graphs of well known curves and surfaces

8/10/2019 Graphs of Well Known Curves and Surfaces

1/13

Graphs of well known curves and surfaces

1. ELLIPTIC PARABOLOID2 2

2 2

x ycz

a b

+ =

2. PARABOLIC CYLINDER ( )2x cz=

3. ELLIPTIC CYLINDER

2 2

2 21

y z

a b

+ =


2/13

4. HYPERBOLIC CYLINDER2 2

2 21

y x

a b =

5. PARABOLIC HYPERBOLOID

2 2

2

2 2

x ycz

a b

=


3/13


4/13

1. Astroid

Cartesian Equation:

x2/3 +y2/3 =a2/3

Parametric Equations:

x(t) =a cos3 t

y(t) =a sin3 tx

y

a

a

a

a


5/13

15. Figure Eight

Cartesian Equation:

x4 =a2(x2 y2)

Polar Equation:

r2 =a2 cos(2)sec4 -4 -3 -2 -1 1 2 3 4

-2

-1

1

2

x

y

Facts:

(a) Also known as the Lemniscate (of Gerono).


6/13

30. Lemniscate of Bernoulli

Cartesian Equation:

(x2 +y2)2 =a2(x2 y2)

Polar Equation:

r2 =a2 cos(2)-1 1

-0.5

0.5

x

y


7/13

3. Cardioid

Cartesian Equation:

(x2 +y2 2ax)2 = 4a2(x2 +y2)

Polar Equation:

r= 2a(1 + cos )

Parametric Equations:

x(t) =a(2cos t cos(2t))

y(t) =a(2sin t sin(2t))

1 2

-1

1

x

y


8/13


9/13

r(t) (cos t)i(sin t)jtk

x

y

r(t)(cos 5t)i(sin 5t)jtk

x

z

y

r(t)(cos t)i(sin t)j0.3tk

x

z

y

Different forms of Helix


10/13

In cylindrical coordinates, the equation describes not just a circle in the xy-

plane but an entire cylinder about the z-axis (Figure 15.37). The z-axis is given by

The equation describes the plane that contains the z-axis and makes an angle

with the positive x-axis. And, just as in rectangular coordinates, the equation de-

scribes a plane perpendicular to the z-axis.

z = z0

u0u = u0

r = 0.

r = a

z

y

x

O

a

ra,

whereasand zvary

zz0,

whereas rand vary

0,

whereas randzvary

z0

0


11/13

6

4

2

2

4

5 5

f(!)=acsc(!)

a = 2 00

6

4

2

2

4

5 5

f(!)=asec(!)

a = 2 00

6

4

2

2

4

5 5

a = 2.00

f

= a

1 cos

6

4

2

2

4

5 5

a = 2.00

f

= a

1 cos

2

2

!=!

4

2

2

!=2!

3

! #$%%&'( )* +),,)- .)%$' #'$/01

=,83>,7%*4 437"!? " @"8%30*4 437"#? "

A,7B=,83>,7%*4 *7+ A,7B@"8%30*4 437"-

C*8+3,3+-

!B37%"8:*4' D %, E! !B37%"8:*4' D %, E!


12/13

3

2

1

1

2

4 2 2 4

b = 3 00

a = 2 00

f

= a

cos b

4

2

2

5 5

b = 4 00

a = 3 00

f = acos b

4

2

2

5 5

b = 4 00

a = 3 00

f = asin b

2

2

5 5

b = 3 00

a = 3 0 0

f

= a

s in b

2

2

5

r2=a2cos(2!)

a = 3 0 0

2

2

5

r2=a2sin(2!)

a = 2 00

F,-" C18:" G H++ :*41" ,2 % F,-" C18:" G I:"7 :*41" ,2 %

!B37%"8:*4' D %, ! !B37%"8:*4' D %, E!

F,-" C18:" G H++ :*41" ,2 % F,-" C18:" G I:"7 :*41" ,2 %

!B37%"8:*4' D %, ! !B37%"8:*4' D %, E!

J")73-0*%"- ;%$" ,7" K3%$ -37" 3- 4"-- 0,)),7

graphs of well known curves and surfaces

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