let's consider a receiver, a light source in our scene and an obstacle causing umbra and penumbra.

Since the radiosity of the receiver is assumed constant, we'll get a mean value shown on the left of the segment.

To cope with this, the first solution used in radiosity was to increase the mesh with smaller patches. But this induces additional computation, and some aliasing problems remain.

The best way to solve the problem is to use a discontinuity mesh, where the extremities of the patches are the limits of the umbra and penumbra.

Let's consider a face of the complex associated with two segments. Each vertex corresponds to a limit of discontinuity in the scene.

First we consider two discontinuities on the left segment. We cut the segment into patches, whose limit correspond to a line in a the dual space going through the corresponding vertex.

 scene dual space

Now we consider the discontinuities on the right segment, the patch extremities induced, and their associated dual lines.

Here we can see the sub-face corresponding to the two patches in purple.

 scene dual space

It allows the computation the form-factors between the patches as described before.
 scene dual space

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