For a single particle, we now examine the coordinate transformations required to obtain E, the emission direction in the global Cartesian coordinate system. First, as shown in Figure 11, we transform the local spherical components and into the local Cartesian components , and , as follows:
Figure 11: Local Spherical-Cartesian Transformation.
Next, the local Cartesian coordinates must be transformed into global coordinates, as shown in Figure 12. This transformation is a simple two-dimensional rotation, obtained using the components of the surface normal N, by projecting the and directions onto the global X and Y axes:
Now, we have E---half of the information we need for an emission. We still need . We proceed with this now.
Figure 12: Local-Global Cartesian Transformation.