Ray Intersection

Figure 5: The base triangle and four displaced microtriangles generated by setting the subdivision parameter, $N$, to 2. The volume for the maximum displacement is also shown.

Our ray intersection test is similar in spirit to intersecting a ray with a height field using a regular grid over the base plane. We will take advantage of an implicit triangular grid formed by the barycentric coordinates. We choose a subdivision amount $N$ (Figure 4) and use dividing lines $\alpha_i = \gamma_i = \beta_i = i/N$ for $i = {0,...,N}$ which creates $N^2$ grid cells for each triangle. Each grid cell generates one displaced microtriangle, as shown in Figure 5. The grid is regular on the base triangle, but due to the interpolated surface normals, it is irregular throughout space. Although it is irregular, our restrictions limit the range of the displacement function $h()$ to the interval $[-m,+M]$ where a traversal algorithm is possible.

Much like standard grid traversal algorithms, there are two phases to the algorithm. First the start point must be initialized. Next the grid must be traversed, checking each cell for an intersection with the triangle it contains. The traversal algorithm will be described first in order to determine the quantities that need to be initialized.