7/28/2019 MATH 37 UNIT 5.4

1/47

LAST TOPIC. . . HOORAY!!!

CYLINDERS andQUADRIC

SURFACES

1These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

2/47

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.

7/28/2019 MATH 37 UNIT 5.4

3/47

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.

7/28/2019 MATH 37 UNIT 5.4

4/47

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.

7/28/2019 MATH 37 UNIT 5.4

5/47

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

5These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

6/47

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

6These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

7/47

Spheres

2222 rlzkyhx Standard equation of a sphere:

Center:

Radius:

l,k,h

r

7These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

8/47

General equation

0222 DCzByAxzyx

The graph in three-dimensionalspace of

is either a sphere, a point or the

empty set.

8These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

9/47

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

by Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

10/47

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.

10These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

11/47

Illustration

directrixgenerator11These lecture slides were created

by Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

12/47

Illustration

A CYLINDER

12These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

13/47

Remark

In the three-dimensional space,the graph of an equation in two

of the three variablesx, y and zis a cylinder.

13These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

14/47

Example.

2522 yx is a cylinder in R3.

Plane curve: on thexy-plane

Directrix: z-axis

14These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

15/47

Example.

2522

yx

5 10

5

10

-5-10

-5

-10

5

10

-5

-10

x

y

z

15These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

16/47

Example.

is a cylinder in R3.

Plane curve: on the yz-plane

Directrix:x-axis

ysinz

16These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

17/47

Example.

ysinz

1

2

x

y

z

2

1

-1

--1

2

-2

-2

-2

17These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

18/47

Example.

is a cylinder in R3.

Plane curve: on thexz-plane

Directrix: y-axis

42 xz

18These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

19/47

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

19These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

20/47

Quadric surfaces

The graph of the second-degreeequation

222

CzByAx FxzEyzDxy

0 JIzHyGxis a quadric surface.

20These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

21/47

Restrictions

Equations that will beconsidered:

222

CzByAx

0 JIzHyGx

These are expressed instandard forms.

21These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

22/47

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

by Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

23/47

x

y

z

Standard forms

Ellipsoid

12

2

2

2

2

2

c

z

b

y

a

x

23These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

24/47

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

24These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

25/47

Standard forms

Elliptic hyperboloid of one sheet

x

y

z

12

2

2

2

2

2

c

z

b

y

a

x

25These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

26/47

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

26These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

27/47

Standard forms

Elliptic hyperboloid of two sheets

x

y

z

12

2

2

2

2

2

c

z

b

y

a

x

27These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

28/47

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

28These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

29/47

Standard forms

Elliptic cone

x

y

z

02

2

2

2

2

2

c

z

b

y

a

x

29These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

30/47

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

30These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

31/47

Standard forms

Elliptic paraboloid

x

y

z02

2

2

2

c

z

b

y

a

x

31These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

32/47

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

32These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

33/47

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

33These lecture slides were createdby Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

34/47

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

7/28/2019 MATH 37 UNIT 5.4

35/47

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.

35These lecture slides were createdby Prof. Babierra.

f

7/28/2019 MATH 37 UNIT 5.4

36/47

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

36These lecture slides were createdby Prof. Babierra.

S f 2

7/28/2019 MATH 37 UNIT 5.4

37/47

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

37These lecture slides were createdby Prof. Babierra.

S f #2

z

7/28/2019 MATH 37 UNIT 5.4

38/47

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

38These lecture slides were createdby Prof. Babierra.

S f #3

7/28/2019 MATH 37 UNIT 5.4

39/47

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

7/28/2019 MATH 37 UNIT 5.4

40/47

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

40These lecture slides were createdby Prof. Babierra.

S f #3z

7/28/2019 MATH 37 UNIT 5.4

41/47

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

7/28/2019 MATH 37 UNIT 5.4

42/47

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

7/28/2019 MATH 37 UNIT 5.4

43/47

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

43These lecture slides were created

by Prof. Babierra.

S f #4z

7/28/2019 MATH 37 UNIT 5.4

44/47

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

7/28/2019 MATH 37 UNIT 5.4

45/47

x

y

zStandard forms

Hyperbolic paraboloid

12

2

2

2

c

z

a

x

b

y

45These lecture slides were created

by Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

46/47

What equation describes the

PRINGLEs shape?

HYPERBOLIC PARABOLOID46These lecture slides were created

by Prof. Babierra.

7/28/2019 MATH 37 UNIT 5.4

47/47

END47These lecture slides were created