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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?

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up vote 3 down vote accepted

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 :)

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I suppose the answer to the first question is to point out that from a point $P$ not on a line $\ell$, there’s a unique line perpendicular to $\ell$ passing through $P$. And that you measure the distance along this line. – Lubin Dec 17 '13 at 16:50
Thank you Igor. I don't understand the picture, but I got something out of the wikipedia article: in the "upper half plane model" for the hyperbolic plane, the positive y-axis is a straight line. With respect to this line the shapes I am asking about in (2), called hypercycles, are the rays through the origin at any angle between 0 and pi. That is good enough for me! But now I would like to know: what is the distance between the hypercycle at angle $\theta$ and the straight line at angle $\pi/2$? – Hyperbolic Asker Dec 17 '13 at 16:54
Look at the distance formula here: – Igor Rivin Dec 17 '13 at 16:56
Take the two points to be $i$ (so, $(0, 1)$ in coordinates) and $(\sin(\theta), \cos(\theta).$ (This will be a point on your equidistant whose closest point is $i.$) – Igor Rivin Dec 17 '13 at 16:57
@HyperbolicAsker Sounds good... – Igor Rivin Dec 17 '13 at 17:02

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).

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