Spherical Coordinates

Pre-requisites: Cartesian Coordinates.

Spherical coordinates, like cylindrical coordinates are another extension of polar coordinates into three dimensional space. All of these coordinate systems have a theta, the angle made with the xz-plane, variable; spherical coordinates has the additional two variables rho, the distance from the origin, and phi, the angle with the z-axis. In formula,

rho = sqrt( x2 + y2 + z2 )
theta = tan-1 y/x
Once again, every point in space now has a triplet (rho, theta, phi) of spherical coordinates, and if we restrict 0 <= theta < 2 pi and 0 <= phi < pi, then every point not on the z-axis (i.e, rho > 0) has an unique spherical coordinate.

Spherical coordinates are useful in describing geometric objects with (surprise) spherical symmetry; i.e., rotational symmetry about the origin. For example, the implicit equation rho = 3 describes a sphere with raidus 3 about the origin. An object with spherical symmetry will not have any mention of theta or phi when you convert it to spherical coordinates.

The following table summarizes all the conversions you will need between cartesian and spherical coordinates:

Cartesian Spherical
x
rho cos(theta) sin(phi)
y
rho sin(theta) sin(phi)
z
rho cos(phi)
sqrt(x2 + y2 + z2 )
rho
tan-1 (y/x)
theta
cos-1(z/rho)
phi

Exercises:

  1. What does the set of all points with rho = 5 look like?
  2. What does the set of all points with phi = pi / 5 look like?
  3. What does a plane look like in spherical coordinates? What about a line?
  4. Convert the point with cartesian coordinates (1,2,3) to spherical coordinates.
  5. Convert the point with spherical coordinates (rho = 2, theta = pi, phi = pi/4) to cartesian coordinates.
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Last modified 1 July 1997