(2) (Circles in the hyperbolic plane)
(a) Show that circles in the hyperbolic plane are Euclidean circles.
Hint: First explain why this is true for circles centered at 0 in the Poincaré disk. Then
use facts about isometries.
(b) Consider the circle in the Poincaré disk centered at 0 of Euclidean radius s. Show that
its hyperbolic radius is $\ln(\frac{1+s}{1-s})$ and its hyperbolic circumference is $\frac{4\pi s}{1-s^2}$.
Hint: Use the fact we showed in class, that the distance from $i$ to $bi$ in the upper half
plane model is $\ln b$. You may also use the fact that in the Poincaré disk (like in the
upper half-plane) hyperbolic distance is conformal, that is, for points $z$ and $z'$ that are
very close together,
$d_{hyp}(z, z') \approx const(z)|z - z'|$,
independent of the direction that $z$ lies from $z'$. This is actually equivalent to the fact
that Euclidean angles are the same as hyperbolic angles.
(c) Use this to derive the formula $2\pi \sinh(r)$ for the circumference of a circle of radius $r$ in
the hyperbolic plane.
