geometrie - formule
TRANSCRIPT
1. Produse de vectori: Produsul scalar: < v1, v2 >= x1x2 + y1y2 + z1z2, ‖v‖ =√
x21 + y2
1 + z22 , cos(v1, v2) = <v1,v2>
‖v1‖·‖v2‖ .
Ortogonalitate: < v1, v2 >= 0. Daca ∆ - dreapta de vector director u, ‖u‖ = 1, atunci Pr∆v1 = (‖v‖ cos(v1, u))u, si
pr∆v1 = ‖v‖ cos(v1, u). Produsul vectorial: v1 × v2 =
∣∣∣∣∣∣
i j kx1 y1 z1
x2 y2 z2
∣∣∣∣∣∣si ‖v1 × v2‖ = ‖v1‖ · ‖v2‖ sin(v1, v2). Avem:
σtriunghi = 12‖v1× v2‖, σparalelogr = ‖v1× v2‖. Coliniaritate: v1× v2 = 0. Produsul mixt: (v1; v2; v3) =< v1, v2× v3 >=∣∣∣∣∣∣
x1 y1 z1
x2 y2 z2
x3 y3 z3
∣∣∣∣∣∣. Avem: Vparalelipiped = (v1; v2; v3), Vtetraedru = 1
6 (v1; v2; v3). Coplanaritate: (v1; v2; v3) = 0.
2. Miscari: Translatie de vector r = x0i + y0j + z0k :
8<:
x′ = x− x0
y′ = y − y0
z′ = z − z0
sau
8<:
x = x′ + x0
y = y′ + y0
z = z′ + z0
. Rotatie de unghi θ ın jurul
originii:(
x′
y′
)=
(cos θ sin θ− sin θ cos θ
)·(
xy
)sau
(xy
)=
(cos θ − sin θsin θ cos θ
)·(
x′
y′
).
3. Dreapta: Un punct si directia:
x = x0 + tly = y0 + tmz = z0 + tn
saux− x0
l=
y − y0
m=
z − z0
n. Doua puncte: l = x1−x0, m = y1− y0,
n = z1 − z0 si se ınlocuieste mai sus.
4. Planul: Un punct si doua directii:
x = x0 + t1l1 + t2l2y = y0 + t1m1 + t2m2
z = z0 + t1n1 + t2n2
sau
∣∣∣∣∣∣
x− x0 y − y0 z − z0
l1 m1 n1
l2 m2 n2
∣∣∣∣∣∣= 0. Trei puncte:
l1 = x1 − x0, m1 = y1 − y0, n1 = z1 − z0, l2 = x2 − x0, m2 = y2 − y0, n2 = z2 − z0 si se ınlocuieste mai sus. Prin taieturi:xa + y
b + zc = 1. Dat de un punct si de normala Ai + Bj + Ck: (x− x0)A + (y − y0)B + (z − z0)C = 0.
5. Intersectii, proiectii, unghiuri:
d(M, D)=‖M0M × vD‖
‖vD‖ , M0 ∈ D, d(M,α)=|AxM +ByM +CzM +D|√
A2 + B2 + C2, d(D1, D2)=
|(M1M2; vD1 ; vD2)|‖vD1 × vD2‖
,M1 ∈ D1,M2 ∈ D2
Perpendiculara comuna: directie v=vD1× vD2 , ec. - la intersectia planelor (M1, v, vD1) si (M2, v, vD2), M1 ∈ D1,M2 ∈ D2.
6. Sfera:
(x− a)2 + (y − b)2 + (z − c)2 = R2 sau x2 + y2 + z2 + mx + ny + pz + q = 0.Cerc=sfera ∩ plan.Planul tangent - prin dedublare: x2 → xx0, x → x+x0
2 .
7. Conice: H(x, y) = a11x2+a22y
2+2a12xy+2a13x+2a23y+a33 = 0. Invarianti: ∆ =
∣∣∣∣∣∣
a11 a12 a13
a12 a22 a23
a13 a23 a33
∣∣∣∣∣∣, δ =
∣∣∣∣a11 a12
a12 a22
∣∣∣∣,
I = a11 + a22.
