Computing the Fréchet distance between piecewise smooth curves

We consider the Fréchet distance between two curves which are given as a sequence of m+n curved pieces. If these pieces are sufficiently well-behaved, we can compute the Fréchet distance in O(mnlog(mn)) time. The decision version of the problem can be solved in O(mn) time. The results are based on an analysis of the possible intersection patterns between circles and arcs of bounded curvature.