High Quality Content by WIKIPEDIA articles! In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem. We briefly present some facts from hyperbolic geometry which are helpful in understanding prime geodesics. There are then 3 types of transformation: hyperbolic, elliptic, and parabolic. The loxodromic transformations are not present because we are working with real numbers. Then an element ? of ? has 2 distinct real fixed points if and only if ? is hyperbolic.

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