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Ray shooting in polygons using geodesic triangulations

Algorithmica volume 12, pages 54–68 (1994)Cite this article

Abstract

LetP be a simple polygon withn vertices. We present a simple decomposition scheme that partitions the interior ofP intoO(n) so-called geodesic triangles, so that any line segment interior toP crosses at most 2 logn of these triangles. This decomposition can be used to preprocessP in a very simple manner, so that any ray-shooting query can be answered in timeO(logn). The data structure requiresO(n) storage andO(n logn) preprocessing time. By using more sophisticated techniques, we can reduce the preprocessing time toO(n). We also extend our general technique to the case of ray shooting amidstk polygonal obstacles with a total ofn edges, so that a query can be answered inO(√ logn) time.

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Author information

Authors and Affiliations

  1. Computer Science Department, Princeton University, 08544, Princeton, NJ, USA

    B. Chazelle

  2. Computer Science Department, University of Illinois, 61801, Urbana, IL, USA

    H. Edelsbrunner

  3. Department of Mathematics, MIT, 02139, Cambridge, MA, USA

    M. Grigni

  4. DEC Systems Research Center, 130 Lytton Avenue, 94301, Palo Alto, CA, USA

    L. Guibas & J. Hershberger

  5. Laboratory for Computer Science, MIT, 02139, Cambridge, MA, USA

    L. Guibas

  6. Computer Science Department, Stanford University, 94305-4055, Stanford, CA, USA

    L. Guibas & J. Snoeyink

  7. School of Mathematical Sciences, Tel Aviv University, 69 978, Ramat Aviv, Israel

    M. Sharir

  8. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, 10012, New York, NY, USA

    M. Sharir

Authors
  1. B. Chazelle
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  2. H. Edelsbrunner
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  3. M. Grigni
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  4. L. Guibas
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  5. J. Hershberger
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  6. M. Sharir
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  7. J. Snoeyink
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Additional information

Communicated by Kurt Mehlhorn.

Work by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-89-21421. Work by Micha Sharir has been supported by ONR Grants N00014-89-J-3042 and N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.

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Chazelle, B., Edelsbrunner, H., Grigni, M. et al. Ray shooting in polygons using geodesic triangulations. Algorithmica 12, 54–68 (1994). https://doi.org/10.1007/BF01377183

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  • DOI: https://doi.org/10.1007/BF01377183

Key words

  • Computational geometry
  • Ray-shooting
  • Triangulation