Abstract
A bounded curvature path is a continuously differentiable piecewise C 2 path with bounded absolute curvature that connects two points in the tangent bundle of a surface. In this note we give necessary and sufficient conditions for two bounded curvature paths, defined in the Euclidean plane, to be in the same connected component while keeping the curvature bounded at every stage of the deformation. Following our work in [3], [2] and [4] this work finishes a program started by Lester Dubins in [6] in 1961.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Ayala, Classification of the homotopy classes and minimal length elements in spaces of bounded curvature paths, PhD. Thesis, University of Melbourne, 2014.
J. Ayala, Shortest bounded curvature paths with self intersections, (2014), arXiv:1403.4930v1 [math.MG].
J. Ayala and J. H. Rubinstein, A geometric approach to shortest bounded curvature paths, (2014), arXiv:1403.4899v1 [math.MG].
J. Ayala and J. H. Rubinstein, Non-uniqueness of the homotopy class of bounded curvature paths, (2014), arXiv:1403.4911v1 [math.MG].
L. E. Dubins, On curves of minimal length with constraint on average curvature, and with prescribed initial and terminal positions and tangents, American Journal of Mathematics 79 (1957), 139–155.
L. E. Dubins, On plane curve with curvature, Pacific Journal of Mathematics 11 (1961), 471–481.
H. Whitney, On regular closed curves in the plane, Compositio Mathematica 4 (1937), 276–284.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ayala, J., Rubinstein, H. The classification of homotopy classes of bounded curvature paths. Isr. J. Math. 213, 79–107 (2016). https://doi.org/10.1007/s11856-016-1321-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-016-1321-x