# Computing the Antipenumbra of an Area Light Source

Seth Teller
Proceedings of SIGGRAPH 1992

## Abstract

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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