math 37 unit 5.4
TRANSCRIPT
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LAST TOPIC. . . HOORAY!!!
CYLINDERS andQUADRIC
SURFACES
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Equation of a plane in 3D
Point on the plane: 0000 z,y,xP
c,b,aNNormal vector to the plane:
0000 zzcyybxxaStandard equation of the plane:
J ust a review2These lecture slides were createdby Prof. Babierra.
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Equation of a plane in 3D
0 dczbyaxGeneral equation of a plane:
ifa, b and c are not all zero,
is a normal vector tothe planec,b,a
J ust a review3These lecture slides were createdby Prof. Babierra.
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Lines in 3D
Let be a line that contains
the point and isparallel to .
L
0000 z,y,xPc,b,aR
atxx 0 btyy 0 ctzz 0PARAMETRIC EQUATIONS ofL
SYMMETRIC EQUATIONS ofL
c
zz
b
yy
a
xx 000
J ust a review4These lecture slides were createdby Prof. Babierra.
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Using parametric equations
tx 21 ty 42 tz 53
At ,0t 1x 2y 3z 321 ,,
At ,2t 3x 6y 13z 1363 ,,
At ,1t 3x 6y 2z
263 ,,At ,
21
t 1x 0y21101 ,,
211z
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Illustration
1 2 3 4
1
23
4
5
-1-2-3-4-5-1
-2
-3
-
12
34 5
-2-3
-4-5
y
z
R
tx21
ty 42
tz 53L
321 ,,
1363 ,,
2
63
,,
21101 ,,
542 ,,R
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Spheres
2222 rlzkyhx Standard equation of a sphere:
Center:
Radius:
l,k,h
r
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General equation
0222 DCzByAxzyx
The graph in three-dimensionalspace of
is either a sphere, a point or the
empty set.
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Illustration
1 2 3 4 5
1
2
34
5
-1-2-3-4-5-1
-2
-3
-4
-5
1
23
45
-2-3
-4-5
x
y
z
9211 222 zyx
211 ,,C 3r9These lecture slides were created
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Cylinder
A cylinder is a surfacegenerated by a line (generator)moving along a given planecurve in such a way that it isalways parallel to a fixed line
(directrix) not lying in the planeof the given curve.
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Illustration
directrixgenerator11These lecture slides were created
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Illustration
A CYLINDER
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Remark
In the three-dimensional space,the graph of an equation in two
of the three variablesx, y and zis a cylinder.
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Example.
2522 yx is a cylinder in R3.
Plane curve: on thexy-plane
Directrix: z-axis
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Example.
2522
yx
5 10
5
10
-5-10
-5
-10
5
10
-5
-10
x
y
z
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Example.
is a cylinder in R3.
Plane curve: on the yz-plane
Directrix:x-axis
ysinz
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Example.
ysinz
1
2
x
y
z
2
1
-1
--1
2
-2
-2
-2
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Example.
is a cylinder in R3.
Plane curve: on thexz-plane
Directrix: y-axis
42 xz
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1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
12
34
5
-2-3
-4
-5
x
y
zExample.
42
xz
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Quadric surfaces
The graph of the second-degreeequation
222
CzByAx FxzEyzDxy
0 JIzHyGxis a quadric surface.
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Restrictions
Equations that will beconsidered:
222
CzByAx
0 JIzHyGx
These are expressed instandard forms.
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Graphs
To graph quadric surfaces,
obtain traces on the following:
xy-plane 0z
yz-plane 0xxz-plane 0y
Level curves (cross-sections)on particular values ofz can
also be used.22These lecture slides were created
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x
y
z
Standard forms
Ellipsoid
12
2
2
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic hyperboloid of one sheet
12
2
2
2
2
2
c
z
b
y
a
x
12
2
2
2
2
2
c
z
b
y
a
x
12
2
2
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic hyperboloid of one sheet
x
y
z
12
2
2
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic hyperboloid of two sheets
12
2
2
2
2
2
c
z
b
y
a
x
12
2
2
2
2
2
c
z
b
y
a
x
12
2
2
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic hyperboloid of two sheets
x
y
z
12
2
2
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic cone
02
2
2
2
2
2
c
z
b
y
a
x
02
2
2
2
2
2
c
z
b
y
a
x
02
2
2
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic cone
x
y
z
02
2
2
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic paraboloid
02
2
2
2
c
z
b
y
a
x
02
2
2
2
c
z
b
y
a
x
02
2
2
2
c
z
b
y
a
x
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Standard forms
Elliptic paraboloid
x
y
z02
2
2
2
c
z
b
y
a
x
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Standard forms
Hyperbolic paraboloid
12
2
2
2
c
z
b
y
a
x
12
2
2
2
c
z
b
y
a
x
12
2
2
2
c
z
b
y
a
x
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Surface # 1.
