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