José Ayala - Dubins Explorer

A planar bounded curvature path is a $C^1$ and piecewise $C^2$ path lying in ${\mathbb R}^2$ having its curvature bounded by a positive constant and connecting two elements of the tangent bundle ${T \mathbb R}^2$. Dubins Explorer is a mathematical software developed with Jean Diaz, which computes several features inherent of bounded curvature paths. The software is part of a biggest research project currently in preparation. Dubins Explorer emphasizes the visual nature of the subject. In particular, Dubins Explorer illustrates the minimal length elements given any two points in ${T \mathbb R}^2$. Dubins Explorer also computes with precision the extent we can perturb the initial and final points in order to detect non-uniqueness of the homotopy classes of bounded curvature paths. Graphical representations of data as heat maps are incorporated into the software in order to visually represent, for example, angular variations. Dubins Explorer also gives a complete panorama (synthesis problem) for the minimal length elements simultaneously for all points in the plane assigning to them their associated path type. Recently, we extended Dubins Explorer for the flat torus and Klein bottle. Currently, we are working on implementing a constructive proof of Dubins´ theorem in homotopy classes on flat surfaces.

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