily has an envelope and this envelope has a locus of characteristic points,we can obtain the envelope and the locus by a similar reasoning, although wemay obtain other surfaces, curves, and points as well. Before we pass tothe question of proof, let us first take a few examples to elucidate some ofthe existing possibilities.
(a) The family of spheres of equalradii R with centers on the Y-axis(Fig. 5-1),
F=x2+(y-u)2-f-z2-R2=0, (1-4)
has as its envelope the cylinder
x2 + z2 = R2 FIG. 5-i162
5-11 ENVELOPES 163
obtained by elimination of u from Eq. (1-4) and
Fu =-2(y - u) = 0. (1-5)Since
Fu. =+2,there are no characteristic points.
(b) For the concentric spheres
F=x2+y2+z2-u2=0, (1-6)we have
Fu = - 2u, Fuu = - 2.
There is no envelope; elimination of u frorn F = 0 and F. = 0 gives thecommon center.
(c) For the spheres
F - (x - u)2 + y2 + z2 - u2 = x2 + y2 + z2 - 2xu = 0, (1-7)
which have their centers on the X-axis and are all tangent to the YOZ-plane (Fig. 5-2), we have
Fu=-2x, Fuu=O.
The equations F = 0 and F. = 0 give y2 + z2 = 0, the point of tangency.
z
z
Fia. 5-2
(d) For the spheres
Fm. 5-3
F = (x - u)2 + y2 + z2 - (a2 + u2) = 0, (1-8)
which have their centers on the X-axis and all pass through the same circlein the YOZ-plane (Fig. 5-3), we have
F. = - 2x, Fuu = 0.
The equations F = 0 and F. = 0 give the circle through which all spherespass.
164 SOME SPECIAL SUBJECTS [CH. 5
(e) For the cylinders
F = (y - u)2 - x3 - x2 = 0,
we findFu = - 2(y - u), Fu,. = 2.
Elimination of u from F = 0, F = 0 gives the envelope x = -1, but alsothe locus x = 0 of the nodal lines.
There exists no theory which systematically accounts for all cases whichmay occur. We have to be satisfied, in the main, with the principle ob-tained by the generalization of the method used for families of planes.However, it is possible to express the simpler applications of the principlein a more careful way, which we shall now give in outline.
Let us suppose that the surfaces S, given by F = 0, have an envelope E.This means that a surface E exists which is tangent to each of the surfaces Salong a curve C. We can label the curves C by means of the same param-eter u which determines the surface S on which it lies. Then E has theequation x = x(t, u), where x = x(t, uo) is the equation of the curve C onthe surface S for which u = uo. When the coordinates x, y, z of x aresubstituted into F(x, y, z, u) = 0 this equation must be identically satisfied.
The vectors x, and xu determine the tangent plane to E at a point P ofC. But the coordinates of these vectors must satisfy the equations
Fzet +Fya +F.at =0,
Fzau+Fya +F:a+F,.=0,or
(Fz + F:p) at + (Fy + F,') ay = 0,
(Fz + F=p) ax + (Fy + F,q) ay + Fu = 0. (1-9)
Let us now suppose that we take such points of space for which F = 0,but Fu 0 0. Then, according to the theory of implicit functions, we cansolve F = 0 for u and obtain an expression of the form u = u(x, y, z).Since
F. + Fp = 0, Fy + F,q = 0, (1-10)
we can find, by substitution of u into these equations, values for p and q.Suppose that we confine our attention to regions where both x,, x,, andp and q are uniquely determined and different from each other, so that thetangent planes to the surfaces S are uniquely determined at each point.
5-11 ENVELOPES 165
Comparison of Eqs. (1-9) and (1-10) shows that when F. 54- 0 the tangentplane to S cannot be a tangent plane to E. This can only happen whenFu = 0. Moreover, if F. = 0 and xt X x, /- 0, and if p, q are uniquely de-termined, then E is actually an envelope of the surfaces S. We have thusfound the theorem:
When the equation F(x, y, z, u) = 0 represents a family of surfaces S,and the equations F = 0, Fu = 0 in a certain region R(x, y, z, u) represent forfixed u a family of curves C which form a surface E for varying u, then thissurface E is an envelope of S which is tangent to the S along the curves C, pro-vided the surfaces S have in R a unique tangent plane at every point of thecurves C.
The curves C are the characteristics of the surfaces S.We can now show that this envelope E actually exists in such regions
R(x, y, z, u) in which the vector product (grad F) X (grad Fu) 0. Insuch regions it is possible to solve the equations F = 0, F. = 0 for twovariables and express them in terms of the two other ones. Let us singleout y and z, so that the result appears in the form:
y = y(x, u), z = z(x, u) (hence here FvFzu - FY,u o 0). (1-11)
Eqs. (1-11) represent a surface which has a tangent plane given at each ofits points by the equations
i+jax+kax = i+jyz+kzz,
i+jay+kau=i+jyu+kzu. (1-12)
Since the Eqs. (1-11), substituted into F = 0, Fu = 0, change theseequations into identities, we also have the relations:
F. + Fzzz = 0, Fu + Fvyu + Fzzu = 0, (1-13)Fzu + Fyuyz + Fuzz = 0, Fuu + Fzuzu = 0.
These equations determine yz, zz and yu, y uniquely, since FyF,u-F=Fyu O,
provided Fuu 0 0. The vectors i + jyz + kzz and i + jyu + kzu are there-fore perpendicular to the vector Fzi + Fsk, which means that thetangent plane to surface (1-11) coincides with the tangent plane to thesurface F = 0 at all points (x, u) common to both surfaces. For u = uowe obtain the characteristic on the surface F(x, y, z, uo) = 0. In otherwords:
In a region R(x, y, z, u) where the vector product (grad F) X (grad Fu) 0 0and Fuu 0 0 there exists an envelope E, locus of the characteristics C.
166 SOME SPECIAL SUBJECTS [CH. 5
We can obtain some more information when we consider such regionsR, of R, where also the triple scalar product (grad F, grad Fu, grad Fuu) X 0.In this case let us solve
F = 0, Fu= 0,F.,.=0
for x, y, z in terms of u, which is possible since the functional determinant ofF, Fu, F,,,u is different from zero. The curve thus obtained lies on the sur-face given by F = 0, Fu = 0, and hence on the envelope. The tangentdirection is given by the vector
du+ jdu+kdu = ixu+.lyu+kzu.
By a reasoning similar to that used before, we find that
Fixu + Fvyu + Fezu + F. = Fxxu + Fvyu + F,zu = 0,
Fzuxu + Fyuyu + Fzuzu + Fuu = Fsuxu + Fvuy,, + Fsuzu = 0,
and these equations for u = Uo express the fact that the curve is tangent tothe characteristic on the surface F(x, y, z, uo) = 0. Hence:
If there exists an envelope E, locus of characteristics C, then in regions R, ofR, where the triple scalar product (grad F, grad Fu, grad Fuu) 0 0, the equa-tions F = 0, Fu = 0, Fuu = 0 together determine a curve which at each of itspoints is tangent to one of the characteristics C.
This curve is called the edge of regression of the envelope; its points aresometimes called the focal points.
As an example of an envelope of surfaces which are not plane we cantake the canal surfaces, already studied by Mange. We obtain them asenvelopes of spheres of constant radius a of which the center moves alonga space curve x = x(s) (s = u is the param-eter). Then
F°Then, in accordance with Eqs. (1-2) and (1-3)we take
Fu = - 2t (y - x) = 0,
these equations show that the characteristic isthe great circle in which a sphere is intersectedby the normal plane to the center curve. The FIG. 5-4
5-1] ENVELOPES 167
FIG. 5-5(a) FIG. 5-5(b)
envelope is the locus of these circles and is the canal surface. SinceFe,. = 0 is equivalent to
(y -
we see that the edge of regression is the locus of the points in which thespheres are intersected by the polar axis of the curve. Hence it splits intotwo curves, which come together at points where a = R (Fig. 5-4). Whena > R, no edge of regression exists. An example of a canal surface is theordinary torus. When the curve x = x(u) is a plane curve, the edge ofregression is in the same plane and is the envelope of the circles of constantradius a with their center on the curve. In Figs. 5-5a, b we find the caseillustrated, where the curve is an ellipse; this figure is the projection onthe XOY-plane of a torus of which the center circle is inclined with respectto the XOY-plane.
The name canal surface is sometimes used for the envelope of any singleinfinite system of spheres. One of the center surfaces (Section 2-9) ofthese surfaces degenerates into a curve.
EXERCISES
1. A necessary condition for a family of curves F(x, y, u) = 0 to have an en-velope is that the equations F = 0, F. = 0 have a common solution. If in a cer-tain neighborhood (xo, yo, uo)F.F,, - FF.. 0, then we obtain the envelope ofthe curves F = 0, F. = 0 by the elimination of u from F = 0, F = 0.
2. Determine the envelope of the following plane curves: (a) the circles ofconstant radius, of which the centers are on a line, (b) the circles (x - a)2 + y2 = b2,b2 = 4am (m constant).
3. A problem in elementary ballistics. A projectile is fired from a gun withconstant velocity vo making an angle a with the horizon. Show that the envelopeof the trajectories, if a varies in a vertical plane, is a parabola. This problem was
168 SOME SPECIAL SUBJECTS [CH. 5
taken up and solved by E. Torricelli in his book on the motion of projectiles(1644).
4. From the points of a parabola y2 = 4x - 1 tangents are drawn to the ellipsex2 y2
a2+ b2 = 1. Find the envelope of the chords connecting the points of contact.
This envelope is called the polar figure of the parabola with respect to the ellipse.5. Show that a plane curve is the envelope of its osculating circles. This is a
case where, for the envelope, the equations F = 0, F. = 0 hold, although two oscu-lating circles do not intersect along an arc of increasing (decreasing) K.
6. Show that a space curve is the edge of regression on the canal surface formedby its osculating spheres.
7. Verify that the edge of regression of a canal surface is tangent to the charac-teristic.
8. Analyze the cases (a) - (e) of this chapter by evaluating (grad F) X (gradand (grad F, grad F,,, grad F..).
9. Find the envelope of the family of spheres which have as their diameters aset of parallel chords of a circle.
10. When the envelope S of a family of surfaces is intersected by a surface S,which is not tangent to the edge of regression E, then the curve of intersection Cof S and S1 has a cusp (or higher singularity) at the point where S intersects E.Hint: Show that at that point x' = 0, when C is given by x = x(u).
11. When a family of surfaces is given by the equation x = x(u, v, a), where ais a parameter and u, v are curvilinear coordinates, then the envelope, if it exists,is found by the elimination of a from x = x(u, v, a) and (x x xa) = 0. See, onthe discussion of envelopes from this point of view, 0. Bierbaum, Festschrift derTechn. Hochschule Briinn, 1899, pp. 117-150.
5-2 Conformal mapping. Two surfaces S and S, are said to be mappedupon one another if there exists a one-to-one correspondence between theirpoints. When S is given by the equation x = x(u, v) and S, by the equa-tion x1 = x.1(u1, v1), then a mapping of a region of S on a correspondingregion of S, is established by the relations
u, = ui(u, v), v, = vl(u, v), (2-1)
where u,, v1 are single-valued functions with continuous partial derivativessuch that the functional determinant a(u,, v,)/a(u, v) s 0. When we useEq. (2-1) to transform the coordinates (u,, v1) of S1 into coordinates (u, v)on the same surface Si, then we obtain the equation of Si in the form x, =x1(u, v). Two points of S and S, which correspond in the mapping are nowindicated by the same values of (u, v) :
x = x(u, v); x1 = x,(u, v). (2-2)
5-2 CONFORMAL MAPPING
We shall discuss the following types of mapping:1. Conformal mapping, in which angles are preserved;
II. Isometric mapping, in which distances and angles are preserved;III. Equiareal mapping, in which areas are preserved;IV. Geodesic mapping, in which geodesics are preserved.
169
Conformal mapping. A mapping of surface S on surface S, (we shalluse this expression although we mean the mapping of a certain region of Son a corresponding region of Si) is conformal if the angle between two di-rected curves through a point P of S is equal to the angle between two cor-responding directed curves through the point P, on S, corresponding to P.When the two surfaces are written in the form (2-2) and all elements of S,corresponding to those on S are indicated by the index 1, then for the angles0, B, between the corresponding elements dx, Sx, d,x, Six the equation holds:0 = B,. If we take as one pair of corresponding elements Sx, Six, the ele-ments (Su arbitrary, Sv = 0), we find that according to Chapter 2, Eq.(2-9):
E du Su + F dv Su E, du Su + F, dv Suds/E Sue ds, E Su
or
du F d_v __ du F, dvids + ids ds,+ ds,
This equation must hold for all directions dv/du, hence
or
d_u = VIE du dv= F
dvds ds,'
F Vide - ' -VEk ds,'
ds, F,VE78
- - FVITE
This equation shows that the ratio p = ds,/ds is independent of dv/du anddepends only on u and v. Hence
ds, = p ds, P = P(u, v), (2-3)or
E, = P2E, F, = P2F, G, = PEG. (2-4)
Conversely, when Eq. (2-3) holds, then because of Eq. (2-4) :
(E du Su + F(du Sv + dv Su) + G du Sv) =cos 9, =
Sscos 6.
ds
170 SOME SPECIAL SUBJECTS [ca. 5
We have thus arrived at the result:A necessary and sufficient condition that a mapping be conformal is that the
elements of arc ds, Ss at corresponding points be proportional.We can also express this property by stating that a small figure (e.g., a
triangle) at a point of a surface is almost similar to the figure which corre-sponds to it in the mapping, and the more similar the smaller the figuresare:
A necessary and sufficient condition that a mapping be conformal is that itbe a similarity "in the infinitesimal."
When p is constant the ratio of similarity is the same for all points. Wethen speak of similitude.
A conformal map of a small region of a surface near a point on a plane istherefore very nearly accurate in the angles as well as in the ratio of dis-tances, although the map may give a very distorted picture of the region inthe large.
From Eq. (2-3) it follows that from ds = 0 follows dsl = 0. If, there-fore, we admit imaginary elements, then we find that in a conformal map-ping isotropic elements (Section 1-12) correspond. Discarding isotropicdevelopables (EG - F2 = 0, see Section 5-6) we have on each surface anet of isotropic curves which we can take as the net of coordinate lines.Then E = G = 0 and
ds2 = 2F du dv, ds; = 2F1 du dv, F1 = p2F. (2-5)
This equation shows that when the isotropic curves correspond, ds and dslare proportional:
A necessary and sufficient condition that a mapping be conformal is that theisotropic curves correspond.
Since this correspondence can be accomplished for any pair of surfaces,we find that two surfaces can always be mapped conformally upon each other.
This has to be understood again in the sense that to a certain region on surfaceS there corresponds a certain region on S, on which it can be conformally mapped.The theory of complex variables establishes exact conditions for the characterof these regions and their boundaries. The fundamental theorem of Riemannstates that any region with a suitable boundary can be conformally mapped ona circle by means of a simple analytic function.*
After having introduced the isotropic curves, let us again change the
coordinate curves on both surfaces by the transformation
u = u1 + iv1, v = ul - ivi. (2-6)
* See, e.g., E. C. Titchmarsh, The theory of functions, Oxford, Clarendon Press,1932, p 207.
5 2j CONFOI MAL MAPPING 171
The first fundamental forms in the new coordinates, written in terms ofthe new coordinates, are now
ds2 = 2F(dui + dvi), dsi = 2F,(dui + dvi), (2-7)
where F and F, are now functions of u, and v,.If a system of curves can be introduced as coordinate curves so that ds2
takes the formds2 =X(du2 + dv2), X = X(u, v), (2-8)
then this system is called an isometric (or isothermic) system. It is neces-sarily orthogonal. Since the length of du is the same as the length of dv,we can sketch the nature of isometric systems by saying that they dividethe surface into infinitesimal squares. Eq. (2-7) expresses the fact that twosurfaces are conformally mapped upon each other when an isometric sys-tem on one surface corresponds to an isometric system on the other surface.
The use of the term "isometric" both for certain systems of curves and fora certain type of mapping does not mean that there is any direct relation betweenthem.
We shall now show how on a surface an infinity of isometric systems canbe obtained. For this purpose, let us suppose that apart from the systemobtained by Eqs. (2-5) and (2-6) and expressed by Eq. (2-8) there existsanother isometric system, given in terms of (u, v) by
a = a(u, v) = constant, 0 = 3(u, v) = constant. (2-9)
Then there exist two functions X(u, v), A(a, p) such that
ds2 = X (du2 + dv2) = A(da2 + d/32).
But (au = 8a/8u, etc.):
dal + d,2 = (aU + O'u) due + 2(a,.a,, + NuO.) du dv + (a! + 02.) dv2.
The necessary and sufficient conditions that the system (2-9) be isometricare therefore:
ati+RL = a;+Qo; 0.
If we write, for a moment, IA = au/Ru = -O./a., then
(µ2 + 1)ao = (/A2 + 1)0U.
Now µ2 + 1 ,E 0, since u = ±i leads to a linear relation between a and /3.Hence either
a, = Nu, au =-/v, (2-10)
172 SOME SPECIAL SUBJECTS
or
ao =-O., a = #V.
[CH. 5
(2-11)
Eqs. (2-10) and (2-11) are known as the Cauchy-Riemann equations, andare the conditions that, in the case (2-11) a + i$, and in the case (2-10)a - ifl, is an analytic function of the complex variable u + iv:
a ± i$ = f(u + iv).*
This leads us to the theorem:When u ^ constant, v = constant form a system of isometric coordinates,
then all other isometric systems are given by
Rf(u + - iv) = a, If (u + iv)
where Rf(u + iv) and I f (u + iv) are the real and imaginary parts of an arbi-trary analytic function of the complex variable u + iv.