8. Cuadrice: Σ2 : (a11x2 + a22y
2 + a33z2 + 2a12xy + 2a13xz + 2a23yz) + (2a14x + 2a24y + 2a34z) + a44 = 0,
D =
0BB@
a11 a12 a13 a14
a12 a22 a23 a24
a13 a23 a33 a34
a14 a24 a34 a44
1CCA, A =
0@
a11 a12 a13
a12 a22 a23
a13 a23 a33
1A. Invarianti: ∆ = det D, δ = det A, ρ = rangD, r = rangA,
p=nr. de patrate pozitive.9. Suprafete cilindrice, conice, de rotatie: Ec. supr. cilindrice care se sprijina pe curba Γ :
{f(x, y, z) = 0g(x, y, z) = 0 si are
generatoarea paralela cu dreapta D :{
P1(x, y, z) = 0P2(x, y, z) = 0 se obtine eliminand λ, µ din sist. a). Ec. supr. cilindrice tangente
la supr. F (x, y, z) = 0 si cu generatoarea paralela cu D :{
P1(x, y, z) = 0P2(x, y, z) = 0 se obtine eliminand λ, µ din sist. b). Conul de
varf V (sist. c)) care se sprijina pe curba Γ :{
f(x, y, z) = 0g(x, y, z) = 0 se obtine eliminand λ, µ din sist. d). Conul de varf V (sist.
c)) tangent la supr. F (x, y, z) = 0 se obtine eliminand λ, µ din sist. e). Suprafata de rotatie generata de rotirea curbei
Γ : f(x, y, z) = 0, g(x, y, z) = 0 ın jurul dreptei D :x− x0
l=
y − y0
m=
z − z0
nse obtine eliminand λ, µ din sist. f).
1
a)
8>><>>:
P1(x, y, z) = λP2(x, y, z) = µf(x, y, z) = 0g(x, y, z) = 0
, b)
8<:
P1(x, y, z) = λP2(x, y, z) = µF (x, y, z) = 0
, c) V :
8<:
P1(x, y, z) = 0P2(x, y, z) = 0P3(x, y, z) = 0
, d)
8>><>>:
P1(x, y, z) = λP3(x, y, z)P2(x, y, z) = µP3(x, y, z)f(x, y, z) = 0g(x, y, z) = 0
,
e)
8<:
P1(x, y, z) = λP3(x, y, z)P2(x, y, z) = µP3(x, y, z)F (x, y, z) = 0
, f)
8>><>>:
(x− x0)2 + (y − y0)
2 + (z − z0)2 = λ2
l(x− x0) + m(y − y0) + n(z − z0) = µf(x, y, z) = 0g(x, y, z) = 0
.
