A Monotonic Convolution for Minkowski Sums

We present a monotonic convolution for planar regions A and B bounded by line and circular arc segments. The Minkowski sum equals the union of the cells with positive crossing numbers in the arrangement of the convolution, as is the case for the kinetic convolution. The monotonic crossing number is bounded by the kinetic crossing number, and also by the maximum number of intersecting pairs of monotone boundary chains, which is typically much smaller. We give a Minkowski sum algorithm based on the monotonic convolution. The running time is O(s + nα(n) log(n) + m), versus O(s + n) for the kinetic algorithm, with s the input size and with n and m the number of segments in the kinetic and monotonic convolutions. For inputs with a bounded number of turning points and inflection points, the running time is O(sα(s) log s), versus Ω(s) for the kinetic algorithm. The monotonic convolution is 37% smaller than the kinetic convolution and its arrangement is 62% smaller based on 21 test pairs.