We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac| ≥ |bc|. We analyze the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order. Lastly, we provide an algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.

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Keywords Involute curve, Self-approaching path, Shortest path, Simple polygon
Persistent URL dx.doi.org/10.4230/LIPIcs.SoCG.2017.21
Conference 33rd International Symposium on Computational Geometry, SoCG 2017
Bose, P, Kostitsyna, I. (Irina), & Langerman, S. (Stefan). (2017). Self-approaching paths in simple polygons. In Leibniz International Proceedings in Informatics, LIPIcs (pp. 211–2115). doi:10.4230/LIPIcs.SoCG.2017.21