We define a variety of quantities for each surface L, where . Where the context to a general surface is clear, we shall omit the subscript L. Geometries are required to form full enclosures, as all particles must be traced eventually to absorption---thus, no ``holes'' are permitted.
We define and store the following quantities for every surface L during the input phase of the program:
(where denotes the cross product).
Note that, the surface normal vector is defined as pointing to the right when proceeding from node to node . This is important because we allow particles to interact with only the front of the surface. Normally, the back of the surface is the solid material from whence the particles derive and which serves to absorb incident particles (recall that a surface has zero thickness, and thus contains no material).
We restrict our analysis to enclosures, i.e., closed geometries such as shown in Figure 3 (frequently in practice, actual ``holes'' exist in geometries---these are modelled as totally absorptive surfaces). Figure 3(a) shows a square, while Figure 3(b) depicts a generalized geometry, with an interior element (which must be supported somewhere in Z---not shown) and shading, i.e., the geometry is convex.
Figure 3: Cross-Sections of Prismatic Geometries.