]> 3.7 Cylindrical and Spherical Coordinates

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.