The collocation method for solving the radiosity equation for unoccluded surfaces
type: Item
instance of: scholarly article
title: The collocation method for solving the radiosity equation for unoccluded surfaces
zbMATH Open document ID: 0914.65137
author: Graeme Chandler
author: Kendall E. Atkinson
published in: Journal of Integral Equations and Applications
publication date: 1999-06-07T00:00:00Z
full work available at URL: CONT10-3.html
review text: The radiosity equation occurs in computer graphics, and its solution leads to more realistic illumination for the display of surface \(S.\) The equation is \[ (I-\mathcal K)u:=u(P) - \frac{\rho(P)}{\pi}\int_S u(Q)G(P,Q)V(P,Q)dS_Q = E(P),\quad P\in S, \tag{1} \] with \(u(P)\) the ``brightness'' or radiosity and \(E(P)\) the emissivity at \(P\in S.\) The function \(\rho(P)\) gives the reflectivity with \(0\leq\rho(P) < 1.\) The function \(G\) is given by \[ G(P,Q)=\left[(Q-P)\cdot\mathbf n_P\right]\left[(P-Q)\cdot\mathbf n_Q\right]/| P-Q|^4. \] In this, \(\mathbf n_p\) is the inner unit normal to \(S\) at \(P,\) and \(\mathbf n_Q\) is defined analogously. The function \(V(P,Q)\) is a ``line of sight'' function. An unoccluded surface is one for which \(V \equiv 1\) on \(S,\) and it is the simplest case investigated here in order to make clearer the behavior of the approximation methods being used. In the numerical solution of (1), the Galerkin method has been the predominant form of numerical solution. In this paper, the collocation method with approximations of all possible orders is investigated. The case of piecewise smooth \(S,\) which is a more practical situation, is considered, and the effect of using linear interpolation to approximate \(S\) is also analyzed. For one particular choice of such interpolation, superconvergence results are obtained when solving on a smooth surface. Numerical results conclude the paper.
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 1240965
zbMATH Keywords: numerical results
zbMATH Keywords: computer graphics
zbMATH Keywords: unoccluded surfaces
zbMATH Keywords: reflectivity
zbMATH Keywords: emissivity
zbMATH Keywords: smooth and piecewise smooth surfaces
zbMATH Keywords: Lambertian diffuse reflector
zbMATH Keywords: radiosity equation
zbMATH Keywords: piecewise linear collocation method
zbMATH Keywords: superconvergence
reviewed by: Nikolay Yakovlevich Tikhonenko
describes a project that uses: BIEPACK
MaRDI profile type: Publication
OpenAlex ID: W2067316110
cites work: Q4124250
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DOI: 10.1216/JIEA/1181074231
description: scientific article; zbMATH DE number 1240965