Each of the above material properties is the probability of the particular interaction. Figure 16 depicts the material properties as functions of the incident cone angle, . At any particular value of , the incident particle has the following probabilities: , , , and for specular and diffuse transmission, specular and diffuse reflection, and absorption, respectively. Each of these curves must be input (or generated) by point value as a function of cone angle.

Figure 16: Material Properties vs. Incident Cone Angle.

Material properties of much greater complexity could easily be implemented in
Monte Carlo analysis (indeed, this is one of its great strengths).
For example, at each incident angle, we could have specified outgoing
directional distributions for transmission and reflection, in which case we
would need to specify a * surface* of outgoing directions for each point
on the abscissa of
Figure 16
(obviously, we would have to discretize
the abscissa intelligently).
However, in most cases, we lack sufficient data to make this level of effort
worthwhile.
So, we restrict our attention to two specific outgoing directional
distributions---specular and diffuse, and represent properties as shown in
Figure 16.
Finally, we note that we have found it sufficient to provide data every
in and to perform linear interpolation between
points.
For efficiency, we store these data together with their differences in tabular
form.
Thus, each material requires a storage space of
words for particle/material interactions.

Our procedure for determining particle/surface interactions is then as follows. After the intersection is obtained, we determine the incident angle. At this incident angle, we look up (from the material property table) the probabilities of interactions. We choose a random number from a uniform distribution (between 0 and 1) of random numbers to determine the specific interaction for this particle. It is necessary to sum the probabilities for this, that is, if the random number is less than , the particle is made to undergo a specular transmission; if the random number is greater than , but less than , then the particle is diffusely transmitted; etc. For efficiency, it is wise to sum all properties during the input phase (i.e., instead of storing , , etc., we store , , etc.) at each angle, so the sums need not be calculated during the solution phase.