Plane curves with the same endpoints are homotopic; an analogous claim for plane curves with the same endpoints and bounded curvature still remains open. We find necessary and sufficient conditions for two plane curves with bounded curvature to be deformed into each other by a continuous one-parameter family of curves also having bounded curvature. We conclude that the space of these curves has either one or two connected components, depending on the distance between the endpoints. The classification theorem presented here answers a question raised in 1961 by L. E. Dubins.
J. Ayala, D. Kirszenblat, J. H. Rubinstein, A geometric approach to shortest bounded curvature paths. Preprint 2015, arXiv 1403.4899
J. Ayala, H. Rubinstein, The classification of homotopy classes of bounded curvature paths. Israel J. Math. 213 (2016), 79-107. MR3509469
J. Ayala, J. H. Rubinstein, Non-uniqueness of the homotopy class of bounded curvature Paths. Preprint 2014, arXiv 1403.4911
E. J. Cockayne, G. W. C. Hall, Plane motion of a particle subject to curvature constraints. SIAM J. Control13 (1975), 197-220. MR0433303 Zbl 0305.53004
L. E. Dubins, On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Amer. J. Math. 79 (1957), 497-516. MR0089457 Zbl 0098.35401
L. E. Dubins, On plane curves with curvature. Pacific J. Math. 11 (1961), 471-481. MR0132460 Zbl 0123.15503
H. H. Johnson, An application of the maximum principle to the geometry of plane curves. Proc. Amer. Math. Soc. 44 (1974), 432-435. MR0348631 Zbl 0295.53001
Z. A. Melzak, Plane motion with curvature limitations. J. Soc. Indust. Appl. Math. 9 (1961), 422-432. MR0133947 Zbl 0102.13703
G. Pestov, V. Ionin, On the largest possible circle imbedded in a given closed curve. Dokl. Akad. Nauk SSSR127 (1959), 1170-1172. MR0107214 Zbl 0086.36104
J. A. Reeds, L. A. Shepp, Optimal paths for a car that goes both forwards and backwards. Pacific J. Math. 145 (1990), 367-393. MR1069892
P. Souères, J.-Y. Fourquet, J.-P. Laumond, Set of reachable positions for a car. IEEE Trans. Automat. Control39 (1994), 1626-1630. MR1287271 Zbl 0925.93042
H. Whitney, On regular closed curves in the plane. Compositio Math. 4 (1937), 276-284. MR1556973 Zbl 0016.13804 JFM 63.0647.01
Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.