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Ray-shooting and isotopy classes of lines in 3-dimensional space

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Algorithms and Data Structures (WADS 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 519))

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Abstract

A uniform approach to problems involving lines in 3-dimensional space is presented. This approach is based on mapping lines in R 3 into points and hyperplanes in 5-dimensional projective space (Plücker space). We improve previously known results on the following problems:

  1. 1.

    Preprocess n triangles so as to efficiently answer the query: “Given a ray, which is the first triangle hit?” (Ray-shooting problem). We discuss the ray-shooting problem for both disjoint and non-disjoint triangles and several space/time trade offs.

  2. 2.

    Efficiently detect the first face hit by any ray in a set of axis-oriented polyhedra.

  3. 3.

    Preprocess n lines (segments) so as to efficiently answer the query “Given 2 lines, is it possible to move one into the other without crossing any of the initial lines (segments)?” (Isotopy problem). If the movement is possible produce an explicit representation of it.

  4. 4.

    Construct the arrangement generated by n intersecting triangles in 3-space in an output-sensitive way, with a subquadratic overhead term.

  5. 5.

    Count the number of pairs of intersecting lines in a set of n lines in space.

Research supported by NSF grant CCR-8901484.

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Author information

Author notes

  1. Marco Pellegrini

    Present address: Comp. Sci. Dept., Strand, King's College, WC2R 2LS, London, United Kingdom

Authors and Affiliations

  1. Courant Institute, New York University, 251 Mercer St. N.Y., 10012, N.Y.

    Marco Pellegrini

Authors
  1. Marco Pellegrini
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Frank Dehne Jörg-Rüdiger Sack Nicola Santoro

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© 1991 Springer-Verlag Berlin Heidelberg

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Pellegrini, M. (1991). Ray-shooting and isotopy classes of lines in 3-dimensional space. In: Dehne, F., Sack, JR., Santoro, N. (eds) Algorithms and Data Structures. WADS 1991. Lecture Notes in Computer Science, vol 519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028246

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  • DOI: https://doi.org/10.1007/BFb0028246

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54343-5

  • Online ISBN: 978-3-540-47566-8

  • eBook Packages: Springer Book Archive

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