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 \(\ensuremath{\mathcal{B}} \subseteq S\) of at least four points, one can always construct a polygon P such that the points of \(\ensuremath{\mathcal{B}}\) 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.
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., Vogtenhuber, B. (2012). Geodesic Order Types. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds) Computing and Combinatorics. COCOON 2012. Lecture Notes in Computer Science, vol 7434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32241-9_19
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DOI: https://doi.org/10.1007/978-3-642-32241-9_19
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