Collision detection algorithm of a continuous type using spherical extreme vertex diagrams

Most collision detection algorithms are ones of a discrete type which utilize convexity and local coherence to verify the closest points. However, the incremental algorithms have such critical drawbacks like failure to detect high-speed collisions and dependence on the time complexity on the time step size of animation. In this paper, in order to find a new method which can resolve the drawbacks, we are concerned with a more restricted but nontrivial version of the problem: given a fixed infinite plane H and a moving convex polyhedron P, compute their collision time. Our algorithm utilizes the spherical extreme vertex diagram that is the embedding of the dual graph P onto the Gauss sphere. Exploiting the diagram, we are able to efficiently compute the sequence of the extreme vertices vis, i = 1, …, m and find the time interval [ti, ti+1] during which vi is the extreme vertex of P. With the sequences of the consecutive time intervals and extreme vertices, we construct a distance function between P and H. Hence, we can compute their collision time within a given tolerance by applying the interval Newton method to the distance function. Moreover, the total time complexity is independent of the time step size of animation.