Spherical coordinate system

The location of a point in three-dimensional space can be represented in various ways, but three numbers are always required. Spherical coordinates have coordinates typically named <math>(r, \theta, \phi)</math> where the radius <math>r</math> range from 0 to <math>\infin</math>, the colatitude <math>\theta</math> range from 0 to π and the azimuth <math>\phi</math> range from 0 to 2π. They describe a point in space as follows: from the origin <math>(0, 0, 0)</math>, go <math>r</math> units along the z-axis, rotate <math>\theta</math> down from the z-axis in the x-z plane (colatitude), and rotate <math>\phi</math> counterclockwise about the z-axis (azimuth or longitude). The name of the system comes from the fact that the simple equation <math>r = 1</math> describes the unit sphere.

There are conversions between Cartesian and spherical coordinates based on trigonometric functions. Both spherical coordinates and cylindrical coordinates are extensions of the two dimensional polar coordinate system. Spherical coordinates are the natural coordinates for physical situations where there is spherical symmetry. In such a situation, one can describe waves using spherical harmonics.

Unlike Cartesian coordinates, spherical coordinates include some redundancy in naming points, especially ones on the z-axis. For instance, (1, 0°, 0°), (1, 0°, 45°), and (-1, 180°, 270°) all describe the same point. Spherical coordinates emphasize distance from the origin. One application is ergodynamic design, where <math>r</math> is the arm length of a stationary person and the angles describe the direction of the arm as it reaches out.

Conversion from spherical to Cartesian coordinates

x = r sinθ cosφ

y = r sinθ sinφ

z = r cosθ

Conversion from Cartesian to spherical coordinates

<math>r = \sqrt{x^2 + y^2 + z^2}</math>

<math>\theta = \operatorname{arccos}\frac{z}{r}</math>

<math>\phi = \arctan\frac{y}{x}</math>

See also

• gimbal lock[?]