Distance from a point to a line in the hyperbolic plane

Question

I have two questions:

1. What is the distance from a point to a line in the hyperbolic plane?

2. Fix a line $L$ in the hyperbolic plane. What does the set of points of distance $d$ from $L$ look like?

Answer 1

I am not really sure I understand the first question (it depends on how you give your points, lines, etc). As for "what it looks like" it is known as (surprisingly :)) an equidistant curve (or a hypercycle, though I had never heard that name before today). The linked article has pictures :)

Answer 2

I met with the following interpretation of the hyperbolic metric and it should answer your questions. Length of curve $\gamma$ parametrized by $t \mapsto (x(t), y(t)) \in \mathbb{R}^2$, $t \in [0,1]$, $y \ge 0$ in hyperbolic metric is: $$L[\gamma] = \int_{0}^{1} \frac{1}{y}\sqrt{\dot{x}^2 + \dot{y}^2}.$$ It's not very difficult exercise to show that shortest curve connecting two points is semicircle with center on line $y=0$ (if $x$-coordinates of both points are equal then the shortest curve is vertical ray).