Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time

Authors:
Erin Wolf Chambers;Éric Colin de Verdière;Jeff Erickson;Sylvain Lazard;Francis Lazarus;Shripad Thite
Affiliations:
Department of Mathematics and Computer Science, Saint Louis University, USA;Département d'Informatique, École normale supérieure and CNRS, Paris, France;Department of Computer Science, University of Illinois at Urbana-Champaign, USA;INRIA Nancy -- Grand Est, LORIA, Nancy, France;GIPSA-Lab and CNRS, Grenoble, France;California Institute of Technology, Center for the Mathematics of Information, USA
Venue:
Computational Geometry: Theory and Applications
Year:
2010

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Abstract

The Frechet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Frechet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (''trees''). We describe a polynomial-time algorithm to compute the homotopic Frechet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.