If, for example, we take a + i$ = (u + iv)$, then the system defined byu3 - 3uv2 = constant, 3u2v - v3 = constant forms an isometric system,if u = constant, v = constant forms such a system.
With the aid of these isometric systems we can map one surface uponanother in an infinity of ways, depending on an arbitrary analytic functionof a complex variable. If, namely, we refer a surface S to an isometricsystem of coordinates (u, v), and another surface likewise to an isometricsystem of coordinates (ul, v1), then the correspondence u = ul, v = vi es-tablishes a conformal mapping of S upon Si.
Rectangular cartesian coordinates (x, y) in the plane form an isometricsystem. Consequently, any mapping of a surface upon the plane such thatan isometric system of the surface corresponds to the lines x = constant,y = constant in the plane is a conformal mapping, and these mappings arethe only conformal mappings of the surface on the plane. Any analytic func-tion of a complex variable performs such a mapping.
EXAMPLE. For a rotation surface (Chapter 2, Eq. (2-13)):
ds2 = (1 + f'2) dr2 + r2 d'P2 = r2/I
1of/2
dr2 + d
This ds2 can be transformed to isometric coordinates (u, v) by the trans-formation:
u= + dr, v=gyp.r
* See, e.g., C. E. Titchmarsh, The theory of functions, Oxford, Clarendon Press,1932, p. 68.
5-2] CONFORMAL MAPPING 173
A rotation surface can therefore be conformally mapped upon the plane bythe transformation
x=qv, .7 =u,
which transforms the meridians rp = constant into lines parallel to theY-axis, and the parallels u = constant into lines parallel to the X-axis.Equally spaced meridians pass into equally spaced lines x = constant, butthe spacing of the parallels is changed.
In particular, if f (r) = ate, r = a cos 0, f (r) = a sin 0, we obtain in
x = 4p, y = J sec 0 d0 = In tan (2 +!E)
a conformal representation of the sphere on the plane, in which p = 0,0 = 0 corresponds to x = 0, y = 0; the equator 0 = 0 to the X-axis; andequally spaced parallels 0 = constant correspond to lines parallel to theX-axis at ever-increasing distance when 0 increases, until North and Southpole are mapped at infinity. This is the Mercator projection (Fig. 5-6),which therefore, in the way in which it is commonly used in our atlases formaps of the world, is quite faithful near the equator, but gives an exag-gerated impression of the dimensions of Arctic regions. Loxodromes onthe sphere are mapped into straight lines on the map (Section 2-2).
150° 0° 30° 60° 900 1200 1500 180°
ti60
30
0°
30
60
0
°
Fla. 5-6
174 SOME SPECIAL SUBJECTS [CH. 5
Gerhard Mercator, latinized for Kremer (1512-1594), belongs with such menas Ortelius and Blaeu to the school of great Flemish-Dutch cartographers ofthe sixteenth and seventeenth centuries. He introduced the "Mercator" pro-jection in his famous world map of 1569 (on which the term "Norumbega" isused for a section of present New England). This projection had alreadyoccasionally been used. Mercator was aware of the conformal character of hisprojection.
Another. conformal map of the sphere on the plane is found by trans-forming by means of
u = In ri, (P = 'Pi
the system of polar coordinates of the plane (ri, Bpi) into an isometric system:
ds2 = dri + rf d(pi = ri(du2 + d
The resulting conformal mapping is accomplished by
u = fsec o dO = In tan 2+47r + c,
or, taking the integrating constant c = In 2a:
rl = 2a tan (2 + 4), 'P1=1P
This is the stereographic projection, and can be obtained by projectingthe points of the sphere from the North pole on the tangent plane at the
FIG. 5-7
5-3] ISOMETRIC AND GEODESIC MAPPING 175
FIG. 5-8
South pole (Fig. 5-7). Meridianspass into straight lines through theSouth pole S, parallels into concen-tric circles with S as center. Thespacing of the meridians is pre-served, but equally spaced parallelsare mapped into circles at ever-increasing distance when 0 decreasesfrom 7r/2 to -ir/2. The projectionis quite faithful near the South pole;the North pole is at infinity. Loxo-dromes on the sphere are mappedinto logarithmic spirals. Stereo-graphic projection from S to thetangent plane at N is obtained bychanging B into -0 (Fig. 5-8).
This projection was known to Ptolemy (c. 150 A.D.), who described it in hisGeographia; it may be due to Hipparch (c. 150 s.c.). It was used for mapprojections by the Flemish mathematician Gemma Frisius (1540). The namewas introduced in a book on optics by the Belgian author F. d'Aiguillon (1613).
Our theory of conformal mapping is due to Gauss, whose paper on the repre-sentation "of a given surface upon another such that the map is similar in itssmallest parts" appeared in 1822 (Werke IV, pp. 193-216; partial Englishtranslation in D. E. Smith, Source book in mathematics, 1922, pp. 463-475).The name isothermic is due to G. Lame (1833); see his Legons sur les coordonneescurvilignes (1859). The name isometric is due to Bonnet.
5-3 Isometric and geodesic mapping. A mapping (2-2) is isometricwhen at two correspondent points the first differential forms are the same.Hence E = E1, F = F1, G = G1, or
E due + 2F du dv + G dv2 = E1 due + 2F1 du dv + G1 Al. (3-1)
We have already met such mappings when we discussed bending (Sec-tion 3-3) and surfaces of constant curvature (Section 4-5). In Section 3-4we mapped a catenoid isometrically on a right helicoid.
The fact that isometry involves the preservation of certain invariantssuch as the Gaussian curvature implies that two arbitrary surfaces cannot,as a rule, be isometrically mapped upon each other. There are classes ofisometric surfaces, such as developable surfaces or surfaces of Gaussiancurvature 1. Surfaces which can be transformed into each other by bend-ing are isometric, and in this case the surfaces are also called applicable.The terms applicable and isometric are often identified, but it is not apriori clear that the existence of an isometric correspondence between two
176 SOME SPECIAL SUBJECTS [CH. 5
surfaces also means that one surface can pass into the other by bending,that is, by a continuous isometric transformation. The relations betweenapplicability and isometry therefore need careful investigation. Resultson the local problem have been reached in the case of surfaces of zerocurvature, where we know that the two concepts are identical (Section4-6). A much larger class of surfaces is covered by a theorem of E. E. Levi,which states that two isometric surfaces of zero or negative curvature arealways applicable provided certain conditions of differentiability are satis-fied. Two isometric surfaces of positive curvature either are applicableto each other or one surface is applicable to a surface symmetrical to theother; in this case the surfaces must be analytic.
Levi's paper, written at the request of L. Bianchi, can be found Atti Accad.Torino 43, 1907-08, pp. 292-302.
The problem of isometric mapping can be conceived in two differentways:
1. Given two surfaces S and S, with given first fundamental forms
ds2 = E due + 2F du dv + G dv2, ds1 = E, du,2 + 2F du, dv, + G, dvl,
to find whether there exists a correspondence
u, = u,(u, v), v, = v,(u, v), (3-2)such that ds2 = dsl.
2. Given a positive definite quadratic form E due + 2F du dv + G dv2, tofind all surfaces for which this form can be considered as the first differentialform. In other words, assuming that there is a surface S for which thegiven form is the ds2, to find all surfaces which can be isometrically mappedon S.
The first problem is sometimes called after Minding. It is the simplerof the two problems, and can be solved in each particular case by means of afinite sequence of differentiations and eliminations. We have solved itin Chapter 4 for the case that S and S, have constant Gaussian curvature,where we found oo3 transformations (3-2) which establish the correspon-dence. However, we cannot conclude from this case, in which equality ofGaussian curvature means isometry, that two surfaces can always beisometrically mapped upon each other if a correspondence (3-2) can befound such that the Gaussian curvature at corresponding points is equal.For instance, the surfaces
x = u cos v, y = u sin v, z = In u (a rotation surface)x = u cos v, y = u sin v, z = v (a right helicoid)
(3-3)
have at corresponding points (u, v) the same Gaussian curvature K =- (1 + u2)-2, but are not isometric.
6-3] ISOMETRIC AND GEODESIC MAPPING 177
Criteria for the possibility of the isometric mapping of two surfaces were firstestablished by F. Minding, Journal fur Mathem. 19, 1839, pp. 370-387. For anexposition of the theory, see L. P. Eisenhart, Differential geometry, pp. 323-325.
The second problem is sometimes called after Bour. We have solved itfor the case that the differential form is of zero curvature, e.g., due + dv2,when the answer is given by the totality of developable surfaces. When weask, in this case, for all solutions of the form z = f(x, y), then the answer isgiven by all solutions of the differential equation rt - s2 = 0 (Exercise 2,Section 2-8). In the case of differential forms of constant curvature wehad to be satisfied with a sketch of those solutions which are surfaces ofrotation (Section 4-6). The great variety of solutions in this particularcase may give some idea of the complexity of Bour's problem. It leads,in the general case, to the study of the solutions of a partial differentialequation of the Monge-Ampere type:
rt - s2 + Ar + Bs + Ct + D = 0, (r = 1322/19x2, etc.)
where A, B, C, D are functions of x, y, z, p and q.* In the case that thecoordinate lines are isotropic:
ds2 = 2F du dv, F = F(u, v),
this equation is (p = ax/1u, etc.) :
rt - s2 - qr(8 In F) - pt(8 In F)_ (8u, In F) (F - 2pq) - pq(e In F) (8 In F).
The solutions of this equation have only been fully investigated in somespecial cases, such as paraboloids of revolution. For F = constant weobtain the developable surfaces.
This problem received attention when in 1859 the Paris Academy proposedit as a subject of a contest. The prize went to Edouard Bour (1832-1866),whose paper, according to Liouville, "could be taken as a beautiful memoir byLagrange." The principal part of this paper was published in the JournalEcole Polytechn. 22 (1862), pp. 1-148. Honorably mentioned were papers byBonnet and Codazzi. For an exposition of Bour's problem see Darboux'Legons III, also L. P. Eisenhart, Differential geometry, Chap. 9.
Geodesic mapping preserves geodesics. An isometric mapping is alsogeodesic, but not every geodesic mapping is isometric. Since two arbi-trary surfaces cannot in general be geodesically mapped upon each other,
* In E. Goursat, Legons sur l'integration des equations aux derivees partielles dusecond ordre (1896), p. 39, such equations of the Monge-Ampere type are investi-gated.
178 SOME SPECIAL SUBJECTS [CH. b
we can ask for such classes of isometric surfaces which can be mappedgeodesically on other classes of isometric surfaces. This problem hasbeen fully solved. We first prove a special case of this theorem.
The only surfaces which can be geodesically mapped upon the plane arethose of constant curvature. This theorem, discovered by Beltrami, isproved by referring the surface to that system of coordinates (u, v) whichin the mapping corresponds to cartesian coordinates in the plane. Thegeodesics of the surface are then linear expressions in u and v, and con-versely, if there exists a coordinate system (u, v) on the surface in which thegeodesics are expressed in the form au + by + c = 0 (a, b, c constants),then it can be geodesically mapped on the plane. This means that theequation of the geodesic lines (Chapter 4, Eq. (2-3a)) must be identicallysatisfied when v" = d2v/due = 0:
r11 = r22 = o, r11 = 2ri2, r22 = 2r12.
The Gaussian curvature K then satisfies the equations (see Chapter 3,Eqs. (3-3)) :
KE = r12r12 - aur12, KF = ri2ri2 - avriz,KG = r12r12 - avri2, KF = ri2ri2 - auri2.
(3-4)
Since a;ur12 = au,r12, we obtain by differentiating the first two equa-tions of (3-4) and using (3-4) again:
EKv - FK K(Ev - Fu) + 2r212 avri2 - rl2aur12 - rizauri2_ -K(EE - 2r212 ri2ri2 - 2KFri2 - r12ri2ri2
+ KEr12 - r212 r12r12 + KFr12
=-K(E, - Fu) + K(-Fr2 + Erl2)-12
Substituting into this equation the expressions for the Christoffel symbols,Chapter 3, Eqs. (2-7), we finally conclude (see Exercise 20, Section 3-3):
EKv - FK = -K(EE - K(Ev - 0.
Differentiating the last two equations of Eq. (3-4), we obtain in a similarway:
FKv - GKu = 0.Hence
K is constant.
5-3] ISOMETRIC AND GEODESIC MAPPING 179
Conversely, if K is constant, then geodesic mapping is possible. Thiscan be shown by mapping the sphere geodesically on the plane by means ofthe relations
x = cot 9 cos'p, y = cot 0 sin p, (3-5)
where 0 and are, as usual, the latitude and longitude of the sphere and(x, y) are rectangular cartesian coordinates in the plane. Indeed, allgeodesics of the sphere lie in planes through the center, and are therefore(compare with Chapter 2, Eq. (1-7)) given by:
A cos 0 cos p + B cos 8 sin w + C sin 0 = 0, (A, B, C, arbitrary constants),
which by virtue of Eq. (3-5) passes into the equation of all straight linesin the plane:
Ax + By + C = 0.
This transformation maps great circles through a point on the sphereinto straight lines through a point in the plane, the meridians P = constantpassing into the lines y = x tan c, the parallels 0 = constant into the circlesx2 + yz = cotz 0. The plane corresponds to one of the hemispheres0 < 0 it/2 or -a/2 < 0 < 0.
As to the problem of the geodesic mapping of arbitrary surfaces, it canbe shown that in general the only geodesic mappings are isometric mappingswith or without similarity. Excepted are two classes of surfaces, whichare specified by the following theorem of Dini-Lie.
Two surfaces can be geodesically mapped upon each other without isometryor similarity when their first fundamental forms, by means of correspondingcoordinates (u, v), can be cast either into the form of Dini:
dsz = (U - V)(dua + dvz), dsi =(1V - UJ(Uz + y), (3-6)
or into the form of Lie:
dsz = (u + V) du dv, de, = u 2v8V du dv - u 4v4z
dvz, (3-7)
where U is a function of u only and V a function of v only.The form (3-6) characterizes the Liouville surfaces (see Section 4-2,
Exercises 18 and 19). The form (3-7) characterizes certain surfaces dis-cussed by Lie. They are real only for some values of V, and the corre-spondence itself is imaginary. The only real surfaces which can be geo-desically mapped on each other by means of a real transformation withoutisometry or similarity are given by Eq. (3-6).
180 SOME SPECIAL SUBJECTS [CH. 5
For the proof of the theorem of Dini-Lie we refer to A. R. Forsyth, Lectureson differential geometry, pp. 243-254, or to Darboux' Lesons III, pp. 40-65.Beltrami's theorem was first published in 1865 (Opere I, pp. 262-280). It wasfollowed by the paper of U. Dini, Annali di Matem, 3 (1869),,pp. 269-293, afterwhich S. Lie gave his supplementary theorem in 1879, see Math. Annalen 20(1882), p. 419. (See also Exercise 11 below.)
EXERCISES
1. Theorem of Tissot. In any real mapping of real surfaces which is not con-formal there exists a unique orthogonal system of curves on each surface to whichcorresponds an orthogonal system on the other surface (Tissot, Nouvelles Annalesde Mathem. 17, 1878, p. 151).
2. A mapping which is both geodesic and conformal is either isometric or asimilitude.
3. Inversion. If for the two points P(x) and Pi(a,) of space the relationx, = a2x/(x x) exists, we say that we have established an inversion. Show that(a) under inversion spheres remain spheres and (b) the mapping established be-tween two surfaces by inversion is conformal.
4. Show that any mapping of one surface upon another which preserves theasymptotic lines also preserves conjugate systems.
5. The sum of the angles of a triangle on a surface of revolution, of which thesides are loxodromes, is equal to two right angles.
6. Two surfaces of constant curvature K1, K2, K1 ' K2, admit a similitude.7. Establish a conformal mapping of the pseudosphere of curvature -1, for
which (Section 4-6):ds2 = du2 + e2u dv2,
upon the plane such that the geodesics pass into the circles x2 + (y - a)2 = P.
8. Equiareal map. The mapping of two surfaces x = x(u, v) and x, = x1(u, v)preserves the areas of corresponding figures if and only if
EG-F2=E,G,-F;.9. Show that if we project every point P of a sphere on the cylinder which is
tangent to the sphere at the equator, and then develop the cylinder into a plane,we obtain an equiareal map of the sphere upon the plane. What is the image ofthe meridians and of the parallels of the sphere?
10. Show that the mapping of the sphere on the plane
x = a sin 0 + f(,p), y =
where f ((p) is arbitrary, is also equiareal.11. Prove the theorem of Dini (hence the part of the theorem of Section 5-3 ex-
pressed by Eq. (3-6)) by selecting on the geodesically mapped surfaces a coordinatesystem according to Tissot's theorem (Exercise 1), so that F = F, = 0. Hint:Show that (E,/G',): (E/G2) is independent of u, and that (G,/E;) : (G/E2) is inde-pendent of v.
5-3] ISOMETRIC AND GEODESIC MAPPING 181
12. Find the complex function by which we obtain the stereographic projectionfrom the Mercator projection.
13. Polyconic projections. These are mappings of the sphere upon the plane inwhich the parallels are mapped into a system of circles whose centers C lie on astraight line corresponding to a meridian (the "central" meridian). In the so-called American polyconic projection:
(a) each parallel is a segment of a circle, the developed base of the cone tangentto the sphere along this parallel, the equator being a straight line,
(b) parallels equally spaced along the central meridian are equally spaced alongthis meridian in the map,
(c) the scale along the parallels remains constant, so that for all parallel linesthe unit of measure is represented by the same distance.
