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  Analytic Geometry: Spherical Coordinates  

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Spherical Coordinates

Spherical coordinates are obtained by using polar coordinates in a plane, adding a vertical axis perpendicular to the plane passing through the pole, and assigning a positive direction to it. The first coordinate of any point P is the distance rho of P from the pole O. The second coordinate of P is the angle theta from the polar axis to the projection of OP into the plane. The third coordinate of P is the angle phi between that positive part of the vertical axis and the line segment OP. If P = O, then rho = 0 and any values of theta and phi can be used. This gives the three coordinates (rho,theta,phi) of any point.

To transform from rectangular Cartesian coordinates (x,y,z) to spherical ones and back, use the following formulas:

x = rho cos(theta)sin(phi),
y = rho sin(theta)sin(phi),
z = rho cos(phi),

rho = ±sqrt(x2+y2+z2),
theta = arctan(y/x),
phi = arccos(z/sqrt[x2+y2+z2]).

The value of phi is usually chosen in the interval between 0 and Pi, inclusive. The sign of rho is determined by which of the values of the arctangent function is chosen:

  Sign of x     Sign of y     Quadrant of theta     Sign of r  
++ I+
++ III
+ II+
+ IV
+ II
+ IV+

The quadrant of theta can always be chosen to make rho positive, if it is so desired.

Common Uses

The most common use for spherical coordinates is in situations where a function has values that are spherically symmetrical; that is, where they depend only on the distance rho from the origin.


  • The equation of a sphere of radius R and center at the origin is

    rho = R.

  • The equation of a right circular cone with vertex at the origin is

    phi = Arctan(m).

  • The equation of a right circular cylinder with radius R and axis the line phi = 0 is

    rho = R csc(phi).

Sometimes a change from rectangular to spherical coordinates makes computing difficult multiple integrals simpler.

Compiled by Robert L. Ward.

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