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Pestov–Ionin theorem

From Wikipedia, the free encyclopedia

A smooth simple closed curve of curvature at most one, and a unit disk enclosed by it

The Pestov–Ionin theorem in the differential geometry of plane curves states that every simple closed curve of curvature at most one encloses a unit disk.

History and generalizations

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Although a version of this was published for convex curves by Wilhelm Blaschke in 1916,[1] it is named for German Gavrilovich Pestov [ru] and Vladimir Kuzmich Ionin [ru], who published a version of this theorem in 1959 for non-convex doubly differentiable () curves, the curves for which the curvature is well-defined at every point.[2] The theorem has been generalized further, to curves of bounded average curvature (singly differentiable, and satisfying a Lipschitz condition on the derivative),[3] and to curves of bounded convex curvature (each point of the curve touches a unit disk that, within some small neighborhood of the point, remains interior to the curve).[4]

Applications

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The theorem has been applied in algorithms for motion planning. In particular it has been used for finding Dubins paths, shortest routes for vehicles that can move only in a forwards direction and that can turn left or right with a bounded turning radius.[3][5] It has also been used for planning the motion of the cutter in a milling machine for pocket machining,[4] and in reconstructing curves from scattered data points.[6]

References

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  1. ^ Blaschke, Wilhelm (1916), "24.II: Kleinster und größter Krümmungskreis einer konvexen Kurve", Kreis und Kugel (in German), Veit, pp. 114–117
  2. ^ Pestov, G.; Ionin, V. (1959), "On the largest possible circle imbedded in a given closed curve", Proceedings of the USSR Academy of Sciences (in Russian), 127: 1170–1172, MR 0107214
  3. ^ a b Ahn, Hee-Kap; Cheong, Otfried; Matoušek, Jiří; Vigneron, Antoine (2012), "Reachability by paths of bounded curvature in a convex polygon", Computational Geometry, 45 (1–2): 21–32, arXiv:1008.4244, doi:10.1016/j.comgeo.2011.07.003, hdl:10754/562037, MR 2842619
  4. ^ a b Aamand, Anders; Abrahamsen, Mikkel; Thorup, Mikkel (2020), "Disks in curves of bounded convex curvature", The American Mathematical Monthly, 127 (7): 579–593, arXiv:1909.00852, doi:10.1080/00029890.2020.1752602, MR 4128552, S2CID 202539477
  5. ^ Agarwal, Pankaj K.; Biedl, Therese; Lazard, Sylvain; Robbins, Steve; Suri, Subhash; Whitesides, Sue (2002), "Curvature-constrained shortest paths in a convex polygon", SIAM Journal on Computing, 31 (6): 1814–1851, CiteSeerX 10.1.1.398.3027, doi:10.1137/S0097539700374550, MR 1954880, S2CID 37983528
  6. ^ Guha, Sumanta; Tran, Son Dinh (2005), "Reconstructing curves without Delaunay computation", Algorithmica, 42 (1): 75–94, doi:10.1007/s00453-004-1141-y, MR 2131830, S2CID 11620890