Can I convert between a rotation about an axis and a rotation according to two angles (all in 3D) without solving a system of nonlinear equations?

Question

I am writing a program that needs to be able to switch between a rotation described by 2 angles to a rotation described an axis and one angle. I found one way to do this from this question, which does the conversion all in one step and has equations of degree 2.
Is there an algorithm that does not require such complex computation all at once? Are there certain assumptions that might simplify the computation?

Answer 1

Nope. You are lucky there is method that will give and answer in finite steps, instead of iterating until convergence. And for a computer the calculation is not much more complex than algebra.