Analytic Evaluation of Collocation Integrals for the Radiosity Equation

Abstract

In this work, we consider solving the radiosity equation using the collocation method. We develop analytic evaluation of the integrations which are needed to setup the linear system in solving the radiosity equation using the collocation method. These integrations are over triangular elements in R ³. Our approach is to use affine transformations to convert integrations over elements in R ³ to integrations over elements in R ² and then to use a change of variables. For this, we introduce functions H _{m ,n ,k} for m , n , k ∈ N ₀ and use these to give our analytic formulas. The analytic evaluations of H _{m ,n ,4} and other relevant integrations are given in detail for some values of m and n . Finally, a performance comparison of the analytic evaluation integration with that of other well-known numerical integration schemes is given. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)