Show that (a) a = N sin 0, when a is the angle with the central meridian of theline connecting a point P of longitude N from the central meridian and latitude 0with its corresponding center C,
(b) The projection can be given by
x = a cot 0 sin a,y = a0 + a cot 0(1 - cos a),
(c) The projection is neither conformal nor equiareal.
40 40
20\ N. I 1\ \ / I I /1 /2070
4050 50
FIG. 5-9
40
182 SOME SPECIAL SUBJECTS [CH. 5
Fig. 5-9 shows a map of the earth in this projection. This projection, ex-tensively used by the U. S. Coast and Geodetic Survey, seems to have beendevised by Superintendent F. R. Hassler (1770-1843) for the charting of thecoast of the U. S. A. See 0. S. Adams, General theory of polyconic projections,Department of Commerce, U. S. Coast and Geodetic Survey, Serial No. 110(1919), especially pp. 143-152.
5-4 Minimal surfaces. We have defined minimal surfaces as surfaceson which the asymptotic lines form an orthogonal system (Section 2-8).This means that the Dupin indicatrix consists of two conjugate rectangularhyperbolas (that is, hyperbolas with perpendicular asymptotes: x2 - y2 =± 1) and that consequently Kl = - K2, and the mean curvature M is zero(Chapter 2, Eq. (7-2)) : Minimal surfaces are surfaces of zero mean curvature.The orthogonality of the asymptotic lines can be expressed by the relation(see Chapter 2, Eq. (7-2) or Exercise 6, Section 2-3) :
Eg - 2Ff + Ge = 0. (4-1)
Examples of minimal surfaces have been found in the plane, the catenoid,and the right helicoid (Section 2-8).
Minimal surfaces are sometimes defined as surfaces of the smallest areaspanned by a given closed space curve. We have seen in the case of geo-desics that this definition by means of a minimum condition of the calculusof variations is not always satisfactory. Consequently we preferred forgeodesics the definition as lines of vanishing geodesic curvature. We shallnow show that our definition of minimal surfaces as surfaces of zero meancurvature has exactly the same relation to the problem of the surface ofminimal area in a given contour C as our definition of geodesics has to theproblem of the shortest distance.
To show this, let a very small deformation of a surface x = x(u, v) begiven by the equation
xl = x + eN, (4-2)
where e is a small quantity and like x and N a function of u and v. Then
x + eN + e N,avxl=x.+eNv+e. ,
and we find for the coefficients of the first fundamental form of the deformedsurface, neglecting terms of higher order in e:
E1 = E - 2ee, F1 = F - 2ef, G1 = G - 2eg. (4-3)
5-4] MINIMAL SURFACES
Hence, introducing the mean curvature M (Chapter 2, Eq. (7-2)):
183
E1G1 - Fl = (EG - F2) - 2e(Eg - 2Ff + Ge) = (EG - F2)(1 - 4eM),
and integrating over an area enclosed by a fixed contour C:
JfVE1G1 dudv=JJ E du dv - 2f feM Edudv.
This formula can be written in the form
JJ
BA = - 2ff feM dA, (4-4)
where dA is the element of area of the original surface and SA is the so-called (first) variation of the area enclosed by the fixed contour C. Thisformula (4-4) may be compared to that for the (first) variation of the lengthof a curve (Chapter 4, Eq. (4-1)) fixed at the ends. Where in previoussections we had occasion to compare the ordinary or geodesic curvature of acurve with the Gaussian curvature of a surface, we have here an analogywith the mean curvature.
The first variation oA vanishes for all a if M = 0, that is, if the meancurvature vanishes. This can be expressed in the following words:
If there is a surface of minimum area passing through a closed space curve,it is a minimal surface.
We can also find this result (compare with Section 4-4) by means of thegeneral rules of the calculus of variations. Let z = f(x, y) represent a surface;then our problem is to find the Euler-Lagrange equation of the variationalproblem (compare Exercise 3, Section 2-3):
S f f 1+p2+g2dxdy=S f fF(z,x,y,p,q)dxdy=0.
Hence according to the rules of the calculus of variations
8F d OF d aF= 0,8z dx Sp dy 8q
or
r(1 + q2) - 2pqs + t(1 + p2) = 0,
which is Lagrange's equation of the minimal surfaces (Exercise 3, Section 2-8)and equivalent to the condition M = 0.
* See F. S. Woods, loc. cit., Section 4-4.
184 SOME SPECIAL SUBJECTS [CH. 5
Let us now introduce imaginary elements and admit the isotropic curvesas parametric lines. Then E = 0, G = 0, F 5-4- 0, and the condition M = 0becomes
Ff = 0 or f = 0. (4-5)
In this case, according to Chapter 3, Eq. (2-7) :
r12 = r12=0,
which, according to the Gauss equations (2-6), Chapter 3, shows thatvanishes. The surface is thus a translation surface (Chapter 3, Eq. (2-14))and its equation can therefore be written
x(u, v) = U(u) + V(v), (4-6)
where U = U(u) and V = V(v) are functions of u and v respectively.Moreover, since the curves u = constant, v = constant are isotropic, thecurves U = U(u) and V = V(v) are isotropic curves. This form (4-6) isdue to Monge. Hence we can state this theorem, in the formulation of Lie:
Minimal surfaces can be considered as surfaces of translation, of which thegenerating curves are isotropic.
In Chapter 1, Eq. (12-8) we have given an explicit expression for thecoordinates of an isotropic curve in terms of an arbitrary function. Eq. (4-6)therefore allows us to give an explicit expression for the coordinates of anyminimal surface in terms of two arbitrary functions: one, f(u) in u only, theother, g(v), in v only. Such surfaces are, as a rule, imaginary. By select-ing the f(u) and g(v) in such a way that U(u) and V(v) are conjugate imagi-nary, we obtain all real minimal surfaces. Their equation can be writtenas follows (compare with Chapter 1, Eq. (12-8)):
x= (u2- 1)f"- 2uf'+2f+(u2- 1)7"-2f'+27,y = i[(u2 + 1)f" - 2uf' + 2f] - i[(u2 + 1)f" - 2uf' + 27],z = 2uf" -2f'+2W"-2f',
(4-7)
where u is the complex conjugate of u and 7 of f. When u = a + i/3,2G=a-i/3, f+i¢,Eq. (4-7) expresses x as x(a, a). The formulas (4-7) are known as theformulas of Weierstrass.
From these equations we can compute the fundamental quantities ofthe minimal surface. We find (writing for the sake of convenience innotation u = v, 7 = g):
5-41
Hence:
MINIMAL SURFACES 185
xu((u2 - i)f,,,f i(u2 + 1)f / 2uf,,,)xv((v2 - 1)g"', i(v2 + 1)g,,, 2vg,,,)
xuu(2uf+ (u2 - 1)f iv, 2iuf"' + i(u2 + 1)f 'v, 2f,,, + 2uf iv)Xuv(O, 0, 0)
xvv(2vg"' + (v2 - 1)gi,,, 2ivg"' + i(v2 + 1)giv, 2vgiv).
E=G=0, F=2(uv+1)2f'lt 9 M
Nu+v u-v -uv+1 (4)
(uv+1'auv+1' uv+1 -8e=2f,,,, f = 0, g=2g,,,
The equation of the asymptotic lines is
fill due + gill dv2 = 0,
and that of the lines of curvature is
f,,, due - g,,, dv2 = 0.These equations show that both the asymptotic lines and the lines of curva-
ture on a minimal surface can be found by means of quadratures.By introducing the lines of curvature as parametric lines by the equa-
tions
du1 = -\/f, du + Vg, dv, i dv1 = V f' du - ,\Ig,,, dv,
where we take g1-" as the conjugate imaginary of v'f"', then with respectto these real parameters the first differential form becomes, since
du; + dv2, = 4V'f"' g"' du dv:
ds2 = ).(dull + dv12), X = X(u1, v1),
and by introducing the asymptotic lines as parametric lines by the equations
due = f"' du + i g"' dv, i dv2 = f"' du - iv'g"' dv,
we obtain the first fundamental form as
ds2 = µ(du2 + dv2), µ = µ(u2, Q.
Both the lines of curvature and the asymptotic lines on a minimal surfaceform an isometric system.
From the expression (4-8) for N we derive by differentiation that
NuNu=N.N=0.
186 SOME SPECIAL SUBJECTS [CH. 5
This means that, if do is the first fundamental form of the spherical image(Section 2-11, Exercise 6; also p. 156) :
dr2 = 2N.u N. du dv = 2pF du dv = p ds2,
where p is some function of position:A minimal surface is mapped conformally on its spherical image. The
isotropic curves of the minimal surface are mapped into the isotropic linesof the sphere. And since conformal mapping preserves isometric systems,both the lines of curvature and the asymptotic lines of the minimal surfaceare mapped into isometric systems on the sphere.
When Eq. (4-6) is replaced by
X = e'°U(u) + e 'V(v), (4-9)
where a is a constant, then the surfaces (4-9) not only continue to representminimal surfaces, but also represent real minimal surfaces when U and Vare conjugate complex. They are called, according to H. A. Schwartz,associate minimal surfaces. From Eq. (4-7) we can thus obtain a set of oo'real minimal surfaces by replacing f by e'°f and g = f by e-'"g. But underthis transformation, according to Eq. (4-8), F and N remain unchanged,as well as eg - f2 and EG - F2. The oo' surfaces (4-9) are therefore iso-metric, and at corresponding points the tangent planes are parallel. Theypass into each other by bending.
When we call the associate surface obtained for a = 7r/2 the adjointminimal surface,
y = iU - IV, (4-10)
then all associate surfaces z = z(u, v)can be expressed as follows in termsof y and the original surface x(a = 0):
z = x cos a + y sin a. (4-11)
We conclude from this that the endpoint of the vector z describes with
FIG. 5-10
varying a an ellipse of which the end points of the vectors x and y mark twovertices (Fig. 5-10). Hence, summing up:
A minimal surface admits a continuous isometric deformation (applica-bility), in which each point describes an ellipse.
Comparison of the equations of the lines of curvature and of the asymp-totic lines shows that in two adjoint minimal surfaces the lines of curvaturecorrespond to the asymptotic lines.
5-41 MINIMAL SURFACES 187
EXAMPLE. Take f = a(u In u - u); then
If
then
f = a In u,
u =-ae`",
f" = au'.
r = 2a(a + a 1),
x = a(a + a 1)(e"* + e-"*) = 2a(a + a1) cos lp = r cos tp,
y = ai(a + a-') (e" - e`11) = 2a(a + a-1) sin tp = r sin lp,
z=4-4lna, or r=2Gzb+e Z°4b=4a,4
which is a catenoid.For the adjoint surface (4-10) we take r1 = 2a(a - a1). Then
x = ia(a - a')(el" - e 1") = r1 sin lp = r1 sin (p1,
y = a(a 1 - a) (0' + e +,o) =-r1 cos (p = r1 cos p1,
z = -2ia In a 21.p = 4alp = (4ir - 4(p1)a,
(c1=7r -gyp)
which is a right helicoid.We have thus shown that a right helicoid and a catenoid are not only
minimal surfaces, but that one surface can pass into the other by a continu-ous sequence of isometric transformations (see Section 3-5).
Minimal surfaces belong to the best-studied surfaces in differential geometry.Their theory was initiated by Lagrange as an application of his studies in thecalculus of variations (1760-1761, Oeuvres I, p. 335). Monge, Meusnier, Le-gendre, Bonnet, Riemann, and Lie contributed to the theory; it was Meusnierwho discovered the two "elementary" minimal surfaces, the catenoid and theright helicoid. Karl Weierstrass (Monatsber. Berlin Akad., 1866) and H. A.Schwartz developed the relationship between the theory of complex analyticfunctions and the real minimal surfaces (see H. A. Schwartz, Ges. math. abh. I).A full discussion of the minimal surfaces, including the history, can be foundin Darboux' Legons I, pp. 267 if.
In the theory of capillarity the importance of the minimal surfaces as surfacesof least potential surface energy was illustrated by the experiments of Plateau,Statique experimentale et thiorique des liquides (1873), who dipped a wire in theform of a closed space curve into a soap solution and thus realized minimal sur-faces as soap films. The problem of Plateau is the problem of determining theminimal surface through a given curve; it has been studied in great generalityby J. Douglas. See Solution of the problem of Plateau, Trans. Amer. Mathem.Soc. 33, 1931; also American Journal Mathem. 61, 1939. See for further detailsR. Courant-H. Robbins, What is mathematics? (1941), p. 385; R. Courant, Actamathematica 72 (1940), pp. 51-98. For soap film experiments with minimalsurfaces: R. Courant, Amer. Math. Monthly 47, 1940, pp. 167-174.
188 SOME SPECIAL SUBJECTS
EXERCISES
[CH. 5
1. Show that, apart from the plane, the right helicoid is the only real ruledminimal surface. Hint: Consider the orthogonal trajectories of the rulings asBertrand curves. (E. Catalan, Journal de mathkm. 7 (1842), p. 203.)
2. Show that Eq. (4-11) represents an ellipse.
3. Show that the problem of finding all minimal surfaces is identical with theproblem of finding two functions p = p(x, y) and q = q(x, y) such that bothp dx + q dy and (p dy - q dx)/v'l + p2 + q2 are exact differentials (Lagrange).
4. Show that all the equations of Weierstrass for real minimal surfaces can becast into the form
x = R[(r2 - 1)f" - 2rf' + 2f] = R f (1 - r2)F(r) dr,
y = R[i(r2 + 1)f" - 2irf' + 2if] = R fi(r2 + 1)F(r) dr,
z = R[2rf" - 2f] = R J 2rF(r) dr,
where F(r) is an analytic function of the complex variable r, and R indicates thereal part; f"' = F.
5. Find the value of F(r) in Exercise 4 which leads (a) to the right helicoid,(b) to the catenoid.
6. Minimal surface of Enneper. Here F(r) = 3. Show that this surface isalgebraic and that its lines of curvature are plane curves of the third degree (En-neper, Zeitschrift fur Mathem. u. Physik. 9 (1864), p. 108).
7. Minimal surface of Henneberg. Here F(r) = 1 - r-'. Show that thissurface is algebraic (Henneberg, Annali di Matem. 9 (1878), pp. 54-57). This sur-face is a so-called one-sided or double surface, which means that without any breachof continuity we can pass from one side of the surface to the other side, as on theMoebius strip.)
8. Minimal surface of Scherk. Here F(r) = 2/(1 -r'). Show that thecartesian equation of this surface is
(cos x)e = cos y,
and show that this surface is also a translation surface with respect to two familiesof real curves. (Scherk, Crelle's Journal f. Mathem. 13 (1835). This surface wasthe first minimal surface discovered after Meusnier's discovery of the catenoid andthe right helicoid.)
9. Show that the only real surfaces which are mapped conformally upon theirspherical image are the spheres and the minimal surfaces.
5-5] RULED SURFACES 189
10. Parallel surfaces. The locus of the points y which are on the normals tothe surface S, x = x(u, v), at constant distance X from x,
y = x + XN,
is called a parallel surface to S. Show that (a) N is the unit surface normal vectorof all parallel surfaces, (b) the parallel surfaces of a minimal surface are surfaces forwhich R, + R2 = constant, R, = Kl ', R2 = K2
5-5 Ruled surfaces. We have occasionally met ruled surfaces in ourdiscussion, but they were always of a particular type, such as developablesurfaces, right conoids (Section 2-2), or the sphere as the locus of imaginarylines (Section 2-8). We shall here present a general theory of ruled sur-faces, excluding from the beginning the ruled surfaces with imaginary lines.
We have already defined (Section 2-2) a ruled surface as a surface gen-erated by the motion of a straight line, its generating line, generator, orruling. When the surface is not developable, it is sometimes called a scroll.There are oo l generators on a ruled surface. Let i = i(u) be the unit vectorin the direction of the generating line passing through a point A (u) of anarbitrary nonisotropic curve C lying on the surface, of which the equationis x = x(u). A generic point P(y) of the ruled surface is given by theequation
y = s(u) + vi(u) = y(u, v); i i = 1. (5-1)
The directed distance AP is given by the parameter v (Fig. 5-11). Thegenerating lines are the parametric curves u = constant. The vectors
Fia. 5-11 FIG. 5-12
190 SOME SPECIAL SUBJECTS [CH. 5
i = i(u), drawn through the center 0 of the unit sphere, describe thedirector cone of the surface (Fig. 5-12); it intersects the sphere in a curve Clwith equation i = i(u), which can be considered as a spherical image of thesurface. The curve C on the surface is called its directrix.* When x is aconstant the surface is a cone; when i is a constant the surface is a cylinder.
Our formulas are somewhat simplified if we take as parameter u not thearc length of C, but that of C1,which is a quantity depending on the natureof the surface only and not on the arbitrary choice of the directrix. Thischoice of u, and the fact that i is a unit vector, leads to the identities:
=0.
The coefficients of the first and second differential form are obtained from
yu = x' + vi', yv = i; i' = di/du, x' = dx/du,yuu = x'' + vi", Y. = i', Y. = 0,
so thatE = x' - G=1, (5-2)
De = (x"x'i) + v(i"x'i) + v(x"i'i) + v2(i"i'i),Df = (i'x'i), g = 0,
D2 = EG - F2 = x' - x' - (x' i)2+2vx' i'+v2. (5-3)
From these equations we derive immediately for the unit normal theexpression
x'Xi+vi.'XIN = D
for the Gaussian curvature the expression
f2 (x/iii 2K EG - F2 (EG - F2)2'
(5-4)
(5-5)
and for the equation of the asymptotic lines
du(e du + 2f dv) = 0. (5-6)
The expression (5-4) for the normal vector shows that the tangent planechanges, in general, when its point of tangency moves along a generating
* We speak of the director cone, but of the directrix (curve) because cone ismasculine and curve is feminine in Latin. It would really. be better in Englishto use the term director in all cases, and then also speak of the Dupin indicator.
