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Searching for empty convex polygons

Algorithmica volume 5, pages 561–571 (1990)Cite this article

Abstract

A key problem in computational geometry is the identification of subsets of a point set having particular properties. We study this problem for the properties of convexity and emptiness. We show that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon. A linear-time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles. This result is then extended to an algorithm for finding empty convex r-gons (r> 3) and for determining a largest empty convex subset. Finally, extensions to higher dimensions are mentioned.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Princeton University, 08544, Princeton, NJ, USA

    David P. Dobkin

  2. Department of Computer Science, University of Illinois at Urbana-Champaign, 61801, Urbana, IL, USA

    Herbert Edelsbrunner

  3. Department of Computer Science, University of Utrecht, P.O. Box 80012, NL-3508, TA Utrecht, The Netherlands

    Mark H. Overmars

Authors
  1. David P. Dobkin
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  2. Herbert Edelsbrunner
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  3. Mark H. Overmars
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Additional information

Communicated by C. K. Wong.

The first author is pleased to acknowledge support by the National Science Foundation under Grant CCR-8700917. The research of the second author was supported by Amoco Foundation Faculty Development Grant CS 1-6-44862 and by the National Science Foundation under Grant CCR-8714565.

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Dobkin, D.P., Edelsbrunner, H. & Overmars, M.H. Searching for empty convex polygons. Algorithmica 5, 561–571 (1990). https://doi.org/10.1007/BF01840404

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  • DOI: https://doi.org/10.1007/BF01840404

Key words

  • Computational geometry
  • Empty convex subsets
  • Analysis of algorithms
  • Combinatorial geometry