Homeomorphism of unit disc onto itself interchanging two points.

Question

I've searched this topic already and found similar question.

But it was not fully answered.

Actually I know the Möbius transformation of form

$\frac{z-a}{1-\bar a z}$ is the automorphism of unit disc and interchanges $0$ and $a$.

But if I want it to change two arbitrary points in the disc. The above mapping fails.

Then how does the mapping which satisfies the condition on the title look like?

I'd like to see the mapping itself. (I mean like $f(z)=z$)