Algorithm

Rather than creating a correct soft-edged shadow from an area source, the algorithm creates a shadow of a soft-edged object from a point source (Figure 1). The penumbra is the shadow of the semi-opaque (outer) object that is not also shadowed by the opaque (inner) object. The transparency of the outer object increases from no transparency next to the inner object to full transparency at the boundary of the outer object. For an isolated object, we can use inner and outer offsets of the real object to achieve believable results. We also need to make the intensity gradient in the penumbra natural. This can be achieved by computing a variable $\tau$ that begins at $\tau=0$ on the penumbra/umbra boundary (the surface of the inner object) and increases linearly with distance to $\tau=1$ on the outer boundary of the penumbra (the surface of the outer object). This can control an interpolation of the illumination attenuation function s. For diffuse spherical lights and occluders with a straight edge, the attenuation is a sinusoid: $s = (1 + sin(\pi \tau - \pi/2))/2$. To mimic this behavior with a polynomial we use the Bernstein interpolant: $s = 3\tau^2 - 2\tau^3$, which has the same values and derivatives as the sinusoid at $\tau=0$ and $\tau=1$.

Figure 1: The inner object is opaque and the outer object's opacity falls off toward its outer boundary.

While almost any inner and outer surfaces are practical for an isolated object, we would like our algorithm to remain simple and robust for multiple objects. For example, if a point is in the penumbra region for two objects we can make a composite attenuation factor based on the individual factors s₁ and s₂. Obvious candidates are addition: s = 1 - ((1-s₁) + (1-s₂)), multiplication: s = s₁ s₂, and thresholding: $s = \mbox{min}(s_2, s_2)$. All will yield continuous intensity transitions and visually pleasing results for the shadow of two objects. However, thresholding is more conservative when many distinct objects are grazed by a shadow ray, resulting in shadows that are never darker than they should be.

The remaining important issue is how the inner and outer objects are generated for an input object. Figure 2 shows why the inner object must be at least as big as the input object to prevent ``light leaks'': any gap between the inner objects would result in a nonzero s and a lightening in the middle of the shadow. For this reason we use the object itself as the inner object. For the outer object, we use an image of the input object that is expanded in a direction natural for the particular primitive. For example, for a sphere, we use a larger sphere. For a polygon, we use a larger polygon. Our rationale for choosing the size of the outer object is shown in Figure 3. We would like the penumbra width in the approximation to be approximately W. Since $W \approx aD/(A-a)$ and b / (A-a) = W/A, a reasonable penumbra size will occur when b = aD/A. Note that the umbra region will be larger than in a physically-based computation, but when the occluder size is large relative to the light size, this should not be too noticeable. An example of the mechanics of the algorithm is discussed in the Appendix.

Figure 2: For this case to work using local computations without light leaking between the objects requires that the inner objects be at least as large as the objects themselves.

Figure 3: Choosing the size of the outer object for a given configuration.