Reachability by paths of bounded curvature in a convex polygon

Let B be a point robot moving in the plane, whose path is constrained to forward motions with curvature at most one, and let P be a convex polygon with n vertices. Given a starting configuration (a location and a direction of travel) for B inside P , we characterize the region of all points of P that can be reached by B, and show that it has complexity O(n). We give an O(n) time algorithm to compute this region. We show that a point is reachable only if it can be reached by a path of type CCSCS, where C denotes a unit circle arc and S denotes a line segment.