5-5] RULED SURFACES 191
line, always passing, of course, through the generating line. We knowthat N is independent of v when the surface is developable. The expres-sion (5-5) for K shows that this is the case when
p = (x'ii') = p(u) (5-7)
vanishes. This function p is called the distribution parameter. We thushave found that the necessary and sufficient condition that the ruled surface(5-1) be developable is that the distribution parameter vanish.
From Eq. (5-5) we conclude, since EG - F2 > 0 for real surfaces, thatthe Gaussian curvature of real ruled nondevelopable surfaces (scrolls) is nega-tive, except along those generators where p(u) vanishes.
Eq. (5-6) shows that the straight lines u = constant form one familyof asymptotic lines. The other family is given by the equation e du +2f dv = 0, or
dvdu
= A + By + Cv2,
where A, B, C are certain functions of u. Hence, comparing with Sec-tion 1-10, we have found the theorem:
The determination of the curved asymptotic lines of a ruled surface dependson a Riccati equation.
Since the cross ratio of four particular integrals of a Riccati equation is con-stant, and v is the directed distance, AP, we can immediately conclude thatthe cross ratio of the points in which four fixed asymptotic lines intersect thegenerating lines is constant.
Further investigation of ruledsurfaces is facilitated by the theo-rem of solid geometry that twogenerating lines which are not : y A;parauei nave a common perpen-dicular.* We exclude here the casethat generating lines are parallel,which means that we exclude cylin-ders. Let us now take two gener-ating lines which are close together,so that they can be given byu = constant, u + Au = constant. ICTheir common perpendicular, as Fio. 5-13
* G. Wentworth-D. E. Smith, Solid Geometry, Boston; Ginn & Co., 1913, p. 306.
192 SOME SPECIAL SUBJECTS [CH. 5
well as the point where it intersects the line u = constant, then assumes alimiting position for Au - 0. To find this position, let us take the gener-ating lines through the points A (x) and A, (x + Ax) of the directrix C, andlet SS1 be the common perpendicular to the generators through A and A1.
Let vector AS be vi, and vector A1S1 be (v + Av)(i + pi), let the unit vectorin the direction of SS1 be u, and the distance SS1 be Au (Fig. 5-13). Thevectors along the sides of quadrilateral AA1S1S, if taken with the appro-priate sense, add up to zero:
Ax+ (v+pv)(i+Ai) - uzu - vi = 0,
or, dividing by Au and passing to the limit Du -> 0:
x'+vi'+Vi -u, = 0.Since u is perpendicular to i and to i + Ai, we find that for Du -* 0:
so that (i and i' being unit vectors) u can be taken as
u=i'Xi.
(5-8)
The vectors i, i', u thus form a set of mutually orthogonal unit vectors.Eq. (5-8) shows how x' is decomposed in the direction of these vectors;hence
c' = u x' = (x'i'i) =-p, (5-9)
v =-x' it. (5-10)
The third equation, v' _ - x' i, does not express the derivative of the v ofEq. (5-10) with respect to u, but is the equation of the orthogonal trajectoriesof the generators F du + G dv = 0 (compare Eq. (5-2)).
Eq. (5-9) gives a new definition of the distribution parameter. Whenthe surface is developable v' = 0. This can be expressed by saying thaton a developable surface two consecutive generators intersect, or moreprecisely, that the distance between two generators (u) and (u + Au)is of higher order than Du.
Eq. (5-10) determines on every generator a certain point S, the centralpoint. The tangent plane at that point is called the central plane; itsunit normal vector is (except when the surface is developable, when thelimiting position of u is perpendicular to the tangent plane):
iXu=iX(i'Xi)=i'. (5-11)
5-5] RULED SURFACES 193
The locus of the points S is calledthe striction line of the surface. Ona tangential developable it coincideswith the edge of regression, on a cyl-inder it is indeterminate. Its tangentline does not have the direction of u,as can be seen in Fig. 5-14, which repre-sents a rotation hyperboloid on whichthe central circle is the striction line.Taking now the striction line as thedirectrix curve, which is always possiblewhen this line does not reduce to apoint (and this happens only when thesurface is a cone), then we have theadditional condition
x' it = 0.
FiG. 5-14
p2 = (x'ii')2 =
The v in Eq. (5-8) is now zero, since(5-8) relates the point A on the direc-
trix to the central point. Hence (see
Eqs. (5-9) and (5-11)):
x'xi= (uxi)a'=Pit.Moreover (see Eq. (5-7)):
x'-x' x' i 01 0
0 0 1
= x' x' - (x' . i)2. (5-12)
We can therefore write the unit normal N, according to Eqs. (5-4), (5-3),and (5-11), in the form
N /P it + Vu, (5-13)
vp2+v2 \/_+ v2
where the square root is positive. If p is the directed angle between thenormal vector at a point (v) and at the central point (v = 0) of a generator,we find
cos _--, sin (p = vv2
hence
tan cp =vp
(5-14)
194 SOME SPECIAL SUBJECTS [CH. 5
This equation expresses the theorem of Charles:The tangent of the directed angle between a tangent plane at a point P of a
generator of a nondevelopable ruled surface and the central plane is proportionalto the distance of P to the central point.
When v runs from - oo to + oo and p > 0, the directed angle p runs from-,r/2 to +lr/2; when p < 0, it runs from +7r/2 to -a/2. Therefore, whena point moves along a generator, the tangent plane turns 180°. The pointwhere the tangent plane has turned 90° is the central point (hence thename). When p > 0 the tangent plane turns counterclockwise, when p < 0it turns clockwise. This allows us to distinguish between left-handed andright-handed ruled surfaces, respectively. The asymptotic tangent plane isperpendicular to the central plane.
For developable surfaces Eq. (5-14) loses its meaning, but from Eq.(5-13) we see that in this case N = u, independent of v. Eq. (5-11), asalready observed, does not hold in this case. Eq. (5-12) here gives
X' . X1 - (X' j)2 = (X' x i). (X' X i) = 0,
which shows (isotropic directrix and isotropic generators have been ex-cluded) that x' X i = 0; the generators are tangent to the edge of regression.
Ruled surfaces were investigated first by Monge (in his Applications), whoestablished the partial differential equation satisfied by all ruled surfaces (it isof the third order), and then geometrically by Hachette. The present theory ismainly due to F. Minding, Journal fur Mathem. 18 (1838), pp. 297-302, andM. Chasles, Corresp. mathem. et phys. de Quetelet 11 (1839); also to Bonnet.To Chasles we owe the names central point and line of striction. The theorem onasymptotic lines can be found in a book by Paul Serret, Theorie nouvelle geo-metrique et mEchanique des courbes a double courbure (1860).
EXERCISES
1. Find the distribution parameter and the striction line of a right conoidx = V Cos u,
y = v sin u, z = f(u).
2. Verify in the case that in Exer-cise 1 f (u) =
v/r2- a2 cos2 u, r and a
constants, that p > 0 means a left-handed ruled surface, and p < 0 a right-handed one. This surface is the cono-cuneu8 (conical wedge) of Wallis (Fig.5-15).
3. The normals to a scroll along agenerator form a hyperbolic paraboloid.
z
FIG. 5-15
5-5] RULED SURFACES 195
4. The Gaussian curvature of a ruled surface is the same at two points of agenerator which are equidistant from the central point.
5. The cross ratio of four points of a generator of a scroll is equal to the crossratio of the four tangent planes at these points (M. Chasles).
6. Find the asymptotic tangent plane of a generating line of a hyperboloid ofone sheet.
7. Show that (a) the striction line of a hyperboloid of revolution is the centralcircle, (b) the rulings cut it at constant angle, and (c) the parameter of distributionis constant.
8. Show that the striction lines of the hyperbolic paraboloid a2 - bz z arethe parabolas in the planes
a3± by3=
9. Show that the striction line of the hyperboloid
x2 2 Z2
a b2 C2= 1
is the space curve formed by the intersection of the hyperboloid and the surfaceof the fourth degree,
a N2Z2(b2 + c + b2z2x2(C2 + a2)2 - 2 IZ = 0.
10. Show that the first fundamental form of a ruled surface can be cast into theform ds2 = due + ((u - a)2 - QZ) dv2, where a and 8 are functions of v alone.Then u = a is the equation of the striction line and
K= _02
[(u - a)2 - N ]2
11. Find the equation of the asymptotic tangent plane along a ruling of a ruledsurface.
12. The points on a generator of a ruled surface can be paired in such a way thatthe tangent planes at these points are perpendicular. If P, P, form such a pair,and S is the central point, prove that the product SP X SP1 is constant for allsuch pairs of points on the same generator.
13. Show that the distribution parameter of a right helicoid is constant. Isthe converse true?
14. A theorem of Bonnet. If a curve on a ruled surface satisfies any of the threeconditions, (a) of being a geodesic, (b) of being a striction line, (c) of intersectingthe generators at constant angles, then any two of these conditions implies the third.
15. A space curve is the line of striction on the surface of its binormals, the recti-fying plane being the central plane and the tangent the common perpendicular ofconsecutive binormals. The distribution parameter is - T.
196 SOME SPECIAL SUBJECTS [CH. 5
5-6 Imaginaries in surface theory. We already have had a few occasionsto refer to imaginaries in surface theory. In Section 2-7 we found thatasymptotic lines in regions of positive Gaussian curvature are imaginary,and in Section 2-8 we integrated the differential equation of the asymptoticlines for the case of a sphere. The asymptotic lines of a sphere were foundto be isotropic lines.
We shall now give a more detailed discussion of isotropic elements on anarbitrary surface. They are defined by the equation ds2 = 0, or
E due + 2F du dv + G dv2 = 0. (6-1)
When EG - F2 5,1 0 this equation defines two directions, which arrangethemselves to the net of isotropic curves (also called minimal curves). Sincethe distance of any two points on such curves is zero, they are the lines ofshortest real distance between their points, and it therefore seems that theycan be considered as geodesics. That this is the case can be shown by in-troducing the net of isotropic curves as the net of parametric lines on thesurface. Then E = G = 0, and
so thatds2 = 2F du dv, F = F(u, v), (6-2)
rii = rig = ri2 = rs2 = 0, r,, = F /F, rs2 = (6-3)
and this shows that the equation of the geodesic lines (Chapter 4, (2-3a))is satisfied for v = constant. By interchanging u and v we also prove thatthe curves u = constant are geodesics. The isotropic curves can be con-sidered as geodesics. We have used the isotropic curves in Section 5-2 forthe introduction of isothermic lines and in Section 5-4 for the investigationof minimal surfaces.
Let us now introduce imaginary surfaces by considering x = x(u, v) as ananalytic vector function in two complex variables. We can maintain mostof the conceptions of real surface theory by defining them by means of theiranalytic expressions, provided EG - F2 0 0. Thus we can define tangentplane, normal, first and second differential forms, asymptotic lines, linesof curvature, and conjugate lines. Geodesics are defined as curves of zerogeodesic curvature, although not of shortest length, except in special cases.
An exception must be made for the case that EG - F2 = 0, when allformulas and definitions in which EG - F2 occur either lose meaning orhave to be revised. Because of Eq. (2-8a), Ch. 2, this condition holds for allcoordinate systems on the surface, if it holds for one of them, and is there-fore a property of the surface itself. Such surfaces, for which at all points
EG - F2 = (x X x9) (xu X x9) = 0, (6-4)
5-6] IMAGINARIES IN SURFACE THEORY 197
are called isotropic surfaces. Since ds2 here is a perfect square, there existsonly one family of isotropic curves on these surfaces. The tangent planeexists at all points and is tangent to the local isotropic cone. The isotropicplanes discussed in Section 1-12 form a special case of isotropic surfaces.
It is convenient to introduce on these surfaces as one set of parametriclines u = constant, the curves for which ds2 = 0. Then ds2 takes the form:
ds2 = E due, E = E(u, v). (6-5)
From Eq. (6-5) we derive the following relations:
xu XU=E) xu x=0, xro x=0, x uro xu = - xuro xro = 0, x,, xon = 0.
so that we find for x,,, the expression
xvo = axu + 01. = Nxv, P = $(u, v).
We use here the proposition that when a a = 0 and b a = 0, then everyvector c for which c a = 0 lies in the plane of a and b. See Exercise 3, Section1-13.
Eq. (6-6) shows that the isotropic curves u = constant are straightisotropic lines. An isotropic surface is therefore a ruled surface. Let ustake an arbitrary nonisotropic curve x = x(t) on the surface. Then, bydefining the isotropic line through every point of the curve by the vectorfield u(t), we can write the equation of the surface as follows:
y = x(t) + µu(t) = y(µ, t)i u u = 0, (6-7)
where µ is a parameter varying along the isotropic lines. Writing
ds2 = E dt2 + 2F dt dµ + G dµ2,we find that since
the coefficienty:=x'+µu', YM=u,
From EG - F2 = 0 follows that F = 0. Now F = x' u, so that the neces-sary and sufficient condition that Eq. (6-7) represent an isotropic surfaceis that x' u = 0. However, this is also the condition that the surface (6-7)is developable. We can indeed determine p as a function of tin such a waythat the curve y(t) of Eq. (6-7) is tangent to the generating lines of the iso-tropic surface. The tangent vector to this curve must in this case havethe direction of u:
x' + µu' + µ'u = Xu, X = x(t).
198 SOME SPECIAL SUBJECTS [CH. 5
The vectors x', u', and u are coplanar, since x' u = u' u = u u = 0.We thus find for µ the value
x' x' u' x'
except for the case that u' = 0, when we have a cylinder with isotropicgenerators.
The identity of the two expressions for y follows from the fact that(x' x u') (x' x u') = 0 because x', u', and u are coplanar and (u x u') (u x u')=
Substitution of this value of µ into Eq. (6-7) gives us the edge of regres-sion of the developable surface (6-7). This curve, having isotropic tan-gents, is an isotropic curve. When it shrinks to a point, the surface is acone.
We can express these results in the theorem:The surfaces for which EG - Fz = 0 are isotropic developables, that is,
they are isotropic planes, isotropic cylinders, isotropic cones, or tangent sur-faces to isotropic curves.
Let us now suppose that the ruled surface (6-7) is not developable, butis still generated by isotropic lines. Then x' u 0 0, so that G = 0,EG - F2 = - F2 0 0. Furthermore :
Yee = x" +, µu", Yµ = U" Y. = 0,
so that g = 0, but f 0. (Also E 0,e 0.)The equation of the lines of curvature takes the form
dt2(Ef - eF) = 0, (6-8)
and the equation (6-3), Chapter 2, for the normal curvature in directiondµ/dt becomes
edt+fdi (6-9)K Edt+Fdµ
The case Ef - eF = 0 leads to surfaces for which K in all directions is thesame, hence to the sphere (Section 3-5), referred to one set of its isotropiclines as parametric curves t = constant. When Ef - eF 0 0 we see fromEq. (6-8) that the surface has only one set of lines of curvature, the isotropicgenerators. Moreover, we find that the equation for the principal curve,tures, Chapter 3, Eq. (7-1), now becomes
e - KE f - KFf - KF 0 - (f - KF)2 = 0,
5-6] IMAOINARIES IN SURFACE THEORY 199
which is a perfect square. Both principal curvatures are equal, and equal tothe normal curvature in the direction of t = constant.
The condition that both curvatures be equal in the real domain leadsback to the sphere, since the condition that the two roots of Eq. (7-1) ofChapter 2 are equal is
4(EG - F2) (eg - P) - (Eg + eG - 2fF)2 = 0, (6-10)
which for F = 0 is identical with
(Eg - Ge)2 + 4EGf2 = 0,or
Eg - Ge = 0, f = 0,
the case of an umbilic.When we admit imaginaries the sphere is not the only possibility, so that
we have established the existence of surfaces for which the two principal curva-tures are equal, although not all normal curvatures are equal. This was theproperty which led Monge to the discovery of these surfaces. However,Monge, concentrating on real figures, recognized only the one real curvewhich exists on these surfaces (since on every isotropic line there is one realpoint). He thus came to the startling result that these surfaces were reallycurves. "Ce r5sultat est extraordinaire," he concluded. At present,having been accustomed by the work of Poncelet and Chasles to the freeacceptance of imaginaries in geometry, we prefer to summarize as follows:
Those ruled surfaces of which the generating lines are isotropic, and whichare not developables, have one set of lines of curvature, the isotropic lines. Atall points the two principal curvatures coincide with the curvature in thedirection of the isotropic lines.
Monge discovered these surfaces in Chap. 19 of his Applications. A detailedstudy can be found in G. Scheffers, Anwendung II, pp. 283-286, 293-295.
EXERCISES
1. Show that the principal curvatures are equal when the fundamental forms Iand II have a factor in common.
2. Show that when we can introduce curvilinear coordinates such that E = 0,G = 0, e = 0, the curves v = constant are straight isotropic lines.
3. Show that the case of Exercise 1 leads to the nondevelopable ruled surfaceswith isotropic lines.
4. Show that when a surface has two families of straight isotropic lines, the sur-face is a sphere or a plane.
5. Show that there are two nondevelopable ruled surfaces with isotropic lineson which a given curve is an asymptotic line (G. Scheffers).
200 SOME SPECIAL SUBJECTS [CH. 5
6. Show that every isotropic cone is a quadratic cone.7. The differential equations of the lines of curvature and the asymptotic lines
can be written without the denominator BEG - F2 common to e, f, and g. Thisallows us to define such lines on isotropic developables. Show that on these sur-faces all curves can be considered lines of curvature and that the isotropic genera-tors are the asymptotic lines. Also show that all points can be considered umbilics.(F. S. Woods, Annals of mathem, 5 (1903-1904), pp. 46-50.)
