# Equidistant curves in the Half-Plane model.

Definition: An equidistant curve can be one of the three following: A hyperbolic circle, a horocycle or an equidistant line. In the Half-Plane model, a hyperbolic circle is represented by an euclidian circle entirely above the boundary line; a horocycle (Which is a circle with infinite radius centered at infinity) is represent by an euclidean circle tangent to the boundary line or a horizontal line (In this case it will be centered at the upper infinity point); an equidistant line is an euclidean line or a portion of an euclidean circle that makes a non-right angle with the boundary line.

Obs.: Equidistant lines are also called hypercycles.

Then I need to prove this: Let $\mathbb{E}$ be an equidistant curve, and let $A,B \in \mathbb{E}$. Show that there is a constant $k=k(\mathbb{E},A,B)>0$ such that for every $C,D \in \mathbb{E}$ the following is true: If $A,B$ are in the arc defined by $C,D$, then the lenght of $CD$ is bigger than $k$.

I made an image of the Half-Plane Model showing all the kinds of equidistant curves.

I tried using that $k$ could be the lenght of $AB$, but I'm not sure how to write this neither if I can use this. Can someone help me? Thanks.

All in an earlier post Definition of hyperbolic lenght. I was wondering about the translation.

What is an equidistand line? I guess you mean a hypercycle. https://en.wikipedia.org/wiki/Hypercycle_(geometry)

But an hypercycle does not have a lot in common with a line. (see the wikipedia article)

Calling an horocycle an "a circle with infinite radius centered at infinity" while sadly quite common is a sad comparison. It is like calling a parabola some kind of ellipse (the comparison is in fact very close see Studying the hyperboloid model, what is represented by the conic sections? )

It looks like you have to proof that lines, hypercycles, horocycles and circles are of infinite length. But not al circles are of infinite length so your proof will fail.

But maybe this has to do with the translation.

The Image is correct, but do add the two kind of lines to it as well :)

Good luck

• but is it true for horocycles and hypercycles? – user286485 Jul 5 '16 at 20:16
• Horocycles and hypercycles have an infinite length – Willemien Jul 5 '16 at 21:47
• how can I prove the statement for horocycles and hypercycles? – user286485 Jul 5 '16 at 23:44
• How is K(E,A,B) defined? – Willemien Jul 6 '16 at 0:00
• sadly all my book says about $k$ is what I typed above "for every C,D ∈ E the following is true: If A,B are in the arc defined by C,D, then the lenght of CD is bigger than k." – user286485 Jul 6 '16 at 0:22