Cylindrical and Spherical
Coordinates
Written by Dr. Julia Arnold
Associate Professor of Mathematics
Tidewater Community College, Norfolk Campus, Norfolk, VA
With Assistance from a VCCS LearningWare Grant
In this lesson you will learn
•about cylindrical and spherical coordinates
•how to change from rectangular coordinates to cylindrical
coordinates or spherical coordinates
•how to change from spherical coordinates to rectangular
coordinates or cylindrical coordinates
•how to change from cylindrical coordinates to rectangular
coordinates or spherical coordinates
Polar Coordinates
The polar coordinates r (the radial coordinate) and (the
angular coordinate, often called the polar angle) are defined
in terms of Cartesian Coordinates by
x  r cos 
where r is the radial distance from the origin,
and  is the counterclockwise angle from the xy  r sin 
axis.
In terms of x and y, r  x 2  y 2
  tan 1
y
x
Cylindrical coordinates are a generalization of two-dimensional
polar coordinates to three dimensions by superposing a height (z)
axis.
A point P is represented by an
ordered triple of r, , z  .
As you can see, this coordinate
system lends itself well to
cylindrical figures.
To change from rectangular to cylindrical:
r  x2  y2
  tan 1
y
x
zz
To change fromx  r cos
cylindrical to
y  r sin  ,
rectangular:
zz
Common Uses
The most common use of cylindrical coordinates is to give the
equation of a surface of revolution. If the z-axis is taken as the
axis of revolution, then the equation will not involve theta at all.
Examples:
A paraboloid of revolution might have equation
z = r2. This is the surface you would get by rotating the parabola
z = x2 in the xz-plane about the z-axis. The Cartesian coordinate
equation of the paraboloid of revolution would be z = x2 + y2.
A right circular cylinder of radius a whose axis is the z-axis has
equation
r = R.
A a sphere with center at the origin and radius R will have
equation
r + z2 = R2.
A right circular cone with vertex at the origin and axis the zaxis has equation
z = m r.
As another kind of example, a helix has the following equations:
r = R,
z = a theta.
http://mathforum.org/dr.math/faq/formulas/faq.cylindrical.ht
ml
Express the point (x,y,z) = (1,  3 ,2) in cylindrical coordinates.
Solution:
r  1  3  2
  3  

  tan 
 n

3
 1 
zz
1
Work it out before you go to the next slide.
Express the point (x,y,z) = (1,  3 ,2) in cylindrical coordinates.
Solution:
r  1  3  2
  3  

  tan 
 n

3
 1 
zz
1
You have two choices for r and infinitely many choices for theta.


Thus the point 1, 3 ,2 can be represented by non unique cylindrical
coordinates. For example
1,

  
3 ,2   2,
,2 
 3 
or
1,

2 

3 ,2    2,
,2 
3 

See picture on next slide.
z
This graph was
done using Win
Plot in the two
different
coordinate
systems.
(1,-sqr3,2)
(2,-pi/3,2)
(-2,-2pi/3,2)
y
x
The animation at the link below shows the points
represented by constant values of the first coordinate
as it varies from zero to one.
http://www.math.montana.edu/frankw/ccp/multiworld/
multipleIVP/cylindrical/body.htm
The animation below the one above shows the points
represented by constant values of the second
coordinate as it varies from zero to 2 pi.
You can also view at this link:
http://www.tcc.edu/faculty/webpages/JArnold/movies.htm
Example 2 Identify the surface for each of the following
equations.
(a) r = 5
(b) r 2  z 2  100
(c) z = r
Solution:
a. In polar coordinates we know that r = 5 would be a circle of radius
5 units. By adding the z dimension and allowing z to vary we create a
cylinder of radius 5.
5
Example 2 Identify the surface for each of the following
equations.
(a) r = 5
(b) r 2  z 2  100
(c) z = r
Solution:
2
2
2
b. This is equivalent to x  y  z  100which we know to be a sphere
centered at the origin with a radius of 10.
10
10
10
Example 2 Identify the surface for each of the following
equations.
(a) r = 5
(b) r 2  z 2  100
(c) z = r
Solution:
c. Since the radius equals the height and the angle is any angle we
get a cone.
z
y
x
Spherical coordinates are a system of curvilinear coordinates that
are natural for describing positions on a sphere or spheroid.
The ordered triple is:
 , ,   (rho, theta, phi)
For a given point P in spherical coordinates
 is the distance between P and the origin
0

