The classification of homotopy classes of bounded curvature paths

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 to check access.

References

  1. [1]

    J. Ayala, Classification of the homotopy classes and minimal length elements in spaces of bounded curvature paths, PhD. Thesis, University of Melbourne, 2014.

    Google Scholar 

  2. [2]

    J. Ayala, Shortest bounded curvature paths with self intersections, (2014), arXiv:1403.4930v1 [math.MG].

    Google Scholar 

  3. [3]

    J. Ayala and J. H. Rubinstein, A geometric approach to shortest bounded curvature paths, (2014), arXiv:1403.4899v1 [math.MG].

    Google Scholar 

  4. [4]

    J. Ayala and J. H. Rubinstein, Non-uniqueness of the homotopy class of bounded curvature paths, (2014), arXiv:1403.4911v1 [math.MG].

    Google Scholar 

  5. [5]

    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.

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

    L. E. Dubins, On plane curve with curvature, Pacific Journal of Mathematics 11 (1961), 471–481.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    H. Whitney, On regular closed curves in the plane, Compositio Mathematica 4 (1937), 276–284.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hyam Rubinstein.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

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

Download citation

Keywords

  • Unit Circle
  • Minimal Length
  • Homotopy Class
  • Curvature Path
  • Proximity Condition