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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 tex2html_wrap_inline1812 and tex2html_wrap_inline1814 by tex2html_wrap_inline1990 , the unit normal to tex2html_wrap_inline1812 by tex2html_wr ap_inline1994 , and the surface element vector tex2html_wrap_inline1996 by tex2html_wrap_inline1998 :

equation322

If we could find a vector field tex2html_wrap_inline2000 , such that tex2html_wrap_inline2002 , the area integral could be transformed into the contour integral tex2html_wrap_inline2004 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 tex2html_wrap_inline1818 is everywhere 1:

  equation336

where

  1. tex2html_wrap_inline2008 is the signed angle between two vectors. The sign is positive if tex2html_wrap_inline2010 is rotated clockwise from tex2html_wrap_inline2012 looking at them in the opposite direction to tex2html_wrap_inline1994 ,
  2. tex2html_wrap_inline2016 represents addition modulo L. It is a circular next operator for vertices,
  3. L is the number of vertices of surface element j,
  4. tex2html_wrap_inline2024 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.



Szirmay-Kalos Laszlo
Mon Oct 21 14:07:41 METDST 1996