License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.10
URN: urn:nbn:de:0030-drops-87236
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Boissonnat, Jean-Daniel ; Lieutier, André ; Wintraecken, Mathijs

The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces

LIPIcs-SoCG-2018-10.pdf (0.7 MB)


In this paper we discuss three results. The first two concern general sets of positive reach: We first characterize the reach by means of a bound on the metric distortion between the distance in the ambient Euclidean space and the set of positive reach. Secondly, we prove that the intersection of a ball with radius less than the reach with the set is geodesically convex, meaning that the shortest path between any two points in the intersection lies itself in the intersection. For our third result we focus on manifolds with positive reach and give a bound on the angle between tangent spaces at two different points in terms of the distance between the points and the reach.

BibTeX - Entry

  author =	{Jean-Daniel Boissonnat and Andr{\'e} Lieutier and Mathijs Wintraecken},
  title =	{{The Reach, Metric Distortion, Geodesic Convexity and the Variation of Tangent Spaces}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{10:1--10:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-87236},
  doi =		{10.4230/LIPIcs.SoCG.2018.10},
  annote =	{Keywords: Reach, Metric distortion, Manifolds, Convexity}

Keywords: Reach, Metric distortion, Manifolds, Convexity
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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