8. The only surfaces whose element of are is an exact differential are isotropicplanes. (C. L. E. Moore, Journal Math. and Physics 4, 1925, p. 169.)
SOME PROBLEMS AND PROPOSITIONS
1. Courbure incline. Through every point P of a curve C, x = x(s), passes aunit vector of a field u1(s). Then we call du,/ds the curvature vector of C withrespect to the field u,. If du,/ds = K,112 (112 unit vector, K, = relative curvature ofC with respect to the field u1), derive the "Frenet formulas" for due/d8 anddu,/ds, where u3 = u, X u2. (Fig. 6-i, u = u,.)
This curvature was introduced by A. L. Aoust, loc. cit., Section 3-4, whocalled it courbure incline. W. C. Graustein, Trans. Am. Math. Soc. 36, 1934, pp.542-585, calls dul/ds the associate curvature vector of ur with respect to C.
Now let two congruences of curves be given on a surface with unit tangentvector field t,, t2 respectively, and let s,, sz be the respective arc lengths. Showthat the projections of dt,/ds2 and of dt2/ds, on the surface normal are equal.
Fro. 6-1 Fio. 6-2
2. Clothoid. Find the cartesian equation of the curve with natural equationsRs = a2 (a, a constant) and, taking the point of inflection as origin and the tan-gent as X-axis, show that the asymptotic points are given by x = y = (a/2)'-/rand -(a/2)V. This curve is also called the spiral of Cornu, and appears in thetheory of diffraction. It was introduced by James Bernoulli (Fig. 6-2). See E.Cesbro, Naturliehe Geometrie, 1901, p. 15.
3. Geodesic torsion. Show that the torsion of the geodesic of a surface withunit tangent vector t is given by u (dN/ds), where u is the vector defined inSection 4-1. This quantity is called the geodesic torsion ro. Show that rp = 0characterizes the lines of curvature.
4. A moving trihedron on a surface. If we introduce along a curve x = x(s)201
202 SOME PROBLEMS AND PROPOSITIONS
on a surface the trihedron (t, u, N) of Problem 3, then Eq. (1-8), Chapter 4, canbe complemented as follows:
= - K,t - r,N, dN = r U.dS =
KgU + K,N, du6 Ts
Prove from these equations that if two surfaces are tangent to each other along acurve and the normals to them in the points of the curve are similarly directed,the curve has the same geodesic curvature and the same geodesic torsion withrespect to both surfaces and the surfaces have the same normal curvature in thedirection of the curve. . (J. Knoblauch, Grundlagen der Differentialgeometrie,Teubner, Leipzig, 1913, p. 56, calls these formulas the general Frenet formulas ofthe theory of curves on surfaces. See also W. C. Graustein, Differential geometry,p. 165, and our Exercise 2, Section 4-8, and the Appendix.)
5. A theorem on Bertrand curves. If a curve C of constant curvature and acurve C of constant torsion are in such a one-to-one correspondence that thetangent at the corresponding points P, P, are parallel, then the locus of thepoints which divide P, P, in a constant ratio is a Bertrand curve. (C. Bioche,Bull. Soc. math. France 17, 1888-89, pp. 109-112. See also A. P. Mellish, loc. cit.,Section 1-13.)
6. Surfaces of constant width. We define an ovaloid as a convex closed sur-face with continuous nonvanishing principal curvatures (K > 0). Such a sur-face has two parallel tangent planes in every plane direction (opposite tangentplanes). When the distance between opposite tangent planes is the same forall directions, we call the ovaloid a surface of constant width (see Section 1-13 onovals and curves of constant width). Prove that for such a surface:
(a) the principal directions at opposite points are equal,(b) the mean curvatures at opposite points are equal,(c) its normals are double, that is, the normal at a point is also the normal at
the opposite point. (A. P. Mellish, loc. cit., Section 1-13; the theory is due toH. Minkowski, Ges. Werke II, pp. 277-279; see also W. Blaschke, Kreis and Kugel,Veit, Leipzig, 1916, pp. 138, 150.)
7. Find the expressions (Chapter 1, Eqs. (12-8)) for the isotropic curves byintegrating the equation
dx2 + dy2 = ds2, s = ix,by considering x and y as the coordinates of the point of the evolute of a curve C1.(Introduce as parameter the angle (p of the normal of C, with the X-axis, seeEnc. Math. Wiss. III D 1, 2, p. 26.)
8. Curvature of asymptotic lines. When Ka is the curvature of an asymptoticline l at a point P of a surface, and Ky that of the branch of the curve of intersectionof the surface and the tangential plane at P, then I Kyl = Kal. (E. Beltrami, 1865,Opere matem. I, p. 255.)
9. A theorem of Van Kampen. The tangent tin an asymptotic direction ata point P of a surface of negative curvature is tangent both to the asymptoticcurve C and the curve of intersection C1 of the surface and its tangent plane at S.Then (provided C, has its K # 0 near P) C, lies between C and t at P. (E. R.Van Kampen, Amer. Journal of Mathem. 61, 1939, pp. 992-994.)
SOME PROBLEMS AND PROPOSITIONS
10. Natural families of curves. The extremals of a variational problem
S fFds=0,
203
where F is an arbitrary function of x, y, z, form a natural family of curvesThrough every point passes (in general) one curve of the family in every direction.Show that for these curves the relation holds
k = s - s= grad in F,and derive from this relation that
(a) the centers of the osculating circles of all curves of a natural family whichpass through a point P lie in a plane 7r,
(b) the osculating planes at P pass through the line through P perpendicularto 7r.
(E. Kasner, Differential-geometric aspects of dynamics, Princeton Colloquium,1909, New York, 1913, 117 pp.).
11. A theorem of Bonnet on ovaloids. If the Gaussian curvature of an ovaloidK >, A-2, then the maximum distance of two of its points <TA. For this theo-rem of 0. Bonnet, Comptes Rend us Acad. Paris 40, 1855, pp. 1311-1313, see W.Blaschke, Differentialgeometrie I, pp. 218-220.
12. A theorem of W. Vogt. If the are ABof a plane curve has the property that K(>0)decreases from A to B monotonically and if Tthe tangent at A does not meet the arc ABelsewhere, then L TAB > L TBA, if TA andTB are the tangents at A and B and T is onthe same side of AB as are AB. (See Fig.6-3.) (W. Vogt, Crelle's Journal fur Mathem.144, 1914, pp. 239-248; S. Katsura, T6hoku AMath. Journ. 47, 1940, pp. 94-95.) FIG. 6-3
13. Liouville's Theorem on conformal transformations of space. The only con-formal mappings of space on itself are inversions, similitudes, or a combination ofboth. (J. Liouville, Note VI to Monge's Applications.) One demonstrationof this theorem can be given by taking in Section 3-4 II1 = H2 = H3 =H.Then H = k (U2 + V2 + W2)-', where U = U(u), V = V(v), W = W(w), k con-stant. (A. R. Forsyth, Diffeiential geometry, p. 427.)
14. W-surfaces. Surfaces for which there exists a functional relationshipf (K,, K2) = 0 between the principal curvatures are called Weingarten surfaces(W-surfaces). In this case we can express K, and K2 as functions of a parameter w.Show that this can be done in such a way that R, and R2 take the form
R, = so(w), R2 = P(w) - w,P'(w),
and that the first and third differential forms can be written
I = \w/2 dug +(P - WP' d,2,
III=22+Vv2
204 SOME PROBLEMS AND PROPOSITIONS
Hint: Use the Codazzi equations (see L. P. Eisenhart, Differential geometry,p. 291).
15. Clothing of surfaces. Consider a piece of cloth made of threads intersect-ing at right angles and thus forming a pattern of small squares. Let it be de-formed in such a way that it lies smoothly on a given surface ("clothes" thesurface). Let us assume that in this process the points of intersection of thethreads are not changed, but that the angle at which they intersect may change.Then the threads form on the surface a net of Cebysev and we can introduce asystem of curvilinear coordinates such that
ds2 = due + 2 cos a du dv + dv2.Show that
3zK =-csca aau av
(Tschebycheff, Sur la coupe des vStements, 1878, Oeuvres II, p. 708.)16. Covariant differential. Given a vector field v(v;), first as function of x, y, z,
then as function of orthogonal curvilinear coordinates u, v, w, or u', i = 1, 2, 3.Show that if v = vlx + v2x + v3x,,,:
+ +dv = Sv'xu + Sv2x, + Sv3xu,where
Sv' = dv' + I k v1 dull (sum on j, k).
The Sv' are called the covariant differentials of v with respect to the system u, v, w.(See for notation Eq. (4-5), Section 3-4.) Also show that the expression forSv' holds for general curvilinear coordinates in space, the I ;k being defined as inExercise 14, Section 3-3.
17. A theorem of Hazzidakis. Onva surface of constant negative curvature -Kthe asymptotic lines form a net of Cebysev (Problem 15). For the area A of aquadrangle formed by these asymptotic lines with interior angles al, as, as, as (all<Zr) the equation holds
KA=2r-al-as-aa- as.(J. Hazzidakis, Crelle's Journal fur Mathem. 88, 1880, pp. 68-73.)
18. Integral torsion. The integral torsion of a curve is defined as fr ds.
This quantity is zero for any closed curve on the sphere, and if on a surface thisproperty holds for all closed curves on it. the surface is a sphere (or a plane).(W. Scherrer, Vierteiiahresschrift Naturforscher Ges. Zurich, 85, 1940, pp. 40-46;B. Segre, Atti Accad. Lincei. 3. 1947. pp. 420-426.)
19. Integral curvature of a curve. This is defined as PcIds. This quantity
is >, 2a for a closed space curve, the sign of equality holding only for plane convexcurves (ovals). (W. Fenschel, Mathem. Annalen, 101, 1929, pp. 238-252.)
20. Umbilics on a closed surface. On every analytical closed surface of genuszero there exist at least two umbilical points. This conjecture of C. Caratht odorywas proved by H. Hamburger, Acta mathematica 73, 1941, pp. 175-332; see G.Bol, Math. Zeitschr., 49, 1944, pp. 389-410.
APPENDIX
THE METHOD OF PFAFFIANS IN THETHEORY OF CURVES AND SURFACES
1. Pfaffians. A linear differential form of the first order in the differ-entials dx1, dx2i . . . , dx of the variables x1, x2, ... , x,,:
v1 dx1 + v2 dx2 + ... + V. dx,, = E Vi dxi, i = 1, 2, ... , n,
where the vi are functions of the xi (which may be constants), is called aPfafan form or Pfafan. We shall write
w(d) = vi dxi (1-1)
where (see Section 3-3, Exercise 13) we omit the E and agree to sum onthe index which is repeated, here i. If no ambiguity exists, we simplywrite w instead of w(d). The vi shall be continuous functions with a suffi-cient number of continuous partial derivatives in the domain X (n-dimensional domain with coordinates xi) considered.
When n = 3 the expression v dx is a Pfaffian. In analogy we can call thevi in Eq. (1-1) the components of the vector v in X. corresponding to w.
The following operations with Pfaffians will be useful.(a) Linear combination. If w1 = vi dxi, W2 = wi dxi, w3 = ui dxi, then
awl + µw2 + vw3 = Ni + µwi + vui) dxi
is a linear combination of the three given Pfaffians. Here the X, µ, v arescalar functions which, as in the case of Eq. (5-5), Section 2-5, can bevectors, e.g., w1a + web = (via + wib) dxi. We can construct linear com-binations of any number k of Pfaffians.
(b) Linear dependence. Such a set of k Pfaffians is linearly independentin Xn, if the corresponding vectors are linearly independent. For instance,in the case that k = 3, this means that the matrix
v1 V2 V3
wl W2 W3
U1 U2 U3
is of rank 3. In the case of k vectors the corresponding matrix must be205
206 APPENDIX
of rank k. In particular, if n = 2 and
wl = pi du + qi dv, w2 = P2 du + q2 dv, (1-2)
these two Pfaffians are linearly independent in X2 if pig2 - p2g1 0.(c) Exterior multiplication. When two Pfaffians
w1(d) = vi dxi, w2(3) = wi Sxi,
where dxi and Sxi represent two different directions in X,,, are combinedin the following way:
[w1, w2] = wI(d)w2(6) -- w2(S)wl(d) = (vitv1 - vpwi) dxi Sxi
_ -I(viwj - v;wi)(dxi Sxj - dx, Sxi) (summed on all i and j)
= (vim; - v,wi)(dxiOx, - dx, Sxi), i < j, (1-3)
then we call [wi, w2] the exterior product of wi and w2. It is alternating,that is
[wl, 02] = -[w2, wl], ([co, w] = 0) (1-4)
Necessary and sufficient condition that w, and w2 are linearly independentis [WI, w2] 0 0.
When n = 2 and w1, w2 are given by (1-2):
[Wi, W2] = (p1g2 - p2g1)(du Sv - dv Su). (1-5)
In the special case wi = du, w2 = Sv, we find [w,, w21 = du Sv - dv Su andwe often write [du dv] instead of [du, Sv]. Hence
[du dv] = du Sv - dv Su,
[W 1, w21 = (viwi - v,wi)[dxi dx1], i < j.
The vector product of v = vie, + v2e2 and w = wiel + w2e2 in the planecan be written in this notation as follows:
v X w = (viw2 - v2wi)el X e2 = [vw] = (V1W2 - v2w1)[eie2].
More general, if v = viei, w = wiei we can define
[vw] = (viw; - v,wi)[eiei], i < j.
This is often called the Grassmann method.
(d) Exterior differentiation. When w = vi dxi, then we define the ex-terior derivative Dw as follows (vi1 = 8vilax;):
THE METHOD OF PFAFFIANS
Dw = [dvi, dxi] = dvi Sx, - Svi dxi =
207
(vi; - v;i) dxj Sxi(summed on all i, j)
_ (v1i - vi1)(dxi &x1 - dxj Sxi) = (v1i - vi;)[dxi dx,], i < j.(1-6)
In particular, if n = 2, w = p du + q dv, we find, by virtue of (1-5) :
Dw = ( au - ap) [du dv]. (1-7)
(e) Some theorems on composition. It can be readily verified that
[w1, Xw2 + /w3] = X[wl, w2] + 1i[w1, W3], (1-8)
D(wl f w2) = Dwl f Dw2, (1-9)
D(pw) = pDw + [dp, w]. (1-10)
When w = dp, then Dw = 0. (1-11)
Here A, µ, p are scalar functions.
2. Invariance. When in Xn we pass from one system of coordinates xito another system x; by means of the transformation
{ xi=ix1l,x2,,...,xn,, i= 1,2,...,n,with Jacobian 010, so that the xi can also be expressed as functions of
the xi, then
dxi = ax; dx';, dx'i = ax, dxi,
and w = vi dxi becomes w = v; dx;, where
8xvlax.
The linear differential form vi dxi is thus transformed into anotherlinear differential form vjdx;. For the exterior differential forms wl =vi dxi = v; dx; and w2 = wi Sxi = wI 5x; we find the exterior product inthe new variables in the form
[w1, w2] = (vai - viwi)[dxi dx'i], i < j,so that this expression also retains its form in the new variables. It istherefore often called the bilinear covariant of W1 and W2. The exterior
208 APPENDIX
derivative Dw shows the same character: Dw = (vjli - v;;)[dx! dx,']. Thenotations w, [WI, w2], Dw are therefore independent of the coordinates, areinvariant notations like the vector notation; w = v1 dxi = v2 dx; = v!' dxi'etc.
For n = 2 (comp. Section 2-1, Eq. 1-11) we find
[du dv] = (u, v,) [dud v']. (2-1)
The expression fw, taken along a curve C in X,,, that is, evaluated forthe case that the xi depend on one parameter xi = xi(t) is a line integral.The notation is an invariant one. We obtain an invariant notation fora surface integral f f f (u, v) du dv if we change the symbol du dv into [du, dv].Indeed, because of (2-1) we find under a change of coordinates
f ff(u, v)[du dv] = fff'(u', v') (u, v) [du' dv'], (2-2)it v
where f'(u', v') = f[u'(u, v), v'(u, v)] and the integral is taken over aregion R in X2 (a surface). This notation therefore extends to doubleintegrals Leibniz' well-known transformation rule of integrals of functionsof one variable:
du =ff(u) ff'(v') du du', f'(v') = f[u(u')].
3. Stokes' theorem. By means of Green's theorem (p. 154) an integralalong a closed curve C in an X2 can be transformed into an integral overthe surface S enclosed by it. We write the theorem as follows (see Eq. 2-2) :
fyi dxl + v2 dx2ff (49x1
- 49x2) [dxi dx2].C S'
(3-1)
For the orientation see Figure 4-14. We take the region S simply con-nected and C sufficiently smooth, also in the case of a general X, e.g., forn = 3, when we have Stokes' theorem as a generalization of Green'stheorem. It can be written as follows:
f vi dxi+ v2 dx2 + v3 dx3
C S'
evil [dxi dx2149x2
a _ 2 dx dx+ ax ax) [ 2 3]2 3
+ 8v1 - av3 [dx3 dx11, (3-2)49x3 49x1)
THE METHOD OF PFAFFIANS 209
where S is an (orientable) region X2 bounded by the closed curve C in X3.If X 3 is ordinary space, [dxl dx2] can be considered equal to dA cos 'ywhere dA is the element of area of S and ti the angle of its normal withthe 7,-axis* (compare also Eq. 5-3).
Both Eqs. (3-1) and (3-2) can be written
fw = ffDw. (3-3)C S
It can be shown that this equation holds for any number of variables xi.It is called Stokes' theorem for X.
