Computing Exact Shadow Irradiance Using Splines

摘要

We present a solution to the general problem of characterizing shadows in scenes involving a uniform polygonal area emitter and a polygonal occluder in arbitrary position by manifesting shadow irradiance as a spline function. Studying generalized prism-like constructions generated by the emitter and the occluder in a four dimensional (shadow) space reveals a simpler intrinsic structure of the shadow as compared to the more complicated 2D projection onto a receiver. A closed form expression for the spline shadow irradiance function is derived by twice applying Stokes' theorem to reduce an evaluation over a 4D domain to an explicit formula involving only 2D faces on the receiver, derived from the scene geometry. This leads to a straightforward computational algorithm and an interactive implementation. Moreover, this approach can be extended to scenes involving multiple emitters and occluders, as well as curved from these prism-like objects. We call them generalized polyhedral splines because they extend the classical polyhedral splines to include curved boundaries and a density function. The approach can be applied to more general problems such as some of those occurring in radiosity, and other related topics.