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

## Abstract

*R*

^{3}into points and hyperplanes in 5-dimensional projective space (

*Plücker space*). We improve previously known results on the following problems:

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

- 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.
Construct the arrangement generated by

*n*intersecting triangles in 3-space in an output-sensitive way, with a*subquadratic*overhead term. - 5.
Count the number of pairs of intersecting lines in a set of

*n*lines in space.

## Keywords

Computational Geometry Query Point Query Time Hyperplane Arrangement Isotopy Class## Preview

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## References

- [AMS91]B. Aronov, J. Matoušek, and M. Sharir. On the sum of squares of cell complexities in hyperplane arrangements. To appear in the Proceedings of the 7th ACM Symposium on Computational Geometry, 1991.Google Scholar
- [AS91a]P. K. Agarwal and M. Sharir. Applications of a new space partitioning technique. To appear in the Proceedings of the 1991 Workshop on Algorithms and Data Structures, 1991.Google Scholar
- [AS91b]B. Aronov and M. Sharir. On the zone of a surface in an hyperplane arrangement. To appear in the Proceedings of the 1991 Workshop on Algorithms and Data Structures, 1991.Google Scholar
- [CE88]B. Chazelle and H. Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. In
*Proc. of the 29th Ann. Symp. on Foundations of Computer Science*, 1988.Google Scholar - [CEGS89a]B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. Lines in space: combinatorics, algorithms and applications. In
*Proc. of the 21st Symposium on Theory of Computing*, pages 382–393, 1989.Google Scholar - [CEGS89b]B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir. A singly exponential stratification scheme for real semi-algebraic varieties and its applications. In
*Proceedings of the 16th International Colloquium on Automata, languages and Programming*, pages 179–193, 1989. In Springer Verlag LNCS 372.Google Scholar - [Cha85]B. Chazelle.
*Fast searching in a real algebraic manifold with applications to geometric complexity*, volume 185 of*Lecture notes in computer science*, pages 145–156. Springer Verlag, 1985.Google Scholar - [Cla87]K.L Clarkson. New applications of random sampling in computational geometry.
*Discrete Computational Geometry*, (2):195–222, 1987.Google Scholar - [CSW90]B. Chazelle, M. Sharir, and E. Welzl. Quasi-optimal upper bounds for simplex range searching and new zone theorems. In
*Proceedings of the 6th ACM Symposium on Computational Geometry*, pages 23–33, 1990.Google Scholar - [dBHO+91][dBHO
^{+}91] M. de Berg, D. Halperlin, M. Overmars, J. Snoeyink, and M. van Kreveld. Efficient ray-shooting and hidden surface removal. To appear in Procedings of the 7th Annual Symposium on Computational Geometry, 1991.Google Scholar - [DK90]D. Dobkin and D. Kirkpatrick. Determining the separation of preprocessed polyhedra: a unified approach. In
*Proc. of the 17th International Colloqium on Automata, Languages and Programming*, pages 400–413, 1990.Google Scholar - [EGS88]H. Edelsbrunner, L. Guibas, and M. Sharir. The complexity of many faces in arrangements of lines and segments. In
*Proceedings of the 4th ACM Symposium on Computational Geometry*, pages 44–55, 1988.Google Scholar - [EMP+82]H. Edelsbrunner, H. Mauer, F. Preparata, E. Welzl, and D. Wood. Stabbing line segments.
*BIT*, (22):274–281, 1982.Google Scholar - [GOS89]L. Guibas, M. Overmars, and M. Sharir. Ray shooting, implicit point location and related queries in arrangements of segments. Technical Report TR433, Courant Institute, 1989.Google Scholar
- [Mat90]J. Matouěk. Cutting hyperplane arrangements. In
*Proceedings of the 6th ACM Symposium on Computational Geometry*, pages 1–9, 1990.Google Scholar - [MO88]M. McKenna and J. O'Rourke. Arrangements of lines in 3-space: A data structure with applications. In
*Proceedings of the 4th annual Symposium on Computational Geometry*, pages 371–380, 1988.Google Scholar - [Pel90]M. Pellegrini. Stabbing and ray shooting in 3 dimensional space. In
*Proceedings of the 6th ACM Symposium on Computational Geometry*, pages 177–186, 1990.Google Scholar - [Pel91]M. Pellegrini.
*Combinatorial and Algorithmic Analysis of Stabbing and Visibility Problems in 3-Dimensional Space*. PhD thesis, New York University-Courant Institute of Mathematical Sciences, February 1991.Google Scholar - [PS91]M. Pellegrini and P. Shor. Finding stabbing lines in 3-dimensional space. In
*Proceedings of the Second SIAM-ACM Symposium on Discrete Algorithms*, 1991.Google Scholar - [Som51]D. M. H. Sommerville.
*Analytical geometry of three dimensions*. Cambridge, 1951.Google Scholar - [SS83]J.T. Schwartz and M. Sharir. On the piano mover's problem: II. General techniques for computing topological properties of real algebraic manifolds.
*Adv. in Appl. Math*, 4:298–351, 1983.Google Scholar - [Sto89]J. Stolfi. Primitives for computational geometry. Technical Report 36, Digital SRC, 1989.Google Scholar
- [Tar83]R.E. Tarjan.
*Data Structures and Network Algorithms*, volume 44 of*CBMS-NSF Regional Conference Series in Applied Mathematics*. SIAM, 1983.Google Scholar