Clasificarea conicelor
∆ 6= 0δ > 0
I∆ < 0elipsa
–2
–1
0
1
2
3
y
–2 –1 1 2 3 4
x
x2
a2+
y2
b2= 1 centru:
{
a11x + a12y + a13 = 0a12x + a22y + a23 = 0
,
axe: a12k2 + (a11 − a22)k − a12 = 0,
ec. axe:∂H
∂x+ k1,2
∂H
∂y= 0
δ < 0 hiperbola
–8
–6
–4
–2
0
2
4
6
y
–8 –6 –4 –2 2 4 6 8
x
x2
a2−
y2
b2= 1 centru:
{
a11x + a12y + a13 = 0a12x + a22y + a23 = 0
,
axe: a12k2 + (a11 − a22)k − a12 = 0,
asimpt.: a22k2 + 2a12k + a11 = 0,
ec. axe/asimpt:∂H
∂x+ k1,2
∂H
∂y= 0
δ = 0 parabola
–6
–4
–2
2y
2 4 6 8 10x
Y 2 = 2pX p = ±
√
−∆
I3 , axa: Y = 0, tg. ın
varf: X = 0
∆ = 0 δ > 0 drepte imaginare intersectia este un punct realδ < 0 drepte realeδ = 0 dreapta
1
Clasificarea cuadricelor
∆ 6= 0 δ 6= 0 p = 3 elipsoid
–1
–0.5
0.5
1
z
–1
–0.5
0.5
1
y
–1
–0.5
0.5
1
x
x2
a2+
y2
b2+
z2
c2= 1
ρ = 4 r = 3 p = 2 hiperboloidcu o panza
–1
–0.5
0.5
1
z
–1
–0.5
0.5
1
y
–1
–0.5
0.5
1
x
x2
a2+
y2
b2−
z2
c2= 1
x
a∓
z
c= λ
(
1 −y
b
)
x
a±
z
c=
1
λ
(
1 +y
b
)
p = 1 hiperboloidcu douapanze
–4
–2
2
4
z
–4
–2
2
4
y
–4
–2
2
4
x
x2
a2−
y2
b2−
z2
c2= 1
δ = 0 p = 2 paraboloideliptic
0.2
0.4
0.6
0.8
1
z
–1
–0.5
0.5
1
y
–2
–1
1
2
x
x2
a2+
y2
b2= 2pz
r = 2 p = 1 paraboloidhiperbolic
–1
–0.5
0.5
1
z
–1
–0.5
0.5
1
y
–1
–0.5
0.5
1
x
x2
a2−
y2
b2= 2pz
x
a∓
z
c= λ · 2p
x
a±
z
c=
1
λ· z
∆ = 0 δ 6= 0 p = 3 punct dublu (y1)2 + (y2)
2 + (y3)2 = 0
ρ = 3 r = 3 p = 2, 1 con circular
–1
–0.5
0.5
1
z
–2
–1
1
2
y
–2
–1
1
2
x
x2
a2+
y2
b2=
z2
c2
δ = 0 p = 2 cilindru elip-tic
–1
–0.5
0.5
1
z
–1
–0.5
0.5
1
y
–1
–0.5
0.5
1
x
x2
a2+
y2
b2= 1
r = 2 p = 1 cilindruhiperbolic
–1
–0.5
0.5
1
z
–1
–0.5
0.5
1
y
–1
–0.5
0.5
1
x
x2
a2−
y2
b2= 1
δ = 0 p, r = 1 cilindru par-abolic
–0.4
–0.2
0.2
0.4
z
0.2
0.4
0.6
0.8
1
y
–1
–0.5
0.5
1
x
(y1)2 = 2 · (y2)
∆ = 0 δ = 0 p = 2 dreaptadubla
(y1)2 + (y2)
2 = 0
ρ = 2 r = 2 p = 1 plane se-cante
(y1)2 − (y2)
2 = 0
δ = 0 p, r = 1 planeparalele
(y1)2 = 1
plane con-fundate
(y1)2 = 0
1
2
10. Curbe plane: Reprezentare: vectorial: r = r(t), t ∈ R, parametric:{
x = x(t)y = y(t) , cartezian explicit: y = y(x),
cartezian implicit: F (x, y) = 0. Elementul de arc: ds = ‖dr‖; parametric: ‖dr‖ =√
x2(t) + y2(t) dt; cartezian
explicit: ds =√
1 + y2(x) dx. Lungimea arcului M0(t0)M1(t1): lM0M1 =∫ t1
t0
ds. Tangenta ın M0(t0): vectorial:
R = r(t0) + λr(t0), parametric:{
x = x(t0) + λx(t0)y = y(t0) + λy(t0)
, cartezian: y − y0 = kT (x − x0). Panta tangentei: parametric:
kT =y(t0)x(t0)
, cartezian explicit kT = y(x0), cartezian implicit kT = −F ′x(x0, y0)F ′y(x0, y0)
. Normala: y − y0 = kN (x − x0), unde
kN · kT = −1. Puncte singulare sunt solutiile sistemului
F (x, y) = 0F ′x(x, y) = 0F ′y(x, y) = 0
. Panta tangentei ın punctele singulare e solutia
ecuatiei: F′′x2 + 2F
′′xy kT + F
′′y2 k2
T = 0, cu ∆ = 4(F′′xy − F
′′x2F
′′y2). ∆ > 0 ⇒ nod, ∆ = 0 ⇒ punct de ıntoarcere, ∆ < 0 ⇒
punct izolat. Curbura ın M0(t0): parametric: k =x(t0)y(t0)− x(t0)y(t0)
(x(t0)2 + y(t0)2)3/2, cartezian explicit: k =
y(x)(1 + y(x)2)3/2
, cartezian
implicit: k = −F ′x2F′′y2 − 2F ′xF ′yF
′′xy + F ′y
2F′′x2
(F ′x2 + F ′y
2)3/2. Raza de curbura: R(M0) =
1|k| .