When a function p(xi) exists such that the vi are the first partial deriva-tives of p with respect to the xi, hence vi = ap/axi, then co = vi dxi = dp.We then call w a total differential. In this case Dw = 0 (see Eq. 1-11).Inversely, if Dw = 0 in a region of Xn in which Stokes' theorem holds,then a line integral fw from point A to a point B in Xn is independent ofthe path and from this can be shown that co is of the form dp. Under propersafeguardst we can therefore express this property as follows:
Necessary and sufficient condition that w be a total differential is that Dw = 0.
In ordinary vector analysis this theorem is usually expressed by saying that
a vector is a gradient vector if and only if its rotation vanishes.
EXERCISES
1. Find out whether the following Pfaffians (in X3, resp. X2) are linearlyindependent.
(a) w1 = xI dxI + X2 dx2 + X3 dxi (b) w1 = x dxI - XIX2 dx2W2 = (xI + x2) dxI - 2x2 dx2 w2 = xix2 dxI - xlx2 dx2W3 = dxI + X3 dx2 - xlx2 dx3
2. Prove that the necessary and sufficient condition that wI and W2 are linearlyindependent is that [w', w2] s 0-
3. A set of n + 1 Pfaffians in X. are always linearly dependent. Prove.
4. Let wi = xi dxl + x2 dx2 + x3 dx3, w2 = X2 dxI - xI dx2.Find [WI, w2], Dwi, and Dw2
* See, e.g., Ph. Franklin, A treatise on advanced calculus, 1940, pp. 380, 382;also R. Creighton Buck, Advanced calculus, New York, Toronto, London, 1956,pp. 338, 346.
t See e.g. Buck, op. cit., pp. 356-357.
210 APPENDIX
5. Prove that for directions dxi and Sxi for which 6 dxi = d Sxi,Dw = &o(d) - dw(S).
6. Prove (a) : necessary and sufficient condition that for a functionf(xl, x2, ... , the relation 6 df = d of holds is S dxi = d Sxi;
(b) that in this case -q;ati/ax; = Haiti/ax;, if the directions dxi and Sxi aregiven by dxi = 6i dt, Sxi = 77i dt.
7. Interpreting x and y as cartesian coordinates in the plane, prove by actual
integration over a convex region that its area can be expressed as
ff [dx dy] = ff(dx by - Sx dy).
8. Consider the Pfaffian w = (x dy - y dx)/(x2 + y2) in the (not simplyconnected) region between the concentric circles x2 + y2 = 2 and x2 + y2 = 4.Show that Dw = 0 in this region, but that fw is not necessarily zero along aclosed path inside this region.
4. Curves in R3. We consider, as in Section 1-1, a curve C with equa-tion x = x(u) in ordinary space with a fixed system of rectangular co-ordinate axes. To each point of C we associate a trihedron (el, e2, e3) ofmutually orthogonal unit vectors ei:
i±j, i,j =1,2,3(4-1)
(in general not along the coordinates axes). The ei are functions of u,and when P moves along C the change of position of P and of the e; isgiven by equations of the form
dx = wiei de; = wiiei, (4-2)
where the wi, wii are differentials of the form fi(u) du, fij(u) du. Since,according to Eq. (4-1) ei de; = -e1 dei = w;i, we find
wij = -wji (hence wil = w22 = w33 = 0).
We therefore meet in Eq. (4-2) six differentials, w1, w2i W3, W12, W13, 0)31.When we select the trihedron in such a way that el = t, e2 = n, e3 = b,we obtain in Eq. (4-2) the Frenet equations of the curve, with
wl = ds = xu xu du, W2 = w3 = 0, w12 = K ds, w13 = 0, w23 = 7 ds.
The fundamental theorem (p. 29) states that the wi, wij, 'given as single-valued functions of u by means of (4-2), determine one and only one curve
THE METHOD OF PFAFFIANS 211
but for its position in space, endowed with a trihedron of mutually or-thogonal vectors ei at each point. For the proof we can follow a reason-ing analogous to that given on p. 29.
5. Surfaces in R3. We now consider a surface S, given by x = x(u, v)as in Section 2-1. To each point P of S we associate a (uniquely defined)trihedron (el, e2, e3) satisfying Eq. (4-1). The e; are here functions ofu and v, and the change from P in the direction dx on the surface is givenby equations of the form
dx = wieidei. - wi;e; wi; _ -wii, i, j = 1, 2, 3, (5-1)
where the six coefficients wi, wij are now Pfaffians in two variables, e.g.wI = pl(u, v) du + qj(u, v) dv, etc.
We select e3 = N (p. 62). Since dx N = 0, the Pfaffian wa = 0,which is the necessary and sufficient condition that e3 = N. Since, accord-ing to (1-3),
[wi, w2] = [el dx, e2. 6x] = (el X e2) (da X Sz)
= e3 (dx X ox) = e3 (z X x,)[du dv]
EG - F2 [du dv], (5-2)
(comp. Eqs. 3-2 and 3-4, Section 2-3), we conclude
[wi, w2] = dA 0 0 ( EG - F2 > 0, p. 59). (5-3)
In Section 3-3 we have introduced the compatibility relations forEq. (2-6), Section 3-2, using also the condition (x,)u. This leadsus to establish the compatibility relations for Eq. (5-1). They expressthat the components of dx and dei with respect to the cartesian axes aresix total differentials, so that by virtue of the theorem at the end of Sec-tion 3, this Appendix:
Ddx = 0, (5-4)
D de; = 0. (5-5)
From Eq. (5-4) follows, using Eqs. (1-8), (1-9) and (1-10),
eiDwi + [Dei, coil = Of or
eiDwi + [wit, wile; = eiDwi + [wji, wile: = 0, (5-6).
hence
Dwi = [wii, w!] 1 (5-7)
212
or, since w3 = 0:
APPENDIX
Dw1 = [w12, w2], Dw2 = [w21, w1] (5-7a)
[w31, (1) 11 + [w32, wi] = 0.
From Eq. (5-5) follows in a similar way
or
Dwij = [wjk, wik] = [wik, wkj]
Dw23 = [W21, w13],
DW31 = [w32, W211,
Dw12 = 1(1)13,(1)321-
(5-8)
(5-8a)
Equations (5-7) and (5-8) are the equations of structure of the wi, wij.Eqs. (5-8), as we shall see, are the equivalent of Gauss-Codazzi equations.
The fundamental theorem (Section 3-6) states that if the single valuedPfaffians in two variables wi, wij with [WI, w2] 0 satisfy the equationsof structure, then they determine one and only surface, given but for itsposition in space, and endowed with a trihedron (el, e2, e3) at each point.For a proof we must refer to the literature, see below.
From Eq. (5-1) we can derive the following expression for the threefundamental forms of the surface (pp. 59, 73, 103):
ds2 = I = dx dx = (wl)2 + (W2)2
II = -dx . e3 = w1W13\+
W2W23
III = de3 de3 = (w13)2 + (W23)2.
6. Gaussian curvature. We introduce the Gaussian curvature by meansof the formula
K [w31, w32]
[WI, w2](6-1)
in which the denominator is the element of area of the surface (Eq. 5-3) :dA = (w1i W2]. When dAs is the element of area of the spherical imagee3 = e3(u, v), Eq. 5-3 shows that dAs = [w13, w23], so that Eq. (6-1)is equivalent to K = dAs/dA (see p. 157). Since
[w13, W23] = [el de3, e2 - Sea] = (el X e2) (de3 X 5e3)
= e3 (N. X [du dv],
THE METHOD OF PFAFFIANS 213
we find, with the aid of the Weingarten equations (see p. 156) :
[w13, W23] =eg - f2
e.3 - (x. X x.) [du dv],EG - F2
from which we derive, using Eq. (5-2) :
eg - f2K EG-F2
Now, according to the third of Eq. (5-8a), the numerator of the fractionin Eq. (6-1) can be written as -Dw12i moreover, W12 can be expressed asa linear combination of w1 and W2 (Exercise 3, Section 3, this Appendix):
W12 = awl + 19w2
By virtue of the first two equations of (5-7a) :
Dwl = a[w1, W2], Dw2 = #[w1, 02],
so that
(6-2)
(6-3)
K [w 1, 1 D CI Dl, wit]wl + 1 Ilw22]
w2] (6-4)
The Gaussian curvature can thus be expressed exclusively in terms ofW1, w2i their first and second derivatives. This Eq. (6-4) is equivalent tothe theorema egregium (p. 111), since bending, a procedure which leavesds2 = (w1)2 + (w2)2 invariant, can be performed in such a way that wland w2 remain themselves invariant (by appropriate selection of the tri-hedron under this transformation). This theorema egregium has beenobtained as a consequence of the third of the equations of structure(5-8a). It can also be shown that the first and second equations of struc-ture (5-8a) are the equivalent of the Codazzi equations (see Exercise 1of Section 8, this Appendix).
7. Curves on the surface. We have, so far, imposed no special condi-tions on the set (el, e2) except that it be situated in the tangent plane.In this plane, however, we still have oo l choices, depending on one param-eter is
el = el cos <p - e2 sin(7-1)
e2 = e1 sin p+e2cosgyp,
which represent rotations about an angle ,p, counterclockwise from (el, e2)
214 APPENDIX
to (ei, e2) if p > 0. The (p is a function of position. Then, if
dS = wsei = dei, de, = wl. el., e3 = e3 = N, (7-2)
we find
del = d(el cos p - e'2 sin (p)
= dell cos ,p - de2 sin rp - el sin p dsp - e3 cos w dcp
= (w12e2 + w13e3) cos p - (W21e1 + w23e3) sin tp- el sin V dcp - e2 cos V dp,
as well as
de1 = w12e2 + w13e3 = w12(el sin tp + e2 cos rP) + w13e3,
so that, equating the coefficients of ei, e2, e3 in both equations, we obtainthe transformation formulas
w12 = w12 + dv, w13 = W13 cos ,p - W23 sin gyp.
Similar operations on dz and e2 give us
(a) wl = W1 costp+W2sinrp (b) w31 = w31COstp+W82sin rp
W2 = -WI sin tp } W2 cos tp W82 = -W31 sin rp + w82 COs tp
(C) w3=W3=0 (d)Wi2=W12+dp.(7-3)
These formulas can be used in selecting trihedra in a way appropriateto the study of special curves C on the surface. Let the tangent unittangent vector t be given, as on p. 130, by
t = el cos B + e2 sin B, u = u(s), V = v(8), 0 = 0(s). (7-4)
Then dx = t ds = colei + w2e2, so that along the curve
w1 = cos 0 ds, w2 = sin B ds, ds = WI cos 0 + w2 sin 0.
We now select eI = t, e2 = u (Section 4-1). With e3 = N this trihedronforms a natural frame defined by W2 = 0, hence w, = ds. Then, if alongthe curve de; = wije;, w;; = wt;(s), we can introduce (see Problem 4,p. 202) geodesic curvature, Kp, normal curvature K, and geodesic torsionTp by the formulas.
W12 = KO ds, W13 = K+ d8, w32 = Tg ds. (7-5)
THE METHOD OF PFAFFIANS 215
When we replace the natural frame by an arbitrary frame through arotation 0, in a clockwise direction in order to obtain the relation (7-4),then we obtain for a general frame (with e3 = N):
Kpds = W12+do, Kn = W13COS9+W23sin 0,(7-6)
Tp ds = W13 sin 0 + W23 COS 9.
The equations of the geodesics with respect to an arbitrary frame are ob-tained in the form Ka = 0 or
W12 + d9 = 0.
From Eqs. (7-6), (6-2), and (6-3) we conclude that the geodesics, aswell as the geodesic curvature, are bending invariants.
Lines of curvature can be defined as lines of zero geodesic torsion:or
or
W13 Sin 0 - w23 COS 0 = W10023 + W2W31 = 0,
W13 = KW1, W23 = KW2,
which express Rodrigues' theorem (p. 94), with K as principal curvature.Asymptotic lines can be defined as lines of zero normal curvature:
W13 COS 0 + W23 sin 0 = W1W13 + W2W23 = 0.
This means that along the asymptotic lines the second fundamental formvanishes.
8. Gauss-Bonnet theorem. Along a closed curve C on the surface,bounding a simply connected region R, as in Fig. 4-15, f do = 2w.
Hence, by virtue of Stokes' theorem, Eq. (6-1) for K, and Eq. (7-6)for Kg:
fKg ds = f (W12 + do) = 27r + fW12 = 2w +fDw12
2w+ f f [Wls, W23]
= -ffKdA+2w,
which expresses the Gauss-Bonnet theorem (p. 155).
The method of the moving trihedron and its systematic study by means ofPfaffians is typical of the work of E. Cartan (1869-1951), who, inspired by
216 APPENDIX
S. Lie and G. Darboux, introduced his method c. 1900 for the investigation ofcontinuous groups. See his Theorie des groupes finis et continus et la geometriedifferentielle (1937). Application to classical differential geometry in W.Blaschke. Einfuhrung in die Differentialgeometrie (1950). In Russian: S. P.Finikov, The method of the exterior forms of Cartan in differential geometry (1948).For Dw = [dvi dxi] we also find the notation dw and w in the literature.
EXERCISES
1. Prove that the first and second equations of structure (5-8a) are equivalentwith the Codazzi equations (3-4), Section 3-3. Take an orthogonal net ofcurvilinear coordinates or, even more simple, take the lines of curvature as para-metric lines.
2. From Eq. (7-6) derive Liouville's formula (1-13, Section 4-1) for thegeodesic curvature.
3. From Eq. (6-4) derive Liouville's formula for the Gaussian curvature ex-pressed in Exercise 13, Section 4-2, 2nd formula.
4. Defining developable surfaces as surfaces for which K = 0, hence[w13, w23] = 0, show that ds2 can be written in the form du2 + dv2.
5. Prove that for two directions on the surface dxi and dx;:
Wjw13 + w2w23 = w1W13 + w2w23
and that conjugate lines are characterized by the vanishing of these expressions.6. Prove that minimal surfaces are characterized by the equations w13 = awe,
w23 = awl, where a is a scalar function. (Take a natural frame along theasymptotic lines, and use the third equation (5-7a.)
7. Prove that surfaces of constant curvature are characterized by the equa-
tion wlw2 = Cw13w23, C a constant.8. A plane curve x = x(s) intersects all lines through the origin at an angle
of 45°. The moving trihedron is formed by t, n, b. Find wi, wij, i = 1, 2, 3.9. On a sphere a trihedron (el, e2, e3) is defined at each point by the outward
surface normal in the direction of e3 and the vectors el, e2 along the meridiansand parallels in the direction of increasing ip and 0 (Fig. 2-1). Find the wi, wii.
ANSWERS TO PROBLEMS
SECTION 1-6
2 = 4(9u4 + 9u2 + 1) 31. (a) K
(9u4 + 4u2 + 1)3'T
9u4 + 9u2 + 1(b) K2 = 1us(u4 + u2 + 1)-3, r = 0, curve lies in plane z = y - x -
2 = ( 'g" - (J ")2 + (g")2(c)
K [1 +({J')2
+ (g')213f 9,,, -r({{ J'g" - f"g')2 + (f")2 + (g")2
(d) K2 = a2(b2 + 4a2 sin4 u/2) (b2 + 4a2 sing u/2)-1,r = -b(b2 + 4a2 sin4 u/2)-1.
(e) T = 3a (1 + u2)-2.
2. (a) Differentiate x = At.3. (a) Differentiate x = At + An, or x b = 0.4. cos 73 = ±a/c.5. do/ds = TRb + R'n.6. c = -(b 2 /a) (el cos u + e2 sin u) + buk.7. If the index 1 refers to C, then ti = b, dsi/ds = rR, one more differentia-
tion gives Kin = -Kn; select t = b1, then differentiation gives rlr = K2.8. When (xoyozo) is the point, then xoy - yox - b(z - zo) = 0 is the plane.
The same property holds for all curves for which x dy - y dx + k dz = 0, k con-stant (see E. Goursat, Cours d'Analyse I, p. 584).
9. Normal plane: x sin 2u + y cos 2u - z sin u = 0. Curve also lies oncylinder ax + z2 = a2.
10. it = to + Kv2n, a = d2s/dt2, v = ds/dt.11. (x-12. Differentiate dcp/ds= tan-i(dy/dx). Check the sign for a circle.13. X = St + Js2(Kn) + 1S3(K'n - K2t + Krb) + .14. co(u) = c1u + c2, circular helix.15. (a) Let u be the unit vector in direction of common perpendicular at Q
and Qi to tangents at P and P1 respectively. Let PQ = v, P1Q1 = v + Av, QQ, = Av.Then u t = 0, u dt = 0, hence u = b, and Ax + (t + At) (v + Av) - u Av -vt = 0, or v = 0. Compare the reasoning on p. 191.
Similarly for (b). Here u is in direction of t, v = 0, hence tangent is commonperpendicular. For this method compare Section 5-5.
16. t V42 + Ji + J3 = J1i + J2j + J3k.J1 = 8(F1, F2)/a(y, Z), J2 = 9(F1, F2)/a(z, X), J3 = 9(Fl, F2)/a(x, Y)-
17. Differentiate b, = b.19. ds, = Idtl = Id,pl; (a) point, (b) arc of great circle, (c) are of small circle.20. ds = Idnl, dsb = Idbl, use Frenet formulas.
217
218 ANSWERS TO PROBLEMS
SECTION 1-111. (y - X) X t = - Ks2b +2. (y - x) b ='Krs3+.
3. Combine y = x + sx' + is2x" + +nl
with result of Ex-ercises 1 and 2.
4. Distance to plane is ± (a x, + p)/ a a; power with respect to sphereis (x, -a) (x,-a)- p2. Power is of order of distance.
