Spherical Coordinates in 3D

Spherical coordinates are interesting to represent spherical shaped surfaces with short equations. The spherical coordinates of a point in space are determined by the radius r, the rotation angle u and the inclination angle v defined as follows:

The radius r is the distance between the point and the origin;
The rotation angle u is the angle measured from the x-axis to the vertical plane through the point and the z-axis;
The inclination angle v is the angle between the z-axis and the straight line through the point and the origin.

The equation of a surface in spherical coordinates is now an equation where the radius is expressed in terms of the rotation angle and the inclination angle. A sphere around the origin with radius 2 for example obtains the simple equation r = 2, with u between 0 and 2*pi and v between 0 and pi.

Here we see the spherical coordinates graphically.

The following surface shows a shell like structure as illustration. It has the equation r = u/2 with u between 0 and 4*pi and v between 0 and pi.

If we limit v between pi/2 and pi, we obtain a half of the shell, so that we can see the inner structure more clearly.