Transversals to line segments in three-dimensional space

Hervé Brönnimann 1 Hazel Everett 2 Sylvain Lazard 2 Frank Sottile 3 Sue Whitesides
1 Department of Computer and Information Science
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
3 TAMU - Department of Mathematics [Texas]
Abstract : We completely describe the structure of the connected components of transversals to a collection of $n$ line segments in $\mathbb{R}^3$. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the non-generic cases and show that $n\geq 3$ arbitrary line segments in $\mathbb{R}^3$ admit at most $n$ connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of $n$ on the number of geometric permutations of line segments in $\mathbb{R}^3$.
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Submitted on : Monday, October 3, 2005 - 7:05:10 PM
Last modification on : Tuesday, August 13, 2019 - 10:50:23 AM
Long-term archiving on : Thursday, April 1, 2010 - 10:36:58 PM

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Hervé Brönnimann, Hazel Everett, Sylvain Lazard, Frank Sottile, Sue Whitesides. Transversals to line segments in three-dimensional space. Discrete and Computational Geometry, Springer Verlag, 2005, 34 (3), pp.381 - 390. ⟨10.1007/s00454-005-1183-1⟩. ⟨inria-00000384⟩

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