5. Helix: y,= sit + ls2Kn + *sl( - K2t + Krb + K'n) + ' , K, r are constants.Connect points on helix and curve for which s, = s (see remark p. 24), then
3
ly, - y1 = 6 K'n + Contact is of order two independent of r. With method
of Exercise 15, Section 1-6, we find u = (rt + Kb)/ K2+ T2, v- 1 = (K2 + r2) 1K.6. (d) aR = s2 + a2.
3
7.da
+ (K2 + T2)da = 0, hence ac = -A cos cs + B sin cs + C,
c = x2 + r2, select A, B, C appropriately (A1 = C1 = D1 = 0, B1 = -K, etc.).8. Fixed axis direction a = e, + e3, cylinder is y(ab - y)2 = b3(x - z)2
(K/,r = ± 1.)10. See Exercise 5, Section 1-6, r = 0, ds, = dR.11.x1=x+ an,nl=n.By differentiation find a = constant. If
ti = t cos a + b sin a, then ds, = 1 - aK = ards cos a sin a
By differentiation of ti find a = constant.12. Start with (1- aK) sin a + ar cos a = 0, a, a constants. Then differentiate
xi = x + an to show n1= n.13. (a) All orthogonal trajectories of the normals (parallel curves) are Ber-
trand mates; (b) t, = b, xl = x + Rn, compare Exercises 6, 7, Section 1-6;(c) for each a we can find a corresponding a.
Take u=tsina-bcosa,v=tcosa+bsina,thenv=uXu'acscaandt = u sin a + u X u' a cot a. Result obtained because a da = (sin a) ds.
15. From x1 = x + an, x = x,- an,, n = n,, follows (1 - ax) (1 + ax,) = cost a.16. (a) Curves identical, (b) curves are normal sections of a cylinder.17. Differentiation gives t, = t, dsl/ds = Tic, ±n1 = n, + TKl = 1.19. From - dx sin 0+ dy cos O = ds sin a or r dO = ds sin a and ds2 = dr2 +
r2 do2 + dx2 follows dz2 = (r2 cot2 01012 - 1) dr2.20, 21. Use x = x(s), y = y(s), x = s cos a; the arc length of the base is
ds, = ds sin a.22. These helices also intersect the generating lines of the cone at constant
angles (loxodromes); their projections on the base intersect all radii at constantangles; from the theorem on the projection of a helix on the plane (Section 1-9)follows that both R and s are proportional to R,, s,, and R, is proportional toSi (Example 2, Section 1-8).
23. If the equation of the paraboloid is z = ar2, then are = s cos a and for theprojection ar 2 = sl cot a. This typifies the circle involute (p. 27).
ANSWERS TO PROBLEMS 219
24. x" = Kn, x"' _ -K2t + KTb - K'n, z"' X x" = -K$ (b + t' , this differ-K //
entiated gives xi, X x", then find (x1° X x") x"'.25. n = Rx", - Kt + rb = Rx"' + R'x", by means of this equation express b,
and then b' = Tn in terms of x', x", x"', x1°.
SECTION 1-13
1. Let y = ±ix + c,,zk + ck+,xk+1 + (ck 0),a = x2 + y2 = Vkx(k+1)/2 + ...
2s = l + y'2 dx = k
+ 1±2ikckz(-')12 +
hence lim s/a = (21/k-) /k + 1.
3. Isotropic plane through a and 0 is given by x a = 0. This property canalso be demonstrated geometrically by means of the isotropic circle (at infinity),on which a is represented by a point P and the plane x a = 0 by the tangentthrough P to the circle.
4. Follows from the fact that (m - i)/(1 + im) = -i is independent of m.( ± 1 1 ) , C = i X I, D= (i X I) (i X ii). Then
(b) A o0, B=0, C 0, D o0; (c) A =0, B=0, C=0, D=0;(d) A=0, B 0, C'0, D'0; (e) A 0, B=0, C 0, D=0.
7. From i i = 0 follows (i 11)2 = (I ii)(i 1) 1, hence either i 1 = 0 ors 1 = 0, which are equivalent. But 1 i = 0 in (12-8) gives f"' = 0, and henceno isotropic (curved) curve. The only possibility is the isotropic straight line.
8. R1 + R2 = (ds/d(p) + (ds,/d(ill) = µ.9. E.g. from XX' + µµ' = 0 follows X = 0; and when -dx + µ d(p = c drp,
then P = rc, or c = µ (Exercise 10).
13. Follows from K1 ds = K ds for evolute and curve, and fn ds =ft ds =
fdx = 0 for an oval. The curvature centroid is given by x= VXK ds)/
I fK ds) = fxK d8.
//
` 14. Cardioid.15. The cosine of both angles is ± (x t)/ x xx.16. The motion is given by i' = -x, y' = -y + ln(-1).
SECTION 2-3
2. (a), (b) u = constant, v = constant are straight lines, (c) hyperbolicparaboloid.
3. (a) ds2 = (1 + p2) dx2 + 2 pq dx dy + (1 + q2) dye, q2 N =(-p, -q, 1). (b) ds2 = dx2 + dye + dz2, Fzdx + Fvdy + Ftdz = 0.
5. r = a sin 8.6. If dv/du =X, use X1X2 = A/C, X1 + X2 = -2B/C.
220 ANSWERS TO PROBLEMS
9. ± 2 / _ -IG dv/ds.10. Since r = u sin a, the projection is of the form r = c exp(k(p), k constant.11. 12 air sec a.12. x = Za(cos u + cos v), y = 2a (sin u + sin v), z = Zb(u + v), introduce
(u + v) as new parameter V.13. Eliminate x1, yi, z1 from xx' + yyl + zzl = 1 xa2 = yb2 = zc2 = 1,
2 z 2 ,
2 yz 2
az+bz+&2=1.
d- U c x, yl ri A
SECTION 2-4
1. (a) Tangent surface of x1= u, x2 = u2, xa = u2. (b) Tangent surface tocircular helix.
4. Differentiate x = pn + qb and find that p2 + q2 is constant.5. Differentiate x = pb + qt and x = pt + qn. The second case is impos-
sible (except for plane curves).6. Now a in (4-1) depends on two parameters u, v. Differentiate with re-
spect to u and with respect to v and eliminate u and v. Take ax + by + cz = 1,with abc constant.
7. Take ys = 0 in (4-5), which gives s2 = 0 and y2 = JKy;.
8. Take y2 = 0 in (4-5), which gives s = 0 andY "
10. Take the slope of the line as parameter - u. Result: x = a cos' u,y = a sine u.
11. Only when the curve is plane (and not a straight line or circle).12. No.
SECTION 2-8
1. II = (r dx2 + 2s dx dy + t dy2)/ 1 + p2 + q2.4. (a) Right conoid with Z-axis as double line, (b) surfaces of rotation with
Z-axis as axis of rotation.6. rt-s2=0.
// 17. KI cos2 a + K2 $1n2 a + Kl cost , a + 2 I + K2 sin2(.+
ANSWERS TO PROBLEMS 221
14. Since N is proportional to F.e, + Fe2 + F,e3, the equations are equivalent
to an equation of the form (6-5b) with u = x, v = yby equations of theform dz= pdx+gdy,Fx+F,p = 0,dF,,+F,(rdx+sdy)+p dF, = 0. More elegant deduction by using (NN't) = 0, which follows fromEq. (9-3).
16. This determinant expresses that 0 (see Eq. (8-5)), which canbe shown by multiplying the terms of the third row by p and q respectively andadding to those of the first and second row respectively. For a systematic dis-cussion of the theory of curvature of surfaces in the form F(x, y, z) = 0, seeV and K. Kommerell, Allgemeine Theorie der Raumkurven and Fl(chen I.
17. Eliminate 9 from x = K tan rp cos a, y = K tan (p sin a, z = K with theaid of Euler's theorem. The (p is the angle of Eq. (5-11).
SECTION 2-11
1. Y. = 0.2. r = T.3. Follows from dN, N2 + N, dN2 = 0 and Rodrigues' theorem.4. Take f =F=O. Then I = E due + G dv2, II = EK, due + GK2 dv2,
III = EKi due + GK2 dv2 (this because of Rodrigues' formula).5. For asymptotic lines III = r2 ds2. For one asymptotic line we have the
+, and for the other the - sign, but to prove this we must take the asymp-totic lines as coordinate lines and compute r for u = constant and v = constantseparately with the aid of the Frenet formulas and Eqs. (2-9) of Chapter 3.
6. (a), (b) Follows from (9-3) and (9-4).7. Follows from (10-2).
8. (b) Let r2 = x2 + y2 + z2, du = - r2 dx + zr dr, express dx2 + dy2 + dz2in due, dv2, dw2.
10. For the roots µ of the equation of Exercise 9 we have µ,,< a2 < µ2 <b2,for those of Eq. (11-5) we have -oo < X, < µ, < X2 < µz < X8 < oo .
SECTION 3-3
5. C is asymptotic because the tangent to C is self-conjugate (C is touchedby the curves of two conjugate families of curves, which can be taken as thecurves u = constant, v = constant).
6. Parabolas in parallel planes.I7. Write 8
49UD
2DI OD2 a
8. Since In sin w = In D - a In E -z
In G (Section 2, Eq. (2-11)) and
cot w = D, we find by differentiating In sin w the required formulas.
11. Follows from Eqs. (2-9) and (2-6).12. Take Eq. (2-8a), Chapter 2, and the transformation equation of eg - f2 in
Section 2-6, p. 78.
222 ANSWERS TO PROBLEMS
SECTION 3-4
1. xx = 1, etc.2, 3. These are orthogonality relations of the sets (ui, u2, us) with respect to
(e,, e2, es) and vice versa (see Section 1-8).
4. P = -h, U2-d82
l u3 + h, dsa u2.
SECTION 4-2
1. d82 = dr2 + r2 drp2, ds,p + r ds ds = 0,
=r2, dr=
d d dBcp? - 2 tan 02. ds2 = a2(d02 + cost B drp2), = 0, a`p = c sect B, substi-ds2 ds Ts ds
tute this into the expression for ds2.3. As in 1, with r = u, p rp sin a (Eq. (1-8), Chapter 2).4. (r2 + a2) (r2 + a2 - c2) d,p2 = c2 dr2 (Section 2-8).5. See Exercise 4, Section 3-3.6. n = ±N.7. According to Example 3, p. 134, u2u' = constant, and sin a = uv'.8. Their osculating planes contain the surface normal (t).9. From dx dN = 0 and n = ± N follows K = 0; any straight line on the
surface is both asymptotic and geodesic.10. From n = ± N and dN + K dx = 0 follows r = 0; (a) and (c) involve (b);
(b) and (c) do not involve (a).
11. D(tt'N) (t x tu)du + (t x Q } (x, X xe); t x xu = (xu x Xv)
aV-
txx,=+(x,xxv)ds;
au (t - x,,)-a12. Take 8,F=0.13. Follows from Chapter 3 (3-7).14. Follows from Exercise 11.15. Use Rodrigues' formula.
18. fdu(U - a)-' ± fdv(V + a) I = constant.
19. For the surfaces of revolution ds2 = u2(dui + dv2), if du, = du 1 + f,'.u
20. Use Eq. (3-8), Chapter 3.22. This equation expresses that the surface normal lies in the osculating
plane.
ANSWERS TO PROBLEMS 223
SECTION 4-8
1. z = -b(ln tan + cos p), 0 G rp G zr.2
2.
3.
5.
6.
Follows from (8-3). See also Problem 4, p. 201.The surface normal is the same.Use the Gauss-Bonnet theorem.Take in Fig. 4-17 side A1A2 as v = 0, AIA3 as v = vo, let u = rp(v)
represent A2A3.
dv -
zThen f K dA = -
Ivodv
e(v) a y'G du = °o a0 o
auo \ au n-o
j°() dv. The integrand of the first integral is unity, the secondu u.. (v)
integrand has to be evaluated with the aid of Exercise 4.
11. (c) Check with Exercise 14, Section 4-2.12. Let u = constant, v = constant be the geodesic distance from u = 0,
v = 0 respectively. Then along the orthogonal trajectories of the parametriclines, for which F du + G dv = 0, E du + F dv = 0, the ds must be du and dvrespectively. Hence E = G = EG - F2. Then we obtain from Chapter 2,Eq. (2-11), E = G = csc2 W.
14. Apply Liouville's formula for the case that K2 belongs to the orthogonaltrajectories of the curves v = constant.
15. The unit normal vector of the rectifying developable is n.
SECTION 5-12. (b) y2 = 4 mx + 4 W.4. x2b4 - 4y2a' = 4xa2b2.5. If where c=x+Rn, then
gives (c - y) n =R.Two osculating circles do not intersect, since the difference R1 - R2 of their
radii is equal to the are of the evolute between their centers (Exercise 10, Section1-11) and is therefore longer than the chord connecting them.
7. Differentiate y = x + Rn +V'az - R2b and show that its tangent istangent to the circle of radius a and center x in the normal plane.
9. x2 + 2y2 + 2z2 = 2a2 if the given circle is x2 + y2 = a2.10. If S, is given by G = 0, then (grad F, grad F,,, grad G) # 0, and Fix' +
F y' + F.z' = 0, Fu,,,y' + 0, Gix' + G y' + G,z' = 0 at points forwhich F. = 0, Fuu= 0.
11. x du + x° dv + x dx = 0 has only a solution # 0 if (auaaa) = 0.
SECTION 5-3
1. Solve the simultaneous equations
(E du + F dv) bu + (F du + G dv) Sv = 0,(El du + F1 dv) bu + (F1 du + G1 dv) Sv = 0,
hence (E du + F dv) (F1 du + G1 dv) - (F du + G dv) (El du + F1 dv) = 0, whichhas two orthogonal solutions unless E1, F1, G1 are proportional to E, F, G.
224 ANSWERS TO PROBLEMS
2. Introduce geodesic coordinates on both surfaces, then ds2 = du2 + G dv2and dsi = 02 du2 + o2G dv2, where o2 is a function of It alone. Then use Exer-cise 4, Section 4-8.
4. It means that the second fundamental forms are proportional.5. Use the Mercator projection. _6. Use Eq. (5-10) and (5-12) of Chapter 4, o'K1 = v'K.7. x = e-", y = v, use Eq. (2-3a) of Chapter 4, which results in xy" - y' -
2(y')3 = 0, y' = dy/dx.9. x = arp, y = a sin 0.
11. Take, in accordance with Tissot's theorem, ds2 = E du2 + G dv2, dsi =E1 du2 + G, dv2; then the equivalence of the geodesic lines (take Eq. (2-1),
Section 4-2) gives a In ( E' G2) = 0,av
In (G' Ez) = 0, or E, = EU-2V-1, G, _
GU-'V-2, U = U(u), V = V(v). Also, if U # V, a In E = a In (U - V)av av
a In G 81n (U - V) ; change of scale on the parametric lines gives the answer.au au12. w = pes.
SECTION 5-4
4. Follows from Eq. (4-7); the left-hand side of the equations follows fromthe right-hand side by partial integration.
5. (a) F(r) = iar-2, (b) F(r) _ -ar-2.6. The equation can be written with appropriate parameter: x = 3u +
3uv2 - u3, y = 3v + 3u2v - v2, z = 3(u2 - v2); the parametric curves are thelines of curvature. Their planes are given by x + uz = 3u + 2u3, y - vz =3v + 2v2.
8. It is also a translation surface with real generators x = u, y = 0, z =-Incosu;x=0,y=v,z=ln cos v.
9. Use the formula of Exercise 4, Section 2-11.10. N y = N N. y, = 0, lines of curvature correspond, Rodrigues' theorem
gives R1=R1-X, R2=R2-X.
SECTION 5-5
1. Take y = feg + vu, u = e1 cos u + e2 sin u; p = f', Z-axis is strictionline.
3. Follows from Eq. (5-14).4. K = -p2/(p2 + v2), if a' i' = 0.5. Follows from Eq. (5-14) and from (tan (p, - tan rp2) cos 91 cos (p2 =
sin (1P1 - rp2).6. This plane is parallel to the corresponding tangent plane on the asymp-
totic cone.7. Write Eq. (5-1) with x(a cos u, a sin u, 0), i(sin u, -cos u, b).8, 9. Find first the asymptotic tangent plane and then the point of tangency
of the perpendicular plane, or find the shortest distance of two generators of thesame kind.
ANSWERS TO PROBLEMS 225
11. (X-x,ui)=0.13. Take one of the helices on the surface as directrix. Compare Exercise 7.14. Take x = x(s) as the striction line. Then (a) x'- i' = 0, (b) N = n,
(c) i t = constant.15. Take y = x + vb, parameter u satisfies du/ds = r.
SECTION 5-6
1. The condition is Eq. (6-10). Use e.g. the elimination method of Syl-vester.
2. From E = 0, e = 0 follows suu N = 0, au xuu = 0, hence xuu = X(N X xv);since G = 0 this means that xuu = Ax-
3. Taking E = 0, G = 0, we can now make e = 0, from which we can show(Exercise 2) that the curves u = constant are straight isotropic lines.
4. In this case we can make E = 0, G = 0, e = 0, g = 0 and K is constant inall directions.
5. There are two isotropic directions in the osculating plane of the givencurve.
6. Intersect the cone with a plane through the vertex.
7. We can make F = 0, G = 0, f = 0, g = 0.8. Here E in Eq. (6-5) is a function of u alone. Take E = 1. Then prove
that x=u,y=v.
