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Algorithmica

, Volume 70, Issue 1, pp 112–128 | Cite as

Geodesic Order Types

  • Oswin Aichholzer
  • Matias Korman
  • Alexander Pilz
  • Birgit Vogtenhuber
Article

Abstract

The geodesic between two points a and b in the interior of a simple polygon P is the shortest polygonal path inside P that connects a to b. It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments. In this paper we use this extension to generalize the concept of the order type of a set of points in the Euclidean plane to geodesic order types. In particular, we show that, for any set S of points and an ordered subset \(\mathcal {B} \subseteq S\) of at least four points, one can always construct a polygon P such that the points of \(\mathcal {B} \) define the geodesic hull of S w.r.t. P, in the specified order. Moreover, we show that an abstract order type derived from the dual of the Pappus arrangement can be realized as a geodesic order type.

Keywords

Order types Stretchability Pappus arrangement Geodesic Simple polygon 

Notes

Acknowledgements

The authors would like to thank Prosenjit Bose, Stefan Langerman, and Pat Morin for the introduction of the topic and for several fruitful discussions on it.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Oswin Aichholzer
    • 1
  • Matias Korman
    • 2
  • Alexander Pilz
    • 1
  • Birgit Vogtenhuber
    • 1
  1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
  2. 2.Universitat Politècnica de CatalunyaBarcelonaSpain

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