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. View Figure

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.

Figure 12 Local-Global Cartesian Transformation. View Figure