Multithreaded Paraboloid Mapping - Practical Rendering and Computation with Direct3D 11 - page 575

Figure 13.6. A visualization of adding the incident and reflected vectors.
(13.3)
(13.4)
(13.5)
This series of equations will allow us to calculate the normal vector when given a lo-
cation on the paraboloid to look up. However, in general we won't have a prior knowledge
of where to look in the paraboloid when generating and sampling its contents. Instead, we
will have the incident vector (taken from the viewing direction) and the reflected vector
(calculated from the paraboloid origin to the vertex being projected) and must determine
the normal vector that will produce the reflected vector from the incident vector. This is
also a trivial operation if we assume that both of these vectors originate at the paraboloid
surface. When this is the case, we can add the two vectors and the resulting vector will
point in the direction of the normal vector, although it will not have a normalized length.
This is shown in Equation (13.6) and Figure 13.6:
(13.6)
By combining the results of Equation (13.5) and Equation (13.6), we can produce a
simple technique to find the normal vector with the information that we will have readily
available to us. We can add the incident and reflected vectors to find our normal vector,