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Geodesic Order Types

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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.

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Notes

  1. It is common to regard the properties defined by orientations of triples as the combinatorial ones. There are further settings on point sets that can be seen as being combinatorial as well, e.g., asking whether the fourth point of a quadruple lies inside the circle defined by the first three ones (see [12]). Also, the circular sequence of a point set is a richer way of describing the combinatorics of point sets, totally implying the order type [9].

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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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Authors and Affiliations

  1. Institute for Software Technology, Graz University of Technology, Graz, Austria

    Oswin Aichholzer, Alexander Pilz & Birgit Vogtenhuber

  2. Universitat Politècnica de Catalunya, Barcelona, Spain

    Matias Korman

Authors
  1. Oswin Aichholzer
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  2. Matias Korman
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  3. Alexander Pilz
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  4. Birgit Vogtenhuber
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Corresponding author

Correspondence to Matias Korman.

Additional information

Research supported by the ESF EUROCORES programme EuroGIGA—ComPoSe, Austrian Science Fund (FWF): I 648-N18 and grant EUI-EURC-2011-4306. M.K. received support of the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia and the European Union. A.P. is recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria.

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Aichholzer, O., Korman, M., Pilz, A. et al. Geodesic Order Types. Algorithmica 70, 112–128 (2014). https://doi.org/10.1007/s00453-013-9818-8

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