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Convex-Straight-Skeleton Voronoi Diagrams for Segments and Convex Polygons

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Abstract

In this paper, we study the convex-straight-skeleton Voronoi diagrams of line segments and convex polygons. We explore the combinatorial complexity of these diagrams, and provide efficient algorithms for computing compact representations of them.

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Notes

  1. A disadvantage of this approach is that relabeling of the input sites may change the diagram. One can adopt the rule of Klein and Wood [16], who break ties by the lexicographic order of the input points, that is, by the actual coordinates of the points, but with such a solution, the Voronoi diagram will not be invariant under rotation of the plane.

  2. A common tangent of two convex polygons is a line that passes through points on the boundaries of both polygons, such that both polygons are completely contained by one of the two closed halfplanes defined by the line. For two convex polygons, there are exactly two common tangents. We refer to the tangent that lies above the line segment s as the upper common tangent, and to the other one as the lower common tangent. If s is vertical, then we arbitrarily choose the left tangent as the upper one.

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Authors and Affiliations

  1. Department of Computer Science, The Technion—Israel Institute of Technology, Haifa, Israel

    Gill Barequet

  2. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India

    Minati De

  3. Department of Computer Science, University of California, Irvine, CA, USA

    Michael T. Goodrich

Authors
  1. Gill Barequet
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  2. Minati De
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  3. Michael T. Goodrich
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Correspondence to Minati De.

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Preliminary versions of this paper appeared in the proceedings of The 3rd Int. Conf. on Algorithms and Discrete Applied Mathematics, 2017 [6], and The 24th Int. Computing and Combinatorics Conference, 2018 [9]. Work on this paper by the first author has been supported in part by BSF Grant 2017684. Work on this paper by the second author has been supported in part by DST-INSPIRE Faculty Grant (DST-IFA14-ENG-75) and IIT Delhi New Faculty Seed Grant. Work on this paper by the third author has been supported in part by NSF Grant 1815073.

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Barequet, G., De, M. & Goodrich, M.T. Convex-Straight-Skeleton Voronoi Diagrams for Segments and Convex Polygons. Algorithmica 83, 2245–2272 (2021). https://doi.org/10.1007/s00453-021-00824-9

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