 
Summary: Proceedings of the Eleventh Eurographics Workshop on Rendering,
Brno, Czech Republic, June 2000, pages 137148.
A ClosedForm Solution for the Irradiance due
to LinearlyVarying Luminaires
Min Chen James Arvo
California Institute of Technology, Pasadena, CA
{chen,arvo}@cs.caltech.edu
Abstract. We present a closedform expression for the irradiance at a point on a
surface due to an arbitrary polygonal Lambertian luminaire with linearlyvarying
radiant exitance. The solution consists of elementary functions and a single well
behaved special function that can be either approximated directly or computed
exactly in terms of classical special functions such as Clausen's integral or the
closely related dilogarithm. We first provide a general boundary integral that
applies to all planar luminaires and then derive the closedform expression that
applies to arbitrary polygons, which is the result most relevant for global illumi
nation. Our approach is to express the problem as an integral of a simple class
of rational functions over regions of the sphere, and to convert the surface inte
gral to a boundary integral using a generalization of irradiance tensors. The result
extends the class of available closedform expressions for computing direct radia
tive transfer from finite areas to differential areas. We provide an outline of the
