Spherical Coordinate System

The spherical coordinate system is defined by a point and two axes. A point P(r,a,b) in spherical coordinates is at distance r from the point and the horizontal angle of the arc to one of the axis a. The vertical angle to the plane, which is perpendicular to the other axis, is b.

This animation shows a point hovering in a 3D space "equipped" with a spherical coordinate system. Note that there are several ways to define the spherical coordinates and the one visualized here is closer to how a telescope is managed - it is first rotated horizontally and then vertically to the desired direction.

Conversion from polar to Cartesian coordinates is done by:

x = r.cos(a).cos(b)
y = r.sin(a).cos(b)
z = r.sin(b)

Another way of defining spherical coordinates is with inclination and azimuth angles. This will produce slightly different conversion equations - some sines will become cosines and vice versa.