2005-16.md

course	course_year	question_number	tags	title	year
Geometry	IB	16	IB 2005 Geometry	3.I.2A	2005

Write down the Riemannian metric on the disc model $\Delta$ of the hyperbolic plane. Given that the length minimizing curves passing through the origin correspond to diameters, show that the hyperbolic circle of radius $\rho$ centred on the origin is just the Euclidean circle centred on the origin with Euclidean $\operatorname{radius~} \tanh (\rho / 2)$. Prove that the hyperbolic area is $2 \pi(\cosh \rho-1)$.

State the Gauss-Bonnet theorem for the area of a hyperbolic triangle. Given a hyperbolic triangle and an interior point $P$, show that the distance from $P$ to the nearest side is at most $\cosh ^{-1}(3 / 2)$.