The following algorithms focus first on the inner section of the double integration, then estimate the outer integration. The inner integration is given some geometric interpretation which is going to be the base of the calculation. This inner integration has the following form:
Nusselt [SH81] has realized that this
formula can be interpreted as projecting the visible parts of
onto the unit hemisphere centered above
,
then projecting the result orthographically onto the base circle of this hemisphere in the plane
of
(see figure 1.3), and finally calculating the
ratio of the doubly projected area and the area of the unit circle (
).
Due to the central role of the unit hemisphere, this method is called
the hemisphere algorithm.
Later Cohen and Greenberg [CG85] have shown that the projection calculation can be simplified, and more importantly, supported by image synthesis hardware, if the hemisphere is replaced by a half cube. Their method is called the hemicube algorithm.
Beran-Koehn and Pavicic have demonstrated in their recent publication [BKP91] that the necessary calculations can be significantly decreased if a cubic tetrahedron is used instead of the hemicube.
Having calculated the inner section of the integral, the outer part must
be evaluated. The simplest way is to suppose that it is nearly constant on
,
so the outer integral is estimated as the multiplication
of the inner integral at the middle of
and the area of this surface element:
More accurate computations require the evaluation of the inner integral
in several points on
and some sort of numerical integration technique should be used for the integral calculation.