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On Landmark Distances in Polygons
Author(s) -
Gotsman C.,
Hormann K.
Publication year - 2021
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/cgf.14373
Subject(s) - polygon (computer graphics) , geodesic , landmark , gradient descent , path (computing) , simple polygon , mathematics , computer science , shortest path problem , combinatorics , point in polygon , algorithm , artificial intelligence , geometry , monotone polygon , graph , artificial neural network , telecommunications , frame (networking) , programming language
We study the landmark distance function between two points in a simply connected planar polygon. We show that if the polygon vertices are used as landmarks, then the resulting landmark distance function to any given point in the polygon has a maximum principle and also does not contain local minima. The latter implies that a path between any two points in the polygon may be generated by steepest descent on this distance without getting “stuck” at a local minimum. Furthermore, if landmarks are increasingly added along polygon edges, the steepest descent path converges to the minimal geodesic path. Therefore, the landmark distance can be used, on the one hand in robotic navigation for routing autonomous agents along close‐to‐shortest paths and on the other for efficiently computing approximate geodesic distances between any two domain points, a property which may be useful in an extension of our work to surfaces in 3D. In the discrete setting, the steepest descent strategy becomes a greedy routing algorithm along the edges of a triangulation of the interior of the polygon, and our experiments indicate that this discrete landmark routing always delivers (i.e., does not get stuck) on “nice” triangulations.