# How to find a möbius transformation mapping $B(p, r)$ to $B(0, 1)$?

I have an open connected $$\Omega$$ of the complex plane and for a point $$p \in \Omega$$ and $$r > 0$$ we have a ball $$B(p, r) \subset \Omega$$. Is there a general möbius transformation to map this to B(0, 1)?

I know I can find a möbius transformation between two sets circles if I have three distinct points from each, but in this case I don't have any points to work with from $$B(p, r)$$. Or could I go with something like $$p_1 = (a+r) + p$$, $$p_2 = (a-r) + p$$, and $$p_3 = (a+ir) + p$$? If yes, how would I proceed?

I am stuck.

Thank you for your time, Isak

You can do that even with a linear transformation: We have $$z \in B(p, r) \iff |z-p| < r \iff | \frac {z-p}r - 0 | < 1 \iff \frac {z-p}r \in B(0, 1)$$ so that $$f(z) = \frac {z-p}r$$ maps $$B(p, r)$$ conformally onto $$B(0, 1)$$.
All Möbius transformations mapping $$B(p, r)$$ onto $$B(0, 1)$$ are given by $$T \circ f$$, where $$T$$ is an automorphism of the unit disk.