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:
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 
and 
axes:
Now, we have E-half of the information we need for an emission.
We still need
.
We proceed with this now.
