Analytic form factor computation

 

The inner section of the form factor integral, or as it is called the form factor between a finite and differential area, can be written as a surface integral in a vector space, denoting the vector between and by , the unit normal to by , and the surface element vector by :

If we could find a vector field , such that , the area integral could be transformed into the contour integral by Stoke's theorem. This idea has been followed by Hottel and Sarofin [HS67], and they were successful in providing a formula for the case when there are no occlusions, or the visibility term is everywhere 1:

 

where

1. is the signed angle between two vectors. The sign is positive if is rotated clockwise from looking at them in the opposite direction to ,
2. represents addition modulo L. It is a circular next operator for vertices,
3. L is the number of vertices of surface element j,
4. is the vector from the differential surface i to the lth vertex of the surface element j.

We do not aim to go into the details of the original derivation of this formula based on the theory of vector fields, because it can also be proven relying on geometric considerations of the hemispherical projection.