A κ-constrained homotopy

We consider curves of bounded curvature with variable initial and final vectors, the so-called κ-constrained curves. Suppose the curvature bound is κ=1/r.

Let d be the distance between the initial and final points in the depicted curves. Suppose d is less than 2r. Then, the larger circular arcs in the two radius r circles intersecting the initial and final points are κ-constrained homotopic.

In the paper below I obtained a classification theorem for the homotopy classes in spaces of κ-constrained curves for any given initial and final positions. A class of curves gets trapped in an eye-shaped region corresponding to the intersection of the two disks whose boundaries are the two radius r circles containing the initial and final points. The trapped curves are in fact elements of an isotopy class of κ-constrained curves .

J. Ayala, On the topology of the spaces of curvature constrained plane curves, Advances in Geometry, Vol 17, No 3, pp. 283-292, 2017.