11.7

Cylindrical and Spherical Coordinates

The Cylindrical Coordinate System

• In a cylindrical coordinate system, a point P in space is represented by an ordered triple

1. is a polar representation of the projection P in the xy-plane.

2. z is the directed distance from to P.

zr ,,

,r

,r

Figure 11.66

Conversions:

Cylindrical to rectangular

Rectangular to cynlidrical

zz

ry

rx

sin

cos

zzx

y

yxr

tan

222

Examples:

1) Convert from cylindrical to rectangular coordinates

2) Find an equation in cylindrical coordinates for the rectangular equation

3) Find an equation in rectangular coordinates for the equation in cylindrical coordinates and sketch its graph.

2,

4,6

222 yxz

cos2r

Spherical Coordinates

• In a spherical coordinate system, a point P in space is represented by an ordered triple

1. is the distance between P and the origin, 0

2. is the same angle used in cylindrical coordinates for r greater than or equal to 0.

3. is the angle between the positive z-axis and the line segment from the origin to the point P

),,(

0

Figure 11.75

Conversions

• Spherical to Rectangular

• Rectangular to spherical:

cos

sinsin

cossin

z

y

x

222

2222

arccos

tan

zyx

z

x

y

zyx

More Conversions

• Spherical to cylindrical (r 0)

• Cylindrical to Spherical (r 0)

cos

sin 222

z

r

22

22

arccoszr

z

zr

More Examples:

1) Convert the point from spherical to rectangular

2) Find an equation in rectangular coordinates and graph.

3) Convert the equation to cylindrical and spherical,

,,9 4

sec2

zyx 22