Shortest paths in arbitrary plane domains

Let \(\Omega\) be a connected open set in the plane and \(\gamma: [0,1] \to \overline{\Omega}\) a path such that \(\gamma((0,1)) \subset \Omega\). We show that the path \(\gamma\) can be ``pulled tight'' to a unique shortest path which is homotopic to \(\gamma\), via a homotopy \(h\) with...

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Bibliographic Details
Published in:arXiv.org 2019-03
Main Authors: Hoehn, L C, Oversteegen, L G, Tymchatyn, E D
Format: Article
Language:English
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Shortest path planning
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Summary:Let \(\Omega\) be a connected open set in the plane and \(\gamma: [0,1] \to \overline{\Omega}\) a path such that \(\gamma((0,1)) \subset \Omega\). We show that the path \(\gamma\) can be ``pulled tight'' to a unique shortest path which is homotopic to \(\gamma\), via a homotopy \(h\) with endpoints fixed whose intermediate paths \(h_t\), for \(t \in [0,1)\), satisfy \(h_t((0,1)) \subset \Omega\). We prove this result even in the case when there is no path of finite Euclidean length homotopic to \(\gamma\) under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a ``shortest'' path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.
ISSN:2331-8422
DOI:10.48550/arxiv.1903.06737