We define the antiumbra and the antipenumbra of a convex areal light source shining through a sequence of convex areal holes in three dimensions. The antiumbra is the volume beyond the plane of the final hole 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
^2) time algorithm that computes this boundary.
The antipenumbra computation is motivated by visibility computations, and might prove useful in rendering shadowed objects. We also present an implemented extension of the algorithm that computes planar and quadratic surfaces of discontinuous illumination useful for polygon meshing in global illumination computations.