86 research outputs found

    Transversals to Line Segments in Three-Dimensional Space

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    We completely describe the structure of the connected components of transversals to a collection of n line segments in R3. We show that n \u3e 3 arbitrary line segments in R3 admit 0, 1, . . . , n or infinitely many line transversals. In the latter case, the transversals form up to n connected components

    Affine configurations and pure braids

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    We show that the fundamental group of the space of ordered affine-equivalent configurations of at least five points in the real plane is isomorphic to the pure braid group modulo its centre. In the case of four points this fundamental group is free with eleven generators.Comment: 5 pages, 1 figure, final version; to appear in Discrete & Computational Geometry, available from the publishers at http://www.springerlink.com/content/384516n7q24811ph

    Transversals to line segments in three-dimensional space

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    http://www.springerlink.com/We completely describe the structure of the connected components of transversals to a collection of nn line segments in R3\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≥3n\geq 3 arbitrary line segments in R3\mathbb{R}^3 admit at most nn 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 nn on the number of geometric permutations of line segments in R3\mathbb{R}^3
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