3.7 Cylindrical and Spherical Coordinates

In three dimensions there are two analogues of polar coordinates.

In cylindric coordinates, x and y are described by r and exactly as in two dimensions, while the third dimension, z is treated as an ordinary coordinate.
r then represents distance from the z axis.

In spherical coordinates, a general point is described by two angles and one radial variable, , which represents distance to the origin:

The two angular variables are related to longitude and latitude, but latitude is zero at the equator, and the variable that we use is 0 on the z axis (which means at the north pole).

We define , so that with r defined as always here by

The longitude angle is defined by , exactly as in two dimensions. We therefore have , and what is y?


3.12 Express the parameters of cylindric and spherical coordinates in terms of x, y and z.

3.13 Construct a spreadsheet converter which takes coordinates x, y and z and produces the three parameters of spherical coordinates; and vice versa. Verify that they work by substituting the result from one as input into the other.