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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?

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    $\begingroup$ I would think you could first transform coordinates through a non-rotation conformal map, and then define spherical coordinates in terms of the new variables to get a new, non-trivial set of orthogonal coordinates. $\endgroup$
    – BaronVT
    Commented Aug 20, 2013 at 17:38

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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).

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