concatenation of rotations in spherical coordinates

Question

Any rotation in $R{^3}$ can be expressed by a point $(\alpha,\beta)$ on the sphere in spherical coordinates. Is there any way to compute the concatenation of two rotation in spherical coordinates without translating back and forth to cartesian?

Answer 1

The set of points of the sphere $S^2$ is 2-dimensional; the set $SO(3)$ of all rotations of 3-space fixing the origin is 3-dimensional. The map from $S^1 \times S^1$ to $S^2$ given by polar coordinates is continuous.

And there's no (bi)continuous bijection between a 2-manifold and a 3-manifold.

So all this goes to show that you first statement is false, so the second is sort of unanswerable.

Sorry to be disappointing.

Answer 2

It may be a terminological issue, by rotations I do not mean all rotation positions of the sphere, I mean rotations around an axis.