Stabbing Oriented Convex Polygons in Randomized O(n2) Time

We present an algorithm that determines, in expected O (n2) time, whether a line exists that stabs each of a set of oriented convex polygons in R3 with a total of n edges. If a stabbing line exists, the algorithm computes at least one such line. We show that the computation amounts to constructing a convex polytope in R5 and inspecting its edges for intersections with a four-dimensional quadric surface, the Plucker quadric.