What is a hyperbolic line in (a) the disc model (b) the upper half-plane model of the hyperbolic plane? What is the hyperbolic distance $d(P, Q)$ between two points $P, Q$ in the hyperbolic plane? Show that if $\gamma$ is any continuously differentiable curve with endpoints $P$ and $Q$ then its length is at least $d(P, Q)$, with equality if and only if $\gamma$ is a monotonic reparametrisation of the hyperbolic line segment joining $P$ and $Q$.

What does it mean to say that two hyperbolic lines $L, L^{\prime}$ are (a) parallel (b) ultraparallel? Show that $L$ and $L^{\prime}$ are ultraparallel if and only if they have a common perpendicular, and if so, then it is unique.

A horocycle is a curve in the hyperbolic plane which in the disc model is a Euclidean circle with exactly one point on the boundary of the disc. Describe the horocycles in the upper half-plane model. Show that for any pair of horocycles there exists a hyperbolic line which meets both orthogonally. For which pairs of horocycles is this line unique?