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Given two points in the upper half-plane with the usual hyperbolic metric, the geodesic between them is found by drawing a circle through them that crosses the real axis at right angles. However, if I give you coordinates for the points, how do I construct such a circle explicitly? I guess this becomes a question in classical plane geometry, but I don't see how to do it.

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The center of the circle lies on the real axis, and also on the perpendicular bisector of the two points, so...

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What if one of the points lies on the interior of the upper half-plane and the other on the real axis. How to describe the geodesic that goes from the interior point to the boundary point (in infinite time)? –  user27347 Nov 13 '12 at 17:06
    
I thought you wanted a Euclidean geometry construction. The same thing works: i.stack.imgur.com/vu9Ry.png –  Rahul Nov 13 '12 at 17:15
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