Closed-Form Expressions for Irradiance from Non-Uniform Lambertian Luminaires Part II: Polynomially-Varying Radiant Exitance

We present new analytical techniques for computing illumination from non-uniform luminaires. The methods are based on new closed-form expressions derived by generalizing the concepts of irradiance tensor and angular moment to rational forms and an arbitrary number of directions, known as rational irradiance tensors and rational angular moments, respectively. The techniques apply to any emission, reflection or transmission distribution expressed as a polynomial over a polygonal surface, and provide a powerful mathematical tool to handle more complex BRDF's. We derive closed-form expressions for irradiance due to polygonal luminaires with polynomially varying radiant exitance, which satisfy a recurrence relation that subsumes Lambert's formula for uniform luminaires. Our formulas extend the class of available closed-form expressions for computing direct radiative transfer from planar surfaces to points, and can find many potential applications in simulating non-Lambertian illumination and scattering phenomena