To render the problem tractable, we restrict our attention to surfaces of spatially constant properties. Thus, a surface will have constant values for all material properties, but a geometry can consist of any number of surfaces. Also, input parameters must be constant over the surface. For example, in radiative transfer, the surfaces are of constant temperature. When discretizing the geometry, this criterion generally is the controlling factor, i.e., how small must the surface be for the input parameters to be well modelled as constant over the surface? It is good procedure to solve the problem at more than one level of surface spatial discretization; then, the final solution is the one that obtains at the finest level of discretization where the answers are invariant (to within a small tolerance) from the previous (coarser) level of surface discretization.
This model results in a spatially constant (over each surface) number flux of emitted particles. In our implementation, if the surface is divided into small subsurfaces, each subsurface will emit a number of particles proportional to its subsurface area (this eliminates biases due to ``hot spots''). In two dimensions, we subdivide each surface into a number of subsurfaces equal to the number of particle emissions, where one particle is emitted from the centroid of each subsurface. In three dimensions, we specify the number of subsurface divisions in two directions and weight the total number of emissions by subsurface area. Figure 13 depicts these strategies for two- and three-dimensional Cartesian surfaces (surfaces in other than Cartesian space are more complicated).