geometry and transformations: geodesics

jenniferwu3232

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Dec 9, 2013
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I have already solved that Φ (z) =z in the geodesic g, but I am stuck on this part of the problem:

Consider a point P ∈H^2. Let h be the complete geodesic that contains P and orthogonally meets g at a point Q. Show that Φ sends h to h. (use the fact that Φ respects angles). Conclude that Φ(P) is the point of the geodesic h that is at the same hyperbolic distance from Q as P, but is on the other side of g. This construction is the hyperbolic analogue of the Euclidean reflection across a straight line.

[We were previously given the following: Let g be a complete geodesic of H^2, which is a semicircle of radius R centered at C=(x_0,0). Consider the map Φ (z)= x_0 + R^2((z-x_0)/(|z-x_0|^2)) and that Φ is the inversion across g.]
 
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