Computing the Antipenumbra of an Area Light Source

Seth Teller Proceedings of SIGGRAPH 1992


We define the antiumbra and the antipenumbra of a convex area light source shining through a sequence of convex areal holes in three dimensions. The antiumbra is the volume from which all points on the light source can be seen. The antipenumbra is the volume from which some, but not all, of the light source can be seen. We show that the antipenumbra is, in general, a disconnected set bounded by portions of quadric surfaces, and describe an implemented $O(n^2)$ time algorithm that computes this boundary, where $n$ is the total number of edges comprising the light source and holes. The antipenumbra computation is motivated by a visibility scheme in which we wish to determine the volume visible to an observer looking through a sequence of transparent convex holes, or portals, connecting adjacent cells in a spatial subdivision. Knowledge of the antipenumbra should also prove useful for rendering shadowed objects. Finally, we have extended the algorithm to compute the planar and quadratic surfaces along which the rate of areal variation in the visible portion of the light source changes discontinuously due to occlusion. These surfaces are relevant in polygon meshing schemes for global illumination and shadow computations.

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