Intersection Searching Amid Tetrahedra in 4-Space and Efficient Continuous Collision Detection

We develop data structures for intersection detection queries in four dimensions that involve segments, triangles and tetrahedra. Specifically, we study two main problems: (i) Preprocess a set of n tetrahedra in R4 into a data structure for answering segment-intersection queries amid the given tetrahedra (referred to as segment-tetrahedron intersection queries), and (ii) Preprocess a set of n triangles in R4 into a data structure that supports triangle-intersection queries amid the input triangles (referred to as triangle-triangle intersection queries). As far as we can tell, these problems have not been previously studied. For problem (i), we first present a "standard" solution which, for any prespecified value n ≤ s ≤ n6 of a so-called storage parameter s, yields a data structure with O*(s) storage and expected preprocessing, which answers an intersection query in O*(n/s1/6) time (here and in what follows, the O*( ) notation hides subpolynomial factors). For problem (ii), using similar arguments, we present a solution that has the same asymptotic performance bounds. We then improve the solution for problem (i), and present a more intricate data structure that uses O*(n2) storage and expected preprocessing, and answers a segment-tetrahedron intersection query in O*(n1/2) time. Using the parametric search technique of Agarwal and Matoušek [3], we can obtain data structures with similar performance bounds for the ray-shooting problem amid tetrahedra in R4. Unfortunately, so far we do not know how to obtain a similar improvement for problem (ii). Our algorithms are based on a primal-dual technique for range searching with semi-algebraic sets, based on recent advances in this area [2, 11]. As this is a result of independent interest, we spell out the details of this technique. As an application, we present a solution to the problem of "continuous collision detection" amid moving tetrahedra in 3-space. That is, the workspace consists of n tetrahedra, each moving at its own fixed velocity, and the goal is to detect a collision between some pair of moving tetrahedra. Using our solutions to problems (i) and (ii), we obtain an algorithm that detects a collision in O*(n12/7) expected time. We also present further applications, including an output-sensitive algorithm for constructing the arrangement of n tetrahedra in R4 and an output-sensitive algorithm for constructing the intersection or union of two or several nonconvex polyhedra in R4.