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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bannai, Eiichi / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Pasini, Antonio / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 17, Issue 3 (Jul 2017)

Issues

On the topology of the spaces of curvature constrained plane curves

José Ayala
  • Corresponding author
  • Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat, Av. Arturo Prat 2120, Iquique, Chile
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Published Online: 2017-07-22 | DOI: https://doi.org/10.1515/advgeom-2017-0015

Abstract

Plane curves with the same endpoints are homotopic; an analogous claim for plane curves with the same endpoints and bounded curvature still remains open. We find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed into each other by a continuous one-parameter family of curves also having bounded curvature. We conclude that the space of these curves has either one or two connected components, depending on the distance between the endpoints. The classification theorem presented here answers a question raised in 1961 by L. E. Dubins.

Keywords: Bounded curvature; regular homotopy; homotopy classes

MSC 2010: Primary 53A04; 53C42; Secondary 57N20

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About the article


Received: 2014-06-02

Revised: 2015-07-23

Published Online: 2017-07-22

Published in Print: 2017-07-26


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0015.

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