I’m considering Poincaré disk model. Let $z$ be any point in $D$, with $|z|>r$, where $0<r<1$. I need to find Poincare distance of this point from the circle $\{z:|z|=r\}.$ What I think is, I can rotate the point and bring it to |z| on real-axis. This rotation will rotate the circle and circle is again the same circle except for rotation of points on it, but since we’re considering distance of a point from a set, this doesn’t affect the distance. Precisely, I need to prove that required Poincare distance is $\rho(0, |z|)-\rho(0,r)$, where $\rho$ denotes the Poincare distance. I am unable to prove this using calculations, is there any way I can justify this without actually calculating this?
It appears apt when I think it in Euclidean way( after rotating the point on x-axis), but that isn’t correct. \ Edit: After calculations, I got that Poincare distance between the point $|z|$ and the point $r$ on x- axis is $\rho(0,|z|)-\rho(0,r)$. But how to ensure that this only is the required distance.