Eecient Intersection Tests for Objects Deened Constructively 1 Intersection Detection Problems

Testing for the existence of intersections is an important part of algorithms for interference detection, collision detection, and the like. We describe three techniques that can be used to implement an eecient intersection detection routine when entities are described constructively; that is, as set combinations of primitive entities. All three techniques are described in a domain where constructive solid geometry is the principal entity description used, although their use in boundary representation schemes are also discussed. The rst technique, called S-bounds, is a method of reasoning about where intersections may be taking place; in practise it is fast, and often suucient. S-bounds can also be used as a general constraint manipulation method over Boolean algebras. The second technique is based on spatial subdivision, and is used mainly to improve the speed of the intersection test. The third technique is employed only on the regions of space that are left by the rst two techniques; it is a specialisation of the \classical" technique of generate-and-test. The combination of these techniques has been implemented as an intersection detection routine which shows a speedup over the \classical" algorithm of about two orders of magnitude. The general intersection detection problem can be formulated as: `Given two subsets A and B of < n , do they have any point in common?' A common use of intersection detection is to solve the interference detection problem, viz.: `Given two objects, O A and O B , do they interfere?' This can be solved by considering the point sets A and B, which consist of all the points in the objects O A and O B , and performing an intersection test in < 3. Collision detection can also be formulated as an intersection test, this time in < 4 Cam84]. In fact, many of the algorithms used in solid 1 Figure 1: Two robots almost interfering geometry, including many of those useful to roboticists, rely heavily on being able to perform intersection tests between entities of various dimensionality|for example, ray tracing considers the intersections of a line with a solid. Entities are often described to a computer constructively; that is, as set combinations of simpler entities. In such systems it is possible to reformulate an intersection test as one of null object detection (NOD): given the two entities A and B, test whether their set intersection A \B is the null set. In the rest of …