University Mathematics Handbook (2015)
III. Analytic Geometry and Vectors
Chapter 9. Cylindrical and Spherical Coordinates
9.1 Cylindrical Coordinates
a. A cylindrical coordinate system combines the polar coordinates in a plane (see II 1.2), with vertical coordinate 
in a space. The cylindrical coordinates of point 
in space are ordered set of three numbers 
, when 
are the polar coordinates of projection 
of point on plane 
(see illustration). Remember that 
is distance, and therefore can be either positive or zero, and angle 
varies from zero to 
.
Formulas of conversion from cylindrical coordinates 
to Cartesian coordinates 
are
To reverse the conversion, from to
b. In cylindrical coordinates, the cylindrical surface equation has a simpler form.
Example: The equation of cylinder 
is
, 
, 
That is, it is the equation of the locus for all sets of points at distance 
from 
-axis.
9.2 Spherical Coordinates
a. Point 
in space is defined by the set of three numbers 
, when 
is the length of vector 
(when 
is the origin), 
is the angle of projection 
with the positive direction of 
-axis on plane 
, and 
is the angle formed by vector with the positive direction of -axis.
are called the spherical coordinates of
point .
b. Relation between spherical and Cartesian coordinates:
1. If is a point of spherical coordinates , then
.
2. If 
are Cartesian coordinates of point , then
when only has positive values, values vary from zero to 
and angle varies from zero to 
.
c. Examples:
1. The spherical coordinates of point (2,1,2) are:
2. The equation of sphere 
in spherical coordinates is
d. Note: Sometimes angle is referred to as the angle between vector and plane 
. In the illustration above, it is 
.
In such a case, is substituted with 
, so the equation of sphere 
in spherical coordinates is
.