Appendix: Example occluders

For a shadow ray
toward
a light with position
and diameter *D*, and
a sphere with center
and radius *R*, we need to
decide whether we are in the penumbra region, and
if so, what is the value of ,
the fractional distance
between umbra and penumbra boundaries. We first compute the
distance *t*_{0} to the point on the shadow ray closest to :
.
If *t*_{0} is negative,
then *s*=1. We then compute *b* by assuming ,
,
so we use

We compute the value of the minimum distance

If this distance is between

For triangles, the outer object is a triangle with rounded corners. The offset is different for each side of the triangle and is proportional to the cosine of projected triangle normal and the vector .

1998-10-27