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Shortest paths for line segments

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Abstract

We study the problem of shortest paths for a line segment in the plane. As a measure of the distance traversed by a path, we take the average curve length of the orbits of prescribed points on the line segment. This problem is nontrivial even in free space (i.e., in the absence of obstacles). We characterize all shortest paths of the line segment moving in free space under the measured 2, the average orbit length of the two endpoints.

The problem ofd 2 optimal motion has been solved by Gurevich and also by Dubovitskij, who calls it Ulam's problem. Unlike previous solutions, our basic tool is Cauchy's surface-area formula. This new approach is relatively elementary, and yields new insights.

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Author information

Authors and Affiliations

  1. Praktische Informatik VI, FernUniversität Hagen, W-5800, Hagen, Germany

    Christian Icking

  2. Institut für Mathematik, Technische Universität Graz, Kopernikusgasse 24, A-8010, Graz, Austria

    Günter Rote

  3. Institut für Informatik, Freie Universität Berlin, Arnimallee 2-6, 1000, Berlin 33, Germany

    Emo Welzl

  4. Courant Institute, New York University, 10012, New York, NY, USA

    Chee Yap

Authors
  1. Christian Icking
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  2. Günter Rote
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  3. Emo Welzl
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  4. Chee Yap
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Additional information

Communicated by Bruce Randall Donald.

This work was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM) and by the Deutsche Forschungsgemeinschaft Grant Ot 64/5-3. Chee Yap acknowledges support from the Deutsche Forschungsgemeinschaft, and partial support from NSF Grants DCR-8401898 and DCR-8401633.

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Icking, C., Rote, G., Welzl, E. et al. Shortest paths for line segments. Algorithmica 10, 182–200 (1993). https://doi.org/10.1007/BF01891839

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  • DOI: https://doi.org/10.1007/BF01891839

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