Hold one sampled emission direction fixed. A translated quad sweeps a translated photon volume. Positive and negative boundary sheets replace the shrinking difference of two volumes; the images below are computed from the same controlled model.
One point on the source edge launches one ray. Integrating along the complete edge sweeps a two-dimensional sheet: edge length × travel distance. Its visible span is not a beam radius. In the limit the sheet has zero thickness in its normal direction; the drawn ray rulings only make its construction visible.
Shifting the quad to the right adds an orange slab and removes a blue slab. Divide by the displacement: their thickness vanishes, while their integrated signed contribution survives. The signs describe a derivative, not two extra physical lights.
∂x₀ 1[x₀−a,x₀+a](x) = δ(x−x₀−a) − δ(x−x₀+a)
Source half-width changes the separation of the sheets. Finite translation Δx sets the comparison slabs’ thickness. Sensor σ blurs the resulting image, including the image of a zero-thickness sheet; it does not thicken the 3D support.
The same Gaussian sensor filter is applied to both renders and to the analytic boundary result. Reduce Δx to see finite differences converge. Narrowing the filter sharpens and raises the lobes; it does not create energy. Translation preserves total power, so the positive and negative integrals cancel.
The support is [x₀−a,x₀+a]×[−b,b]×[0,H], with b=0.65 and H=2.2. Light travels in +z from the quad; an orthographic sensor at z=0 receives backscattered light. Both legs attenuate, giving exp(−2σt z). Images are shown per unit emission × scattering × constant phase prefactor; that prefactor is normalized out, not assumed to describe a physical nonzero scattering coefficient at zero extinction. This is a conditional fixed-direction, single-scattering example in a homogeneous medium without occlusion, not a complete angular integral for a general area light.
Let C(u;a,s)=Φ((u+a)/s)−Φ((u−a)/s) and A=(1−exp(−2σt H))/(2σt), with limiting A=H when σt=0. The filtered image is I=A C(u−x₀;a,s) C(v;b,s). Its derivative is A[Gₛ(u−x₀−a)−Gₛ(u−x₀+a)] C(v;b,s). The dashed curve uses the one-sided difference [I(x₀+Δx)−I(x₀)]/Δx. Gaussian CDFs are evaluated numerically; the derivative uses the analytic Gaussian density.
In general, differentiation produces both an interior term and boundary terms. Moving a light can change attenuation, geometry factors, BSDFs, visibility and the sampling map. Those derivatives cannot be discarded just because a fixed support looks translated. The boundary-only result is exact for the controlled model stated here.
Computed conditional volume render and analytic signed derivative. Moving-domain derivatives: interior and boundary terms Related work: differentiable rendering with edge sampling