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The upper envelope of piecewise linear functions: Algorithms and applications

Abstract

This paper studies applications of envelopes of piecewise linear functions to problems in computational geometry. Among these applications we find problems involving hidden line/surface elimination, motion planning, transversals of polytopes, and a new type of Voronoi diagram for clusters of points. All results are either combinatorial or computational in nature. They are based on the combinatorial analysis in two companion papers [PS] and [E2] and a divide-and-conquer algorithm for computing envelopes described in this paper.

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Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862. Work by the third author has been supported by the Office of Naval Research Grant N00014-82-K-0381, National Science Foundation Grant No. NSF-DCR-83-20085, by grants from the Digital Equipment Corporation and the IBM Corporation, and by a research grant from NCRD, the Israeli National Council for Research and Development.

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Edelsbrunner, H., Guibas, L.J. & Sharir, M. The upper envelope of piecewise linear functions: Algorithms and applications. Discrete Comput Geom 4, 311–336 (1989). https://doi.org/10.1007/BF02187733

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  • DOI: https://doi.org/10.1007/BF02187733

Keywords

  • Line Segment
  • Convex Hull
  • Voronoi Diagram
  • Piecewise Linear Function
  • Combinatorial Complexity