Bifurcations of Voronoi diagrams and its application to braid theory. (English) Zbl 1050.37025
Summary: We study bifurcations of Voronoi diagrams on the plane. The generic bifurcations of Voronoi diagrams for moving points are classified into four types.
A braid has an associated family of Voronoi diagrams. If we admit only three types among four in the associated generic family of Voronoi diagrams, the braid type reduces to a braid with \(k\) half twists for some integer \(k\).
A braid has an associated family of Voronoi diagrams. If we admit only three types among four in the associated generic family of Voronoi diagrams, the braid type reduces to a braid with \(k\) half twists for some integer \(k\).
MSC:
| 37G99 | Local and nonlocal bifurcation theory for dynamical systems |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 52C99 | Discrete geometry |
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
References:
| [1] | DOI: 10.2307/1969218 · Zbl 0030.17703 · doi:10.2307/1969218 |
| [2] | DOI: 10.1145/116873.116880 · doi:10.1145/116873.116880 |
| [3] | Birman J. S., Ann. Math. Studies 82, in: Braids, links, and mapping class groups (1974) |
| [4] | Edelsbrunner H., EATCS, Monographs on Computer Science 10, in: Algorithms in combinatorial geometry (1970) · Zbl 0634.52001 |
| [5] | DOI: 10.1007/978-94-015-9319-9 · doi:10.1007/978-94-015-9319-9 |
| [6] | D. Siersma, Geometry in Present Day Science, eds. O. E. Barndorff-Nielsen and E. B. V. Jensen (World Scientific, 1999) pp. 187–208. |
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