Efficient Intersection Tests for Objects Defined Constructively

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 ca.n be used to implement a.n efficient intersection detection routine when entities are described constructively; that is, as set combina­ tions 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 first technique, called S-bounds, is a method of reasoning about where intersections may be taking place; in practise it is fast, a.nd often sufficient. S-bounds can also be used as a general constraint manipulation method over Boolean algebras. The second technique is based on spatial subdivision, a.nd is used mainly to improve the speed of the intersection test. The third tech~ nique is employed only on the regions of space that are left by the first 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 ma.gnitude. 'This paper is a slightly longer version of one that appeared in the Inlt.rn.a'ional Journal of Robotics Rt.Sl'an;h 8(1)3-25, February 1989 1 INTERSECTION DETECTION PROBLEMS 1 Intersection Detection Problems The general intersection detection problem can be formulated as: 'Given two subsets A and B of R'\ do they have any point in common?' A common use of intersection detection is to solve the interference detection problem, viz.: 'Given two objects, OA and 0B, do they interfere?' This can be solved by considering the point sets A and B, wh.ich. consist of all the points in the objects OA and DB, and performing an intersection test in ~. Collision detection can also be formulated as an intersection test, this time in i'­ [Cam84}. In fa-ct, IDAUy of the algorithms used in solid geometry, including ma.ny of those useful to rohoticists, rely heavily on being able to perform intersection tests between entities of various dimensiona.lity-for example, ray tracing considers the intersections of a line with a solid. Entities are orten described to a computer constructively; that is, as set combinations of simpler entities. In such systems it is possible to reformu­ late an intersection test aa one of null object detection (NOD): given the two entities A and B, test whether their set intersection An B is the null ~et. In the rest of this paper we will concentrate on the solution of NOD in the context of a constructive solid geometry (eSG) modelling system; that i~, a system in which objects are described to a computer as set combinations of primitive objects (SUCh as parameterised cuboids and cylinders) and which keeps tbese description internally as its principal representation of the ob­ jects. However, the techniques described here can also be of use in other systems, such as some boundary representation (B-rep) systems which also keep a record of the steps used to construct tbe objects, and tbe modifi­ cations reqUired will be outlined aa appropriate. (For a discuf;ision of the various types of solid modelling systems available, see [RV82, RV83J.) Tbe algorithm to be described consists of three separable parts. The first part is based on a new method for reasoning about tbe parts of space that could be occupied by the set we are testing for nullity, caUed the S· bounds method, and because this metbod is new its description occupies mucb of this paper. The use of this method is sometImes enough in itself to partially decide the NOD problem, as it can give the answer "definitely null". (As an example, the two robots in figure 1 are shown to be not intersecting just by using the S-bound method.) The second method is baaed ou a spatial subdivision technique used in computer graphics; this is also a partial decision procedure, with possible a.nswers "definitely null", "definitely not null". and '"don 't know". Both of these two techniques also localise the problem; that is, reduce the 'volume' of 'space' in which it is necessary to search for evidence of uon-nullity. The third method is the exhaustive method that is used when aU else fails, and is based on the 1 INTERSECTlON DETECTlON PROBLEMS 2 Figure 1: Two robots almost interfering folklore of computational geometry; unlike the other two methods, it is also highly dependent on the types of geometric features allowed in the system. In the a.uthor's implementation these three methods operate in cascade, with ea.ch sta.ge a.ttempting to answer the NOD problem itself and only passing on pa.rts of the original problem that it ca.nnot tackle. Much of the efficiency of the overall implementation results from the cannol of this cascade. The rest of this paper is organised as follows. The theory behind the S-bounds method is described in §2 in a fairly general way. §3 describes how this theory is used to prune the search space (or the NOD problem, using the theory of redundancy [TiI84]. §4 a.nd §5 provide the details of the spatial subdivision and the exhaustive steps respectively, §6 gives some examples of interference detection within the RQBMOO geometric modelling system, and §7 provides a summary. 2 THE THEORY OF S-BOUNDS 3 2 The Theory of S-bounds