Computing shortest homotopic cycles on polyhedral surfaces with hyperbolic uniformization metric

The problem of computing shortest homotopic cycles on a surface has various applications in computational geometry and graphics. In general, shortest homotopic cycles are not unique, and local shortening algorithms can become stuck in local minima. For surfaces with a negative Euler characteristic that can be given a hyperbolic uniformization metric, however, we show that they are unique and can be found by a simple locally shortening algorithm. We also demonstrate two applications: constructing extremal quasiconformal mappings between surfaces with the same topology, which minimize angular distortion, and detecting homotopy between two paths or cycles on a surface.

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