is the same angle theta used in
cylindrical coordinates for r  0

is the angle between the
positive z-axis and the line
segment OP, 0    
The figure at right shows the
Rectangular coordinates (x,y,z) and
The spherical coordinates
, , 
P (x,y,z)

O
z
Conversion Formulas:
Spherical to Rectangular:
x   sin  cos  , y   sin  sin , z   cos 
Rectangular to Spherical:

y
z
  x  y  z , tan   ,   arccos  2
 x  y2  z 2
x

2
2
2
Spherical to cylindrical ( r  0 ):
r 2   2 sin 2  ,    , z   cos 
Cylindrical to spherical ( r  0 ):



2
2 
 r z 
  r 2  z 2 ,    ,   arccos 
z




Example 3
A. Find a rectangular equation for the graph represented by the
cylindrical equation
r 2 cos 2  z 2  1
B. Find an equation in spherical coordinates for the surface
represented by each of the rectangular equations and identify the
graph.
2
2
2
1. x  y  z
2. x2  y 2  z 2  4 z  0
Answers follow
Example 3
A. Find a rectangular equation for the graph represented by the
cylindrical equation
r 2 cos 2  z 2  1
z
r 2 cos 2  z 2  1
y
x
r 2 (cos 2   sin 2  )  z 2  1
z
r 2 cos 2   r 2 sin 2   z 2  1
x2  y2  z 2  1
x
y
Example 3
B. Find an equation in spherical coordinates for the surface
represented by each of the rectangular equations and identify the
graph.
2
2
2
x   sin  cos  , y   sin  sin , z   cos 
1. x  y  z
x2  y2  z2
 2 sin 2  cos 2    2 sin 2  sin 2    2 cos 2 
 2 sin 2  (cos 2   sin 2  )   2 cos 2 
z
 2 sin 2    2 cos 2 
 2 sin 2    2 cos 2 
 2 sin 2 
1
2
2
 cos 
tan 2   1
tan   1

 3
4
,
4
A double cone.
x
y
Example 3
B. Find an equation in spherical coordinates for the surface
represented by each of the rectangular equations and identify the
graph.
2. x2  y 2  z 2  4 z  0
x   sin  cos  , y   sin  sin , z   cos 
x 2  y 2  z 2  4z  0
 2 sin 2  cos 2    2 sin 2  sin 2    2 cos 2   4  cosz  0
 2 sin 2  (cos 2   sin 2  )   2 cos 2   4  cos   0
 2 sin 2    2 cos 2   4  cos  0
 2 (sin 2   cos 2  )  4  cos   0
 2  4  cos  0
2
 4 cos 

  4 cos
A sphere
x
y
Rectangular to cylindrical:
r  x2  y2
  tan
Cylindrical to
rectangular:
1
y
x
x  r cos 
y  r sin 
Spherical to Rectangular:
Rectangular to Spherical:
x   sin  cos  , y   sin  sin , z   cos 
  x2  y2  z 2 , tan  
Spherical to cylindrical ( r  0 ): r 2 
Cylindrical to spherical ( r  0 ):

y
z
,   arccos 
 x2  y 2  z 2
x





 2 sin 2  ,    , z   cos 



2
2 
 r z 
  r 2  z 2 ,    ,   arccos 
z
For comments on this presentation you may email the author
Dr. Julia Arnold at jarnold@tcc.edu.