Consider the situation shown in Figure 4, where we trace a particle from to in a simple enclosure with 14 surfaces. The intersection calculations must be done over all surfaces, and the unique ``valid'' point of intersection identified. (Note that the straight line of the particle trajectory, in general, intersects every straight line defining every surface in the enclosure.) Specifically, Figure 4 shows emitting and intersecting surfaces as numbers 13 and 12, respectively (recall that we disallow intersections from the back of a surface---i.e., surface 3; in a well-specified geometry, there is always an intersection prior to one from behind---here, the prior intersection is that of surface 12).
Figure 4: Particle Intersection.
The ``valid'' intersection is identified as the one of minimum distance from the point of emission to the point of intersection (for example, surface 12 instead of surface 2), but this comprises a daunting problem numerically, of order . Specifically, we seek ways to limit this search to render the problem tractable. Emissions are done from all surfaces and each emission requires testing all surfaces for the unique, ``valid'' intersection.
Strategies to limit the search are depicted in Figure 5 for a general surface L. Among them are:
Figure 5: Conditions to Invalidate a Potential Intersection.