A small parameter patch becomes a physical patch. Its projected area depends on both the photon construction and the camera direction. The inverse Jacobian converts uniform parameter density into camera-query density.
Mint and gold arrows are its parameter derivatives. Their cross product gives oriented physical area. Varying photon angle or distance changes this area and its normal.
a = ∂X/∂u, b = ∂X/∂v
The white ray changes the projection of that patch. As the ray becomes tangent, J approaches zero and its inverse grows. A large inverse is poor conditioning, not additional energy.
J = |(a × b) · d| = |a × b| |n · d|
The illustrated uniform density lives in the named parameter coordinates. Sphere: X(θ,φ)=rω, |a×b|=r²sinθ. Cone: X(φ,r)=rω with θ fixed, |a×b|=r sinθ. Plane: X(u,t)=u eₓ+tω, with a fixed unit emitter edge direction. Parameter changes also change the units of J.
For a transverse sheet crossing, 1/J enters the camera-query change of variables. A complete estimator additionally needs sampling PDFs, emission, phase/BSDF factors, attenuation, visibility and every valid branch. At exact tangency the regular local formula fails; this schematic reports the singularity rather than calling a clamp physically exact.
Independent geometric model of the change-of-variables used by photon sheets. The graph is computed from the same tangent vectors as the 3D patch.