Curvature-constrained motion within a limited workspace

We pose the following questions: Given two points within a closed area W C R2, each with a prescribed direction of motion in it, (i) what is the shortest path of bounded curvature that joints them and lies completely in W? (ii) what is the minimum number of cusps one needs to design a path in W? This kind of questions appear in various applications, such as robot motion planning. The proposed approach makes use of a tool dubbed the Reflective Unfolding operator which has a clear geometric interpretation and provides an interesting means for solving other trajectory design problems. In this text, the approach is applied to the following problem: for a car moving with bounded curvature and possible reversals, given the starting and target directions of motion at the center of a disc D of some radius R, design the shortest possible path fully lying in D. The path produced by the algorithm turns out to also be of the lowest complexity (the minimum number of cusps).