In three-dimensional spherical coordinates, which are often used in vector analysis, the coordinates are as follows:
  • The radial coordinate r, describing the distance from the origin, and ranging from zero to infinity.
  • Two angular coordinates: (1)the polar angle θ(theta), which starts at the positive z-axis and ranges from zero to π(pi); and (2)the azimuthal angle φ(phi), which is restricted to the xy plane, starts at the positive x-axis, and ranges from zero to 2π. Both are, of course, measured in radians.
Occasionally, a different notation is used with θ being used to denote the azimuthal rather than the polar angle, and vice versa. It is equivalent to the one given here. Also (thanks unperson), the polar angle can be measured starting from the equator rather than the pole, with a range of values from -π/2 to π/2.

To transform a vector in spherical coordinates to Cartesian coordinates, use

  • x = r*sin(θ)*cos(φ)
  • y = r*sin(θ)*sin(φ)
  • z = r*cos(θ)

The Greek letters used in the formulae for map projections also represent the angles1 of spherical coordinates, but of course, they mean different things. Just for simplicity's sake, mind you.

1The radius is, of course, constant.

Spherical coordinates are a generalisation of polar coordinates to three dimensions. Spherical coordinates are based upon a set of three orthogonal coordinate axes, called, as always, x, y, and z. From this, we can define the spherical coordinates of a point by drawing a line from the origin of the coordinate system to the point under consideration.

  • The radius is the length of the line joining the point to the origin. It thus ranges from zero to infinity.
  • The polar angle is the angle between the line and the positive z-axis. It ranges from 0 to π.
  • The azimuthal angle is the angle between the projection of the line onto the x-y plane and the positive x-axis (going counter-clockwise). It ranges from 0 to 2π.

There are two prevailing conventions for the notation used for spherical coordinates. The one commonly used by mathematicians denotes the triplet (radius, polar angle, azimuthal angle) as (ρ, φ, θ)A. The one commonly used by physicists denotes the same triplet as (r, θ, φ)B. This can be a source of confusion. For the rest of this writeup, the latter convention will be used.

In terms of Cartesian coordinates, the spherical coordinates are given by:
r = (x2 + y2 + z2)1/2
θ = arccos(z/r)
φ = arctan(y/x)

and in terms of spherical coordinates, the Cartesian coordinates are:
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos φ

The volume element in spherical coordinates is:
dV = r2sin θ dr dθ dφ

Sometimes, especially in cartography, the latitude is used instead of the polar angle, where the latitude of a point is defined as the angle between the line to that point and the x-y plane. It therefore runs from -π/2 to π/2 and is equal to (π/2 - θ).

Spherical coordinates are best used where there is some measure of spherical symmetry to the problem. Alternatives to spherical coordinates are cylindrical coordinates and Cartesian coordinates (also called rectangular coordinates).

A: C.H. Edwards and D.E. Penney, Calculus with Analytic Geometry, Fifth Edition, 1998.
B: G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists, Fifth Edition, 2001.

This writeup is copyright 2001-2004 D.G. Roberge and is released under the Creative Commons Attribution-NoDerivs-NonCommercial licence. Details can be found at .

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