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Transversals to line segments in three-dimensional space

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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Contributor : Sylvain Lazard <>
Submitted on : Monday, October 3, 2005 - 7:05:10 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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