Are spherical coordinates unique orthogonal coordinates on sphere?

Question

Spherical coordinates on unit sphere are defined by the following transformation:

$$\begin{cases}x=\sin\theta\cos\varphi\\ y=\sin\theta\sin\varphi\\ z=\cos\theta\end{cases}$$

Are these coordinates the only possible orthogonal coordinate system on sphere, up to rotation and coordinate scaling?

Answer 1

Definitely not. You could also use stereographic projection from the plane, with the standard $(x,y)$ coordinates on the plane, or any other orthogonal coordinates on the plane, for that matter. Stereographic projection is conformal (see http://en.wikipedia.org/wiki/Stereographic_projection).