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(x^{2}+y^{2}+z^{2}),

theta = arctan(y/x),

phi = arccos(z/sqrt[x^{2}+y^{2}+z^{2}]).

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 | – |

– | – |
I | – |

– | – |
III | + |

+ | – |
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.

Examples:

- 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.