A photon sheet intersects a receiver along a curve. Give the camera its own swept sheet and the two footprints can meet at isolated points. Move the sheets and watch the intersection determinant change.
Two transverse surfaces in 3D meet in a 1D curve. A separately sampled camera hit almost surely misses that curve. The construction needs another analytically integrated coordinate, a camera-side sweep, or a finite kernel.
2D photon sheet ∩ 2D receiver → 1D curve
Two footprint curves on the receiver meet at discrete points. The geometry-only inverse determinant is shown for each transverse branch. Tangency makes this local formula singular; visibility, BSDFs and sampling densities still decide actual transport.
J = |nS · (nP × nC)|
The receiver is the unit sphere. Signed-distance constraints are gP=z−h and gC=x cosθ+z sinθ−k. Their unit normals are nP=(0,0,1), nC=(cosθ,0,sinθ), and nS=x/R. For one constraint, the surface coarea factor is |∇S gP|=|nS×nP|=r/R. For two independent constraints, ∫S f δ(gP)δ(gC) dA = Σ f(xi)/|nS·(nP×nC)|. The example reports every root, including the branch a real opaque receiver might hide.
Here z=h, x=(k−h sinθ)/cosθ, and y=±√(1−h²−x²). No real roots means no intersection; a double root is tangency. This is a test of intersection geometry, not a validated surface-radiance estimator. In a complete derivation, parameter derivatives, incidence factors and PDFs must be transformed consistently; replacing all of them by this geometric factor would be incorrect.
Experimental construction study, connected to RadianceLab’s surface-estimator investigation. The drawn sheet is a support, not an extra physical object.