APPENDIX
SECTION 3
1. (a) independent, (b) dependent (w2 = x2w1)4. [w1, W2] = -(xi + x2)[dxl dx2] + x2x3[dx3 dx11 + xix3[dx2 *31,
Dwi = 0, DW2 = -2[dxl dx2].5. Follows from: dw(S) - 3w(d) = (dvi) 3x; + v; d 3x; - (6v;) dx; -
v; 6 dx; = [dv; dx;] + v;(6 dx; - d ox;).6. Follows from: d Of - 3 df = (f,; - f;,)[dx; dxil + f;(d Ox; - 6 dx;),
fti = of/ax:, f,, = a2f/ax;ax,.8. fw = 2T along the circle x2 + y2 = a2, 2 < a < 4.
SECTION 8
4. From [W31, W321 = 0 follows Dw12 = 0, hence W12 can be written dcp. Take,p = const. and the orthogonal trajectories as new coordinates.
8. wl = eBdO\/2,w2 = 0,W12 = d9, w13 = W23 = 0-9. wl = dB, w2 = duo cos 0, W12 = -sin 0 d<p, W23 = -cos 0 dp, W31 = dB.
INDEX
Abbott, 105Adams, 182Aerostation, 76Ampere, 53Angle, of contingency, 14
curvature, 202of directions on surface, 59
Aoust, 119, 201Appell, 96Applicable surfaces, 175Area, 63, 65, 182, 183Asymptotic curves, definition, 77, 215
curvature, 202on developable surfaces, 90differential equation, 77, 96geometric interpretation, 96on minimal surfaces, 182, 185parametric, 99on right conoid, 89on ruled surfaces, 87, 88, 96on sphere, 87on surfaces of revolution, 92torsion of, 194
Asymptotic directions, 77Asymptotic tangent plane, 194
Ballistics, 167Barbier, 50, 51Beltrami, 104, 146, 152, 153, 160, 178,
180, 202Bending invariant, 112Bernoulli, James, 201Bernoulli, John, 132Bertrand, 42, 54, 103Bertrand curves, 42, 43, 188, 202Bianchi, 22, 126, 147, 176Bierbaum, 168Bilinear covariant, 207Binormal, 15, 22, 73, 195Bioche, 201Blaeu, 174Blaschke, 35, 49, 51, 123, 160, 202,
216
Bol, 204Bolyai, J., 152, 153Bonola, 153
Bonnet, 103, 111, 126, 130, 135, 153,156, 177, 187, 195, 203
Bonnet's formula for geodesic curva-ture, 135
Bour, 177Braude, 32v. Braunmuhl, 144Brill, 92, 121Brioschi, 112Buck, 209
Canal surfaces, 166, 167, 168Caratheodory, 204Cartan, 22, 215, 216Catalan, 188Catenary, 42Catenoid, 88, 120, 121, 122, 175, 187,
188Cauchy, 25, 124Cauchy-Riemann equations, 172Nbysev (Tschebycheff), net of, 204Center, of curvature, 14, 15, 21
of principal curvature, 94Center surfaces, 94, 95Central plane, 192Central point, 192Cesiro, 32, 201Characteristic line, 66, 162
point, 66, 162Characteristics, 165, 166Chasles, 103, 194, 195, 199Christoffel, 108Christoffel symbols, 107, 108, 113Circle, 1, 8, 26Circular helix (see: Helix, circular)Clairaut, 134, 135Clothing of surfaces, 204Clothoid, 201Codazzi, 111, 117Codazzi equations, 111, 113, 115, 119,
135Cohn-Vossen, 83Collinear, 11Combescure, 22Cone, 56, 67, 69, 70, 90, 134
circular, 44, 56, 61, 63
226
INDEX 227
Confocal quadrics, 101Confocal system, 104Conformal (see: Mapping)Congruence, 95, 142Conjugate lines (curves), 96, 97,
Sect. 2-10, 180Conjugate point, 143, 144Connectivity, 160Cono-cuneus, 194Conoid, of Plucker, 93
right, 61, 98, 103, 194Coolidge, 152Coordinate curves, 57Contact, 23, Sect. 1-7, 41, 51Coplanar, 11Courbure inclinee, 201Courant, 187Covariant differential, 204Coxeter, 153Cross-ratio, 37Cubical parabola, 21, 73Curvatura integra, 155-160Curvature, center of (see: Center)
constant, 32, 43directions (see: Lines of curvature)first, 13, 15, 18, Sect. 1-4Gaussian (see: Gaussian curvature)geodesic, 127, Sect. 4-1, 157, 159,
182integral, 155-160integral (absolute), 204mean, 83, 104, 182, 183normal, 74of plane curve, 15, 129radius of, 14, 76second (see: Torsion)tangential (see: Curvature, geodesic)third, 23total (curve), 23total (surface), 83 (see: Gaussian
curvature)total (surface), 155 (see: Curvature,
integral)Curvature centroid, 52Curvature vector, 13, 73
associate, 201of a curve with respect to a field, 201geodesic, 74, 127, Sect. 4-1normal, 74, 75, Sect. 2-5, 127tangential (see: Curvature vector,
geodesic)
Curve, of constant slope (see: Helix)of constant width, 50of double curvature, 18in isotropic plane, 47, 51orbiform, 50plane, 14, 32, 129of zero width, 50
Curvilinear coordinates, 57in space, 114, Sect. 3-4, 119
Cuspidal edge, 68Cyclic point, 49Cycloid, 28, 42Cylinder, 66, 91Cylindroid, 93
D'Aiguillon, 175Darboux, 22, 36, 39, 76, 103, 109, 119,
144, 147, 156, 180, 187Definite quadratic form, 77Developable helicoid, 69, 70, 71Developable surfaces, Sect. 2-4, 82
applicable to plane, 70, 146asymptotic curves on, 90condition, 91, 93isotropic, 198, 200line of striction on, 193lines of curvature on, 90tangent planes of, 64, 66
Diameter, 52Differential equation of space curves,
44Differential equations of surface theory,
108Differential parameters, 160Dini, 179, 180Director cone, 190Directrix curve, 190Distribution parameter, 191, 195Douglas, 187Dupin, 53, 82, 96Dupin indicatrix, 84, Sect. 2-7
Edge of regression, 67, 73, 146, 166,Sect. 5-1
Eisenhart, 37, 39, 103, 120, 126, 177,203
Elasticity surface, 65Element of are, 59Ellipsoid, 64, 65, 92, 102, 143Elliptic, coordinates, 102
paraboloid, 64point, 77, 79, 84
228 INDEX
Enneper, 104, 188Envelope of plane curves, 167
of planes, Sect. 2-4of straight lines in plane, 73of surfaces, Sect. 5-1
Epicycloid, 28, 35Equations of structure, 212Euclid, 150Euler, 26, 51, 76, 82, 132Euler's theorem on curvature, 81, 82,
Sect. 2-6on polyhedrons, 160
Euler-Lagrange equations, 142, 183Evolute, 40, 52, 71, 135, 202Exterior derivative, 206Exterior products, 206Extremals, 142, Sect. 4-4, 203
Fenschel, 204Fiedler, 51Field of geodesics, 142Flatland, 105Flexion, 18Focal points, 166Formulas of Frenet, 18, 22, Sect. 1-6Forsyth, 103, 180, 203Franklin, 3, 4, 5, 17, 63, 133, 209Frenet, 19, 54, 210Fresnel, 65Fundamental form, first, 59, 75
second, 73, 75, 79, 91, 118third, 103
Fundamental theorem for curves, 29,210
for surfaces, 124, Sect. 2-6, 212
Gauss, 53, 105, Sect. 3-1, 111, 130, 137,155, 156, 158, 161, 175
Gauss-Bonnet theorem, 155, Sect. 4-8,215
Gauss equations for xi,, 108, 115for K, 113, 114, 119
Gaussian curvature, 83, 212, 216bending invariant, 112, 175for catenoid, 120for developable surfaces, 91equation, 113, 114, 119, 138geometrical interpretation, 157for right helicoid, 120for ruled surfaces, 190, 195
Gemma Frisius, 175
Generating line, 189Generator, 189Genus, 160Geodesic circle, 140Geodesic coordinates, 136, Sect. 4-3Geodesic curvature, 127, Sect. 4-1, 157,
159, 182, 216Geodesic curves, geodesics, 131,
Sect. 4-2, Sect. 4-4, 215on circular cone, 134as curves of shortest distance,
Sect. 4-4on cylinders, 134differential equation, 132, 133on ellipsoid, 143in plane, 133, 134on right helicoid, 134on sphere, 134on surface of revolution, 134
Geodesic distance, 140Geodesic ellipses, 160, 161Geodesic hyperbolas, 160, 161Geodesic mapping, 177, Sect. 5-3Geodesic parallels, 137Geodesic polar coordinates, 137Geodesic polygon, 157Geodesic torsion, 201Geodesic triangle, 158, 160Goursat, 177, 205Gradient, 116Grassmann, 206Graustein, 202Green's theorem, 153, 154Gullstrand, 82
Halsted, 153Hamburger, 204Hassler, 182Hazzidakis, 204Helicoid, developable, 69, 70, 71Helix, 33, Sect. 1-9, 44
circular, 1, 9, 12, 17, 21, 22, 32, 34,39, 42, 62, 65, 70, 71
spherical, 35Henneberg, 188Hilbert, 83, 122, 124, 147, 153Hipparch, 175Hyperbolic, paraboloid, 64, 78, 194,
195point, 78, 79, 84
Hyperboloid, 64, 96, 193, 195
INDEX 229
Hypocycloid, 28, 73Hypothesis, of the acute angle, 151,
160of the obtuse angle, 151of the right angle, 151
Imaginaries (surface), Sect. 5-6Imaginary curves, 44, Sect. 1-12Ince, 30, 37Indicatrix of Dupin, 84, Sect. 2-7Inflection, point of, 12Integral geometry, 48Invariant notation, 208Inversion, 180Involute, 39, 40, 50, 137
of circle, 9, 28
Isometric, correspondence, 145,Sect. 5-3
system, 171, 175Isothermic system, 171, 175Isotropic, cone, 45, 46, 200
curve 44, 46, 47, 87, 170, 184, 196,Sect. 5-6, 202
developable, 198, 200line, 45, 51, 87, 147, 198, 199plane, 45, 46, 52, 197, 198, 200
Jackson, 49Jacobi, 144Joachimsthal, 103Jordan, 51
Kant, 152Kasner, 51, 203Katsura, 203Kneser, 49Knoblauch, 202Kommerell, 209Kowalewski, 32Kremer, 174
Lagrange, 25, 53, 177, 183, 187, 188Lame, 116, 119, 160, 175Lancret, 34, 41, 53Lavoisier, 76Legendre, 94, 151, 158, 187Leibniz, 137, 208Levi, E. E., 176Levi Civita, 126Lie, 36, 39, 179, 180, 187Liebmann, 123, 124
Linearly independent, 205Line integral, 208Lines, asymptotic (see: Asymptotic
curves)conjugate, 96, 97, Sect. 2-10, 180
Lines of curvature, definition, 80, 215on developable surfaces, 90differential equation, 80, 91, 92, 94geometric interpretation, 93, 201on isotropic developables, 200on minimal surfaces, 185, 186parametric, 81on quadrics, 102, 103on right conoid, 90on sphere, 86near umbilics, 82
Liouville, 53, 54, 131, 132, 135, 177, 216Liouville's formula, 131, 154, 161
theorem on conformal mapping, 203Liouville surfaces, 135, 179Lipschitz, 29, 133LobaSevskii, 152, 153Logarithmic spiral, 26, 42, 65Loxodrome, 34, 43, 60, 65, 173, 175,
180, 206
Mainardi, 111Malus, 53Mannheim, 43Mapping, Sects. 5-2, 5-3
conformal, 169, 180, 186, 188equiareal, 169, 180
Mapping, geodesic, 169, 177isometric, 145, 175
Matrix, 205Mean curvature, 83, 104, 182, 183Mellish, 51, 52, 202Mercator, 173, 180Mercator projection, 173Meridians, 60, 61, 87Meusnier, 53, 76, 187, 188Meusnier's theorem, 76, Sect. 2-5, 81,
127Minding, 123, 130, 145, 147, 176, 177,
194Minimal curves, 45 (see: Isotropic
curves)Minimal surfaces, 88, 89, Sect. 5-4
adjoint, 186associate, 186
Minkowski, 202
230 INDEX
Moebius, 188Monge, 25, 41, 53, Sect. 1-14, 93, 94,
95, 105, 131, 156, 166, 187, 199, 203Monge-Ampere equation, 177Monkey saddle, 83, 85Montgolfier, 76Moore, 202Moving trihedron (curve), 16, 18, 20,
22(surface), 106, 116, 202
Natural equations, 26, Sect. 1-8families, 203
Natural frame, 214Navel point, 81Needle problem, 51Net, 57Normal (surface), 62
to isotropic plane, 46Normal curvature vector, 74, 75,
Sect. 2-5, 127Normal plane, 19Normal section, 76Null curves, 45
Olivier, 42One-sided surface, 188Orbiform curves, 50Oriented curve, 6Ortelius, 174Orthogonal, directions, 59, 60
trajectories, 65Osculating, circle, 14, 42
helix, 42plane, 10, 11, 12, 19sphere, 25, 32, 41, 71, 168
Oval, 47, Sect. 1-13, 204Ovaloid, 202
Parabola, cubical, 21, 73Parabolic curve, 92
point, 77, 79, 85, 90Parallel curves, 65, 137
surfaces, 189Parallels, 61, 87Parametric curves, 56Pedal, curves, 52
surfaces, 189Pfaffian, 205-216Pitch, 2Plane curve, 14, 32, 129Plateau, 187
Pliicker, 93Point of inflection, 12Polar, axis, 25
developable, 41, 71, 135figure, 168
Polyconic projection, 181, 182Poncelet, 45, 53, 199Principal, normal, 13, 42, 73
curvatures, 80Profile, 60Projection, Mercator, 173
polyconic, 181, 182stereographic, 174, 175
Pseudosphere, 148, 149, 150, 180Ptolemy, 175Puiseux, 45
Radius, of curvature, 14of normal curvature, 84of osculating circle, 14of osculating sphere, 25of torsion, 17
Reciprocal sets of vectors, 119Rectifying, developable, 72, 73, 161
plane, 19, 72, 73, 161, 195Regular plane, 45Rhumb line, 60Riccati equation, 36, 37, 39, 43, 126,
191Ricci-Curbastro, 119Riemann, 152, 170, 187Right conoid, definition, 61
distribution parameter, 194first fundamental form, 61second fundamental form, 89
Right helicoid, asymptotic lines, 89curves on, 65definition, 13, 62first fundamental form, 62geodesics on, 134isometric mapping, 120, 175, 176lines of curvature, 89as minimal surface, 89, 120, 121, 187,
188tangent plane, 63
Robbins, 187Rodrigues, 53, 94, 117, 156Rotation surface (see: Surface of
revolution)Ruled surface, asymptotic lines, 190,
191
INDEX 231
definition, 62, 72, 188, Sect. 5-5developable, 191, 194Gaussian curvature, 190, 195left handed, 194right handed, 199striction line, 193tangent planes, 194
Ruling, 189
Saccheri, 150Saddle point, 78Saint Simon, 94Saint Venant, 18, 34, 54Salkowski, 43Scalar product, 6Scheffers, 26, 33, 53, 126, 199Scherk, H. F., 188Scherk, P., 49Scherrer, 204Schwartz, 186, 187Scroll, 189Segre, 204Serret, J. A., 18, 19, 54Serret, P., 194Sextactic point, 49Similitude, 178, 180Singular point, 55Smith, D. E., 175, 191Sphere, asymptotic lines, 87
as closed surface, 123conformal maps, 173-175, 188envelopes of, 162, 163, 166equiareal map, 180first fundamental form, 60fundamental theorem, 124-126geodesic map, 179geodesics, 134, 140, 143under inversion, 180isometric map, 146, 149isotropic lines, 87latitude and longitude, 56lines of curvature, 86loxodromes, 60second fundamental form, 86umbilics, 86, 122
Spherical, curves, 32helix, 35image (indicatrix), 23, 42, 104, 186,
212Spiral of Cornu, 201Stackel, 132
Stereographic projection, 174, 175
Stokes, 208, 209Striction line, 192, 195Struik, 51Study, 47, 51, 52Surface of constant curvature, 144,
Sects. 4-5, 4-6, 158, 175, 178, 179,180
Surface integral, 208Surface of revolution, asymptotic lines,
87, 88of constant curvature, Sect. 4-6of constant width, 202first fundamental form, 61geodesics, 134lines of curvature, 87as Liouville surfaces, 135loxodromes, 180meridians and parallels, 60, 61minimal surfaces, 88second fundamental form, 87
Tangent, 7Tangent plane, 62Tangential developable, 64, 67, 71, 146Tensor, calculus, 119
notation, 114, 115, 116Theorema egregium, 111, 113, 213Tissot, 180Titchmarch, 170Topological, equivalent, 158, 159Torsion, of asymptotic curves, 104
geodesic, 201integral, 204of space curves, 16, Sect. 1-5
Torus, 79, 159, 166Total curvature (see: Curvature, total)Total differential, 209Tractrix, 149, 159Translation surfaces, 103, 109, 184Triangular curve, 51Trihedron (see: Moving trihedron)Triple scalar product, 11Triply orthogonal systems, 100,
Sect. 2-11, 115Tschebycheff, 204
Umbilic, 81, 82, 122, 123on closed surfaces, 204parabolic, 82
U. S. Coast and Geodetic Survey, 182
232
Vallee, 18Vallee Poussin, de Ia, 3, 6Van Kampen, 202Vector, 5, 205Vector product, 11, 206Vertex, 48, 49Viviani, 10Vogt, 203Voss, 43
INDEX
Wallis, 194Weierstrass, 184, 187, 188Weingarten, 108Weingarten equations, 108, 135, 156,
213W-surfaces, 203Wentworth, 190Woods, 183, 187, 188