12.6

Quadric surfaces

Review:

0Vector equation of a line L: t r r v 0 point on the line, direction , ,a b cr v

0 0 0Parametric scalar  equation of a lin , , e L: x x at y y bt z z ct

0 1 0 0 1( ) ( ) with 0 1 is the line which goes from t o t P t P P t P P r

0Vector equation of a plane: 0 n r r0 on the plane

, , normal to planea b c

r

n

Scalar equation of the plane: ax by cz d

1 2

1 2

1 2

Angle between two planes with normal , : cos

n n

n nn n

0| (Q ) |Distance between a point Q and a plane:

r n

n

0| (Q ) |Distance between a point Q and a line:

r v

| v |

2 2 2

2 2 2Ellipsoid 1

x y z

a b c

Surfaces in 3-space:

2 2 2

2 2 2

Hyperboloid of one sheet

1x y z

a b c

2 2 2

2 2 2: 1 any

horizontal traces are ellipses

x y kz k k

a b c

2 2 2

2 2 2

Hyperboloid of two sheet

1x y z

a b c

2 2 2

2 2 2: 1, | | 1

horizontal traces are ellipses

x y kz k k

a b c

2 2

2 2(Elliptical paraboloid)

x yz

a b

Horizontal traces are ellipses (notice 0)z

2 2

2 2(Hyperbolic paraboloid)

x yz

a b

Horizontal traces are hyperbolas

Vertical traces are parabolas

2 22

2 2Cone

x yz

a b

Horizontal traces are ellipses

Vertical traces are hyperbolas

Notice: If 0x straight linesy

zb

2 2

2 21

x y

a b

Cylinders:

2 2

2 21

y x

a b

2y ax

VII

IV

II

III

VI

VIII

V

I

2 2) 2 3a x y z 2 2

or 6 3 2

x y z

elliptic paraboloid

2 2

or 4 1

y zx

hyperbolic paraboloid

opens in the positive directionx

2 2) 4 4 0b x y z

Draw a picture of the following surfaces:

2 2 24 4 ___ 4 6 ___ 36 ___ ___x y y z z 9

2 224 2 4 3 4x y z

364

2 22 2 3

11 4 1

y zx

ellipsoid center: 0,2,3

4

2 2 24 4 4 24 36 0x y z y z Draw a picture of the surface :

notice e.g. 0y

Summary:

12

2

2

2

2

2

c

z

b

y

a

x

2 2 2

2 2 21

x y z

a b c

2 2 2

2 2 21

x y z

a b c

2 2 2

2 2 2

z x y

c a b

Ellipsoid

Hyperboloid of one sheet

Hyperboloid of two sheets

Cone elliptic

all variables present

one variable

not squared

2 2

2 2

z x y

c a b

2 2

2 2

z x y

c a b

Paraboloid elliptic

Paraboloid hyperbolic

all variables present

all variables squared

one variable not present cylinder opening in the direction of the missing variable2 2

2 21 Elliptic cylinder

x y

a b

2 2

2 21 Hyperbolic cylinder

x z

a b

2 Parabolic cylinderz ax

13.1

Curves in Space

A curve in space (or the plane):

( ), ( ), ( ) or

( ) , ,

x f t y g t z h t

t f g h f g h

r i j k

in the plane:

( ) ( ) ( )t f t g t r j j

This is a vector valued function (as opposed to scalar valued)

Example: ( ) cos( ),sin(t), tt t r

2 2Notice that ( ) ( ) 1 x t y t

A circular helix:

i.e. the curve lies on a cylinder

( ) 4cos(2 ) 9sin(2 t)t t r i jA curve in the plane:

2 2

2 2Notice that 1

4 9

x y The particle travels along an ellipse

(0) 4r i = (4,0) Travels counter clockwise

as 0 2 the particle goes twice around the ellipset

Trefoil knot:

3 3 32 2 2

2 cos cos , 2 cos sin ,sint t tt t t r

For an animation go to :http://math.bu.edu/people/paul/225/trefoil_knot.html

, ,t f t g t h tr

lim lim ,lim ,limt a t a t a t a

t f t g t h t

r

, ,t f t g t h t r

, ,

b b b b

a a a a

t dt f t dt g t dt h t dt r

For a vector valued function:

Take limits, derivatives and integrals component wise:

Example:

2

0

sinLet , , ln 1 . Find limt

t

tt e t t

t r r

0

0 0

sinlim lim , , ln 1t t

tt e

t r 1,1,0 i j

1 2Let sin , 1 , ln 1 3 .t t t t r Find tr

2

1, ,

1t

t

r

1/221

1 22

t t

1

31 3t

2

2

1

1 sin t t t t t dt i j k

2 2

1 3, ,

1 31 1

t

tt t

2 2 2

2

1 1 1

, 1 , sint dt t t dt t t dt

22 32

1 13

tt dt

8 1

3 3

7

3

2

1

1t t dt 1

0

1u udu

(a)

(b)

1u t du dt

1 0t u

2 1t u

1

3/2 1/2

0

u u dt 1

5/2 3/2

0

2 2

5 3u u

2 2

5 3

6 10

15 15

16

15

Examples:

2

1

sint t dt u t sindv t

du dt 1

cosv t

2 2

1 1

1cos cos

tt t dt

udv uv vdu

2

2

1

1cos sin

tt t

2 2

2 1 1 1cos 2 sin 2 cos sin

2 1

1 0 1 0

3

2

2

1

7 16 31 sin , ,

3 15t t t t t dt

i j k

Differentiation Rules:

1. d

t t t tdt

u v u v

2. d

c t c tdt

u u

3. d

f t t f t t f t tdt

u u u

4. d

t t t t t tdt

u v u v u v

5. d

t t t t t tdt

u v u v u v

6. d

f t f t f tdt u u

( ) ( )c t dt c t dt u u

( ) (t)) ( ) ( )t dt t dt t dt (u v u v

Integration rules:

Geometry of the derivative vector :

x

y

zP

Q

tr

t hr

t h t r r

secant vector

x

y

zP

Q tr

t hr

t h t

h

r r

tangent vector

0limh

t h tt

h

Tangent vector

r rr

, ,t f t g t h t f t g t h t r i j k

The to a smooth curvetangent line 0at t t

0 0 0 0It passes through the point ( ) , , ,t f t g t h tr

0and has the same direction as the tangent vector at t

0 0 0 0'( ) , , .t f t g t h t r

So the equation of the tangent line is:

0 0) '( )t s t s r( r

Or in parametric form:

0 0 0 0 0 0( ) '( ), y ( ) '( ), z ( ) '( )x f t sf t g t sg t h t sh t

A drunken bee travels along the path ( ) cos(2 ),sin(2 t), t

for 10 seconds. It then travels at constant speed in a straight line for 10

more seconds. At what point does the bee end up?

t t Problem : r

Question: What is speed and velocity?

Velocity is a vector: ( )t v r'(t) Speed is a scalar: | ( ) |tv

velocity: ( ) 2sin(2 ), 2cos(2 t),1t t v Speed = | ( ) | 9 3t v

0At time 10, it travels along the tangent line with speed 3 ft/sect

0 0tangent line: s ) ( )t s t r( v

after 10 more seconds, it arrives at

0 0) 10 ( )

cos(20),sin(20),10 10 2sin(20), 2cos(20),1

t t

r( v

cos(20) 20sin(20),sin(20) 20cos(20),20

17.85, 9.07, 20 Question: What is distance traveled?