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Theoretical and practical results on straight skeletons of planar straight-line graphs

Published: 13 June 2011 Publication History

Abstract

We study straight skeletons and make both theoretical and practical contributions which support new approaches to the computation of straight skeletons of arbitrary planar straight-line graphs (PSLGs). We start with an adequate extension of the concept of motorcycle graphs to PSLGs, with motorcycles starting at the reflex vertices of a PSLG, which allows us to generalize well-known results on the relation between the straight skeleton and the motorcycle graph to arbitrary PSLGs: the edges of the motorcycle graph cover a specific subset of the edges of the straight skeleton, and they form the basis of 3D slabs such that the projection of the lower envelope of those slabs to the plane forms the straight skeleton. As an immediate application we sketch how to use a graphics hardware for computing (approximate) straight skeletons of PSLGs. Further, we present and analyze a novel wavefront-type algorithm which bridges the current gap between the theory and practice of straight-skeleton computations. Our algorithm handles arbitrary PSLGs, is easy to implement, and is fast enough to handle complex data: it can be expected to run in O(n log n) time in practice for an n-vertex PSLG; its worst-case complexity is O(n2 log n). Extensive experimental results confirm an average runtime of 20 n log n µs on a standard PC for virtually all of our 13500 datasets of different characteristics. As also confirmed by our experiments, this constitutes an average gain in performance by a multiplicative factor of n, or at least one to two orders of magnitude, relative to the speed of the implementation provided by CGAL for closed polygons.

References

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O. Aichholzer and F. Aurenhammer. Straight Skeletons for General Polygonal Figures in the Plane. In A. Samoilenko, editor, Voronoi's Impact on Modern Science, Book 2, pages 7--21. Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, Ukraine, 1998.
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  • (2023)Straight skeleton filling methods for thin-walled partsThe International Journal of Advanced Manufacturing Technology10.1007/s00170-023-12665-7130:1-2(267-282)Online publication date: 23-Nov-2023
  • (2021)Straight Skeleton Based Automatic Generation of Hierarchical Topological Map in Indoor Environment2021 IEEE International Intelligent Transportation Systems Conference (ITSC)10.1109/ITSC48978.2021.9564514(2229-2236)Online publication date: 19-Sep-2021
  • (2021)Implementing Straight Skeletons with Exact Arithmetic: Challenges and ExperiencesComputational Geometry10.1016/j.comgeo.2021.101760(101760)Online publication date: Feb-2021
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  1. Theoretical and practical results on straight skeletons of planar straight-line graphs

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    cover image ACM Conferences
    SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry
    June 2011
    532 pages
    ISBN:9781450306829
    DOI:10.1145/1998196
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    Publication History

    Published: 13 June 2011

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    Author Tags

    1. experiments
    2. implementation
    3. motorcycle graph
    4. straight skeleton
    5. wavefront propagation

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    SoCG '11
    SoCG '11: Symposium on Computational Geometry
    June 13 - 15, 2011
    Paris, France

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    Overall Acceptance Rate 625 of 1,685 submissions, 37%

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    Cited By

    View all
    • (2023)Straight skeleton filling methods for thin-walled partsThe International Journal of Advanced Manufacturing Technology10.1007/s00170-023-12665-7130:1-2(267-282)Online publication date: 23-Nov-2023
    • (2021)Straight Skeleton Based Automatic Generation of Hierarchical Topological Map in Indoor Environment2021 IEEE International Intelligent Transportation Systems Conference (ITSC)10.1109/ITSC48978.2021.9564514(2229-2236)Online publication date: 19-Sep-2021
    • (2021)Implementing Straight Skeletons with Exact Arithmetic: Challenges and ExperiencesComputational Geometry10.1016/j.comgeo.2021.101760(101760)Online publication date: Feb-2021
    • (2021)A path planning method of lattice structural components for additive manufacturingThe International Journal of Advanced Manufacturing Technology10.1007/s00170-021-07092-5Online publication date: 4-May-2021
    • (2016)Design Procedure for Hollow-mask Type Illusionary Solid “Deluding Arrows with shading” Built on the Computation of Geometric Shape and Shading EffectJournal of Graphic Science of Japan10.5989/jsgs.49.4_349:4(3-9)Online publication date: 2016
    • (2016)A Faster Algorithm for Computing Straight SkeletonsACM Transactions on Algorithms (TALG)10.1145/289896112:3(1-21)Online publication date: 25-Apr-2016
    • (2013)A faster algorithm for computing motorcycle graphsProceedings of the twenty-ninth annual symposium on Computational geometry10.1145/2462356.2462396(17-26)Online publication date: 17-Jun-2013
    • (2012)On computing straight skeletons by means of kinetic triangulationsProceedings of the 20th Annual European conference on Algorithms10.1007/978-3-642-33090-2_66(766-777)Online publication date: 10-Sep-2012
    • (2011)Motorcycle graphsJournal of Experimental Algorithmics (JEA)10.1145/1963190.201957816(1.1-1.17)Online publication date: 26-Aug-2011
    • (2011)Generating realistic roofs over a rectilinear polygonProceedings of the 22nd international conference on Algorithms and Computation10.1007/978-3-642-25591-5_8(60-69)Online publication date: 5-Dec-2011

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