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I was trying to apply Poincare's Polygon theorem, for that I had to give a pairing of sides, i.e., to have an isometry of the hyperbolic plane that will take a side of a polygon to another side (of the same length). So my question is that -

Given any two hyperbolic line segments of equal hyperbolic length, is there an isometry of the hyperbolic plane that sends one of the line segments to the other?

Its is easy to see that there will be mobius transformations that will take one line segment to the other, but I couldn't see why will there be one such transformation that will also fix the upper half-plane (i.e. will be an isometry of the hyperbolic plane).

A proof of existence will be enough for me.

Agneedh Basu
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    Hi and welcome to MSE! I want to remind you that it is generally preferred you include context when asking a question here (which can include: where this problem came from, your own attempts, and a specific idea as to where you're stuck) - it also lets us help you better! As is, your question is little more than an isolated problem, and thus likely to get a lot of downvotes and closed. Feel free to edit the context into your post though! Here's some useful links: asking a good question and a MathJax reference to format your work. – PrincessEev Mar 24 '19 at 05:06
  • To amplify Eevee's comment ... Are you looking for an "abstract" proof that such an isometry exists? (What fundamental notions of isometries do you know?) Are you looking at a particular model —Poincaré disk? Upper Half-Plane?— and wanting to determine an explicit transformation function? (What do you know about such functions?) Explaining exactly what you know and/or want helps answerers tailor their responses, without wasting time (yours or theirs) telling you things you already know or using approaches you haven't seen. – Blue Mar 24 '19 at 05:31
  • I would use the hyperboloid model instead. Then the existence of such isometry is as obvious as on a sphere. – Zeno Rogue Mar 24 '19 at 23:50

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A Möbius transformation is uniquely defined by 3 points and their images. If you have $z_1\mapsto z_1'$ and $z_2\mapsto z_2'$ mapping the endpoints of the line segments, then add $\overline{z_1}\mapsto \overline{z_1'}$, i.e. map the complex conjugates for one point and its image. If the segments $(z_1,z_2)$ and $(z_1',z_2')$ are indeed of equal length, then the map defined by these three points will also map $\overline{z_2}\mapsto \overline{z_2'}$ and it will have a representation using real coefficients only, so that it preserves the real axis.

If some other reader wants the same for the Poincaré disk, use inversion in the unit circle instead of complex conjugate i.e. reflection in the real axis. The idea is that in a way, the upper and the lower half plane in the half plane model, or the inside and the outside (including the point at infinity) of the disk in the disk model, are algebraically pretty much equivalent. It makes sense to think of a hyperbolic point in the half plane model not as a single point in the upper half plane, but as a pair of points reflected in the real axis. Using this helps adding additional constraints for the Möbius transformation.

MvG
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