11. Curbe spatiale: Reprezentare: vectorial: r = r(t), t ∈ R, parametric:
x = x(t)y = y(t)z = z(t)
, cartezian explicit:{
y = y(x)z = z(x) ,
cartezian implicit:{
f(x, y, z) = 0g(x, y, z) = 0 . Elementul de arc: ds = ‖dr‖; parametric: ‖dr‖ =
√x2(t) + y2(t) + z2(t) dt,
cartezian explicit: ds =√
1 + y2(x) + z2(x) dx. Lungimea arcului M0(t0)M1(t1): lM0M1 =∫ t1
t0
ds.
Triedrul lui Frenet
Muchii: tangenta (TM0(Γ)), cu vect. dir. t, normala principala (Dn(Γ)), cu vect. dir.n, binormala (Db(Γ)), cu vect. dir. b. Vectori directori: t = xi + yj + zk
∣∣t0
, b =
Ai + Bj + Ck∣∣t0
, n = li + mj + nk∣∣t0
, unde: A =∣∣∣∣y zy z
∣∣∣∣t0
, B = −∣∣∣∣x zx z
∣∣∣∣t0
, C =∣∣∣∣x yx y
∣∣∣∣t0
,
l =∣∣∣∣y zB C
∣∣∣∣t0
, m = −∣∣∣∣x zA C
∣∣∣∣t0
, n =∣∣∣∣x yA B
∣∣∣∣t0
. Versori: t0 =
t
‖t‖ , b0
=b
‖b‖ , n0 =n‖n‖ .
Relatii: b× t = n, t× n = b, n× b = t. Plane: planul normal (PN (Γ) = (M0, n, b)), planulosculator (Po(Γ) = (M0, n, t)), planul rectificant (Pr(Γ) = (M0, t, b)).
Curbura: ρ(M0) =√
A2 + B2 + C2
(x2(t0) + y2(t0) + z2(t0))3/2; vectorial:
‖r(t0)× r(t0)‖‖r(t0)‖3 ; raza de curbura: R =
1ρ. Torsiunea:
τ(M0) =∆
A2 + B2 + C2, unde ∆ =
∣∣∣∣∣∣
x y zx y z...x
...y ...
