# Prove that a pseudo-hyperbolic ball is a Euclidean ball. Find the radius and center of the Euclidean ball.

We have that the pseudo-hyperbolic metric in the open unit disk $\mathbb D$ is defined by
$$\rho(z,w) = |\phi_w(z)|, \qquad \phi_w(z) = \frac{w - z}{1 - \overline w z}$$ where $z,w \in \mathbb D.$ It is also know that $\phi_w(z)$ is an automorphism of the unit disk.

Then for $r$, $0 \lt r \lt 1$ and $\alpha \in \mathbb D$, the set $$P(\alpha,r) = \{ z \in \mathbb D : \rho(z,\alpha) \lt r \}$$ is the pseudo-hyperbolic ball with center $\alpha$ and radius $r.$

I'm able to show that $P(\alpha,r)$ = $\phi_\alpha(r\mathbb D).$ Then it follows (at least intuitively) that since $\phi_\alpha$ maps circles to circles that $\phi_\alpha(r\mathbb D)$ is a Euclidean ball, which implies that $P(\alpha,r)$ is a Euclidean ball.

I have two problems that I've been struggling with:

(1): I'd like to have a few algebraic steps in order to show more rigorously that $\phi_\alpha(r\mathbb D)$ is indeed a Euclidean ball.

(2): More importantly, determining explicitly the center and radius of the Euclidean ball. I've found online (numerous places) that the Euclidean center $\beta$ and radius $R$ are given by $$\beta = \frac{(1 - r^2)\alpha}{1 - r^2 |\alpha|^2} \qquad \text{and} \qquad R = \frac{r(1 - |\alpha|^2)}{1 - r^2 |\alpha|^2}$$

respectively. Every article says that the center and radius can be determined from a straightforward calculation but never actually shows the calculation. (I'm beginning to think it's a secret.) Rudin remarks in his book Function Theory of the Unit Ball that $z \in P(\alpha,r)$ if and only if $|\phi_\alpha(z)| \lt r.$ He then says that if we square this and use the definition of $\phi_\alpha(z)$, then a little manipulation will give the desired formulas for the center and radius. I've tried squaring $|\phi_\alpha(z)| \lt r$ and then working forward and backwards to come up with the algebraic steps to solve for the center and radius but have not been successful. Maybe there is also a more sophisticated way to find the center and radius using the fact that $\phi_\alpha(z)$ is an automorphism of the unit disk

I'm hoping that someone could help me out with problems (1) and (2). Any help would be greatly appreciated!

Let $w \in B(0,1)$, then we can write $w = |w|e^{i\theta}$ for some $\theta \in \mathbb{R}$. Consider the ball $B(0,r)$, then what line through the origin will $\phi_{|w|}(B(0,r))$ be symmetric about?
Once you figure this out, you should be able to identify the two intersection points of the circle and line, say $\alpha$ and $\beta$. Then, by the symmetry, the true euclidean center will be the midpoint between $\alpha$ and $\beta$ and its radius will follow. Note then, once you have the euclidean center and radius, the only difference between $|w|$ and $w$ is the rotation $e^{i\theta}$, which does not affect the radius and will completely determine the Euclidean ball with center: midpoint $m_{\alpha,\beta}$.
As for justification of circles to circles, note that: $P(w,r) = \phi_w(P(0,r))$. Using what you have above with this should be sufficient.