| In this work, we consider solving the radiosity equation using the collocation method. We provide a physical setting for radiosity and derive the radiosity equation based on the explanation. We also provide the general framework of projection methods in Chapter 3.; In Chapter 4, we present a mathematical framework to analyze the concept of a visual event. Using the framework, we investigate the regularity of the radiosity solutions for basic geometric models which are most frequently encountered in three-dimensional computer graphics. We also discuss the rate of convergence of the collocation method solutions u ∈ D r+1(S). We show that interpolation error for u ∈ Dr +1(S) converges at the same rate as for the functions v ∈ Cr+1( S) even though Dr+1 u may not be continuous.; In Chapter 5 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 R3. We use affine transformations to convert integrations over elements in R3 to integrations over elements in R2. In Chapter 6 we define the function Hm,n,k for m, n, k ∈ N 0, the set of all non-negative integers. The analytic evaluations of Hm,n,k and other relevant integrations are given in detail for some values of m and n. The evaluation of these integrations is an essential part of the integrations discussed in Chapter 6.; In the final Chapter 7 we compare the performance of the analytic evaluation method of integration with that of other well-known numerical integration schemes. In order to provide numerical testing results for the centroid method, i.e., the piecewise constant collocation method, using the analytic evaluations, we use geometric models with given true solutions. The design decisions for the software package which has been used for the testing are summarized in the end. This package is quite general, both as regards the surface, subdivision scheme and the method of solution. |
