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Japanese Conference on Discrete and Computational Geometry and Graphs

JCDCGG 2013: Discrete and Computational Geometry and Graphs pp 44–64Cite as

On Polygonal Paths with Bounded Discrete-Curvature: The Inflection-Free Case

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8845))

Abstract

A shortest path joining two specified endpoint configurations that is constrained to have mean curvature at most \(\varsigma \) on every non-zero length sub-path is called a \(\varsigma \) -geodesic. A seminal result in non-holonomic motion planning is that (in the absence of obstacles) a \(1\)-geodesic consists of either (i) a (unit-radius) circular arc followed by a straight segment followed by another circular arc, or (ii) a sequence of three circular arcs the second of which has length at least \(\pi \) [Dubins, 1957]. Dubins’ original proof uses advanced calculus; Dubins’ result was subsequently rederived using control theory techniques [Sussmann and Tang, 1991], [Boissonnat, Cérézo, and Leblond, 1994], and generalized to include reversals [Reeds and Shepp, 1990].

We introduce and study a discrete analogue of curvature-constrained motion. Our overall goal is to show that shortest polygonal paths of bounded “discrete-curvature” have the same structure as \(\varsigma \)-geodesics, and to show that properties of \(\varsigma \)-geodesics follow from their discrete analogues as a limiting case, thereby providing a new, and arguably simpler, “discrete” proof of the Dubins characterization. Our focus, in this paper, is on paths that have non-negative mean curvature everywhere; in other words, paths that are free of inflections, points where the curvature changes sign. Such paths are interesting in their own right (for example, they include an additional form, not part of Dubins’ characterization), but they also provide a slightly simpler context to introduce all of the tools that will be needed to address the general case in which inflections are permitted.

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Notes

  1. 1.

    In an earlier draft [18], the authors proposed an alternative definition which had some deficiencies that are resolved by the definition used in this paper.

  2. 2.

    We ignore for the present the fact that successive maximal discrete circular arcs of opposite orientation could share an edge. In this case we are free to impose disjointness of arcs by assigning the shared edge to just one of the two arcs.

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Acknowledgements

We thank Sergey Bereg, Stefan Foldes, Irina Kostitsyna and Joe Mitchell for discussions.

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Authors and Affiliations

  1. Courant Institute, New York, USA

    Sylvester Eriksson-Bique

  2. University of British Columbia, Vancouver, Canada

    David Kirkpatrick

  3. University of Helsinki, Helsinki, Finland

    Valentin Polishchuk

Authors
  1. Sylvester Eriksson-Bique
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  2. David Kirkpatrick
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  3. Valentin Polishchuk
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Correspondence to David Kirkpatrick .

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Editors and Affiliations

  1. Tokyo University of Science, Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications, Tokyo, Japan

    Hiro Ito

  3. Tokai University, Tokyo, Japan

    Toshinori Sakai

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Eriksson-Bique, S., Kirkpatrick, D., Polishchuk, V. (2014). On Polygonal Paths with Bounded Discrete-Curvature: The Inflection-Free Case. In: Akiyama, J., Ito, H., Sakai, T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2013. Lecture Notes in Computer Science(), vol 8845. Springer, Cham. https://doi.org/10.1007/978-3-319-13287-7_5

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  • DOI: https://doi.org/10.1007/978-3-319-13287-7_5

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