Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time |
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| Authors: | Erin Wolf Chambers, ric Colin de Verdi re, Jeff Erickson, Sylvain Lazard, Francis Lazarus,Shripad Thite |
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| Affiliation: | aDepartment of Mathematics and Computer Science, Saint Louis University, USA;bDépartement d'Informatique, École normale supérieure and CNRS, Paris, France;cDepartment of Computer Science, University of Illinois at Urbana-Champaign, USA;dINRIA Nancy – Grand Est, LORIA, Nancy, France;eGIPSA-Lab and CNRS, Grenoble, France;fCalifornia Institute of Technology, Center for the Mathematics of Information, USA |
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| Abstract: | The Fréchet 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 Fréchet 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 Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles. |
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| Keywords: | Homotopy Similarity of curves Metric space Homotopic Fré chet distance Geodesic leash map Punctured plane |
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