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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 θ 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, ρ , which represents distance to the origin: ρ 2 = x 2 + y 2 + z 2 .

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

We define ϕ by cos ⁡ ϕ = z ρ , so that with r defined as always here by r 2 = x 2 + y 2 , we have sin ⁡ ϕ = r ρ .

The longitude angle θ is defined by tan ⁡ θ = y x , exactly as in two dimensions. We therefore have x = r cos ⁡ θ = ρ sin ⁡ ϕ cos ⁡ θ , and what is y ?

Exercises:

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.

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