Abstract
LetP be a convex polytope withn facets in the Euclidean space of a (small) fixed dimensiond. We consider themembership problem forP (given a query point, decide whether it lies inP) and theray shooting problem inP (given a query ray originating insideP, determine the first facet ofP hit by it). It was shown in [AM2] that a data structure for the membership problem satisfying certain mild assumptions can also be used for the ray shooting problem, with a logarithmic overhead in query time. Here we show that some specific data structures for the membership problem can be used for ray shooting in a more direct way, reducing the overhead in the query time and eliminating the use of parametric search.
We also describe an improved static solution for the membership problem, approaching the conjectured lower bounds more tightly.
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Part of the work on this paper by Jiří Matoušek was supported by Humboldt Forschungs-stipendium. Otfried Schwarzkopf acknowledges support by the ESPRIT II Basic Research Action of the European Community under Contract No. 3075 (project ALCOM). This research was done while he was employed at Freie Universität Berlin. Furthermore, part of this research was done while he visited INRIA-Sophia Antipolis.
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Matoušek, J., Schwarzkopf, O. On ray shooting in convex polytopes. Discrete Comput Geom 10, 215–232 (1993). https://doi.org/10.1007/BF02573975
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DOI: https://doi.org/10.1007/BF02573975