z
∣∣∣∣∣∣t0
, vectorial:(r(t0); r(t0);
...r (t0))
‖r(t0)× r(t0)‖2 ; raza de torsiune: T =1|τ |
13. Suprafete: Reprezentari: vectorial: r=r(u, v), (u, v) ∈ D, parametric:
x=x(u, v)y=y(u, v)z=z(u, v)
, cartezian explicit: z=f(x, y),
cartezian implicit: F (x, y, z) = 0. Plan tangent: vectorial: (R−r; r′u, r′v) = 0, unde r′u = x′ui+y′uj+z′uk, r′v = x′vi+y′vj+z′vk;
parametric:
∣∣∣∣∣∣
X − x0 Y − y0 Z − z0
x′u y′u z′ux′v y′v z′v
∣∣∣∣∣∣M0
= 0, sau:D(y, z)D(u, v)
∣∣∣∣M0
(X−x0) +D(z, x)D(u, v)
∣∣∣∣M0
(Y −y0) +D(x, y)D(u, v)
∣∣∣∣M0
(Z−z0) = 0, unde
D(y, z)D(u, v)
∣∣∣∣M0
=∣∣∣∣y′u z′uy′v z′v
∣∣∣∣M0
,D(z, x)D(u, v)
∣∣∣∣M0
=∣∣∣∣z′u x′uz′v x′v
∣∣∣∣M0
,D(x, y)D(u, v)
∣∣∣∣M0
=∣∣∣∣x′u y′ux′v y′v
∣∣∣∣M0
; cartezian explicit: p(X−x0)+q(Y −y0)−(Z−z0)=0,
unde p =∂z
∂x
∣∣∣∣M0
, q =∂z
∂y
∣∣∣∣M0
; cartezian implicit: F ′x|M0(X−x0) + F ′y
∣∣M0
(Y −y0) + F ′z|M0(Z−z0) = 0. Dreapta normala:
vectorial: n = r′u × r′v, parametric:(X−x0)D(y,z)D(u,v)
∣∣∣M0
=(Y −y0)D(z,x)D(u,v)
∣∣∣M0
=(Z−z0)
D(x,y)D(u,v)
∣∣∣M0
, cartezian explicit:X−x0
p=
Y −y0
q=
Z−z0
−1,
cartezian implicit:(X−x0)F ′x|M0
=(Y −y0)F ′y
∣∣M0
=(Z−z0)F ′z|M0
. Prima forma fundamentala: I = E du2 + 2F du dv + Gdv2;
determinantul primei forme fundamentale: ∆I = EG − F 2. Calculul E, F,G : parametric: E = x′u2 + y′u
2 + z′u2,
F = x′u x′v + y′u y′v + z′u z′v, G = x′v2 + y′v
2 + z′v2; cartezian implicit: E = 1 + p2, F = pq, G = 1 + q2; cartezian explicit: E =
1+(
F ′xF ′z
)2
, F =F ′x F ′y(F ′z)2
, G = 1+(
F ′yF ′z
)2
. A doua forma fundamentala: II = Ldu2 +2M du dv +N dv2; determinantul
3
celei de-a doua forme fundamentale: ∆II = LN − M2. Calculul L,M,N : parametric: L =1√∆I
∣∣∣∣∣∣
x′u y′u z′ux′v y′v z′vx′′u2 y
′′u2 z
′′u2
∣∣∣∣∣∣,
M =1√∆I
∣∣∣∣∣∣
x′u y′u z′ux′v y′v z′vx′′uv y
′′uv z
′′uv
∣∣∣∣∣∣, N =
1√∆I
∣∣∣∣∣∣
x′u y′u z′ux′v y′v z′vx′′v2 y
′′v2 z
′′v2
∣∣∣∣∣∣; cartezian implicit: L =
z′′u2√
1 + p2 + q2, M =
z′′uv√
1 + p2 + q2,
N =z′′v2√
1 + p2 + q2. Elementul de arie: dσ =
√EG− F 2 du dv, vectorial: dσ = ‖r′u × r′v‖ du dv, cartezian explicit:
dσ =√
1 + p2 + q2 dx dy. Aria suprafetei: A =∫∫
D
dσ. Natura unui punct M0: ∆II(M0) > 0 ⇒ punct eliptic,
∆II(M0) = 0 ⇒ punct parabolic, ∆II(M0) < 0 ⇒ punct hiperbolic;E
L=
F
M=
G
N⇒ M0 - punct circular. Curburile
suprafetei se obtin rezolvand ecuatia (EG− F 2)k2 − (EN + GL− 2FM)k + (LN −M2) = 0; avem: curburile principale:
k1, k2, curbura medie:k1 + k2
2, curbura Gauss k1 · k2.
4