Following a finite element approach, an appropriate set of functions can be defined having broken down the surface into , ,..., surface elements:
If the emission E and the diffuse coefficient are assumed to be constant on the elemental surface and equal to and respectively, equation 1.79 will have the following form:
According to the definition of basis function , the radiosity of patch k is:
Substituting this into equation 1.81 and using the definition of g(p,p') in equation 1.70, we get:
Let us introduce the patch-to-patch form factor as follows:
Note that this is the usual definition taking into account the interpretation of f(p,p') in equation 1.59.
Dividing both sides by , the linear equation is then:
This is exactly the well known linear equation of original radiosity method (equation 1.10). Now let us begin to discuss how to define and use other, more effective function bases.