Traces
xy-plane:0z 144
22
yx
yz-plane:0x
xz-plane:0y
1
164
22
zy
xy-plane
1 2 3 4 5
1234
5
-1-2-3-4-5-2-3
-4
-5
x
y
11644
222
zyx
0164
22
zx
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Surface # 1.
yz-plane: xz-plane:
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
x
z
1164
22
zy 0164
22
zx
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
y
z
34These lecture slides were createdby Prof. Babierra.
z
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11644
222
zyx is an ELLIPTIC
HYPERBOLOID
of one sheet.
2 4 6
2
46
-2-4-6-2
-4
-6
24
6
-4-6
x
y
zSurface # 1.
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f
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Surface # 2.
194
222
zy
xTraces
xy-plane:0z 14
22
y
x
yz-plane:0x
xz-plane:0y
1
94
22
zy
19
22
zx
xy-plane
1 2 3 4 5
1234
5
-1-2-3-4-5-2-3
-4
-5
x
y
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S f 2
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Surface # 2.
yz-plane:
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
y
z
xz-plane:
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
x
z
194
22
zy
EMPTY
19
22 z
x
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S f #2
z
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is an ELLIPTICHYPERBOLOID of
two sheets.
194
222zy
x
Surface # 2.
1 2 3 4 5
1
23
4
5
-1-2-3-4-5-1
-2
-3-4
-5
12
34
5
-2-3
-4
-5
x
y
z
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S f #3
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Surface # 3.
Traces
xy-plane:0z 022 yx
yz-plane:0x
xz-plane:0y
0
9
22
zy
xy-plane
1 2 3 4 5
1234
5
-1-2-3-4-5-2-3
-4
-5
x
y
09
222
z
yx
09
22
zx
yx;yx
39These lecture slides were createdby Prof. Babierra.
S f #3
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Surface # 3.
yz-plane:
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
y
z
xz-plane:
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
x
z
09
2
2 zy
xz;xz 33
09
2
2 zx
POINT
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S f #3z
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is an ELLIPTIC
CONE.
Surface # 3.
1 2 3 4 5
1
23
4
5
-1-2-3-4-5-1
-2
-3-4
-5
12
34
5
-2-3
-4
-5
x
y
z
0
9
222
z
yx41These lecture slides were createdby Prof. Babierra.
S f #4
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Surface # 4.
122
zyxTraces
xy-plane:0z 122 yx
yz-plane:0x
xz-plane:0y
12
zy
12 zx
xy-plane
1 2 3 4 5
1234
5
-1-2-3-4-5-2-3
-4
-5
x
y
42These lecture slides were createdby Prof. Babierra.
S f #4
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Surface # 4.
yz-plane:
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
y
z
xz-plane:
1 2 3 4 5
1234
5
-1-2-3-4-5
-2-3
-4
-5
x
z
12 zy 12 zx
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by Prof. Babierra.
S f #4z
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122
zyx is a HYPERBOLIC
PARABOLOID.
Surface # 4.
1 2 3 4 5
1
2
3
4
5
-1-2-3-4-5-1
-2
-3
-4
-5
12
34
5
-2-3
-4
-5
x
y
z
44These lecture slides were created
by Prof. Babierra.
St d d f
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x
y
zStandard forms
Hyperbolic paraboloid
12
2
2
2
c
z
a
x
b
y
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What equation describes the
PRINGLEs shape?
HYPERBOLIC PARABOLOID46These lecture slides were created
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END47These lecture slides were created