TITLE: Are spherical coordinates unique orthogonal coordinates on sphere?
QUESTION [5 upvotes]: 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?

REPLY [9 votes]: 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).