Extra path freedom can make a steady-state primitive awkward to query. A fixed arrival time spends one of those degrees of freedom: a ball becomes a sheet; a four-parameter support can become a volume.
In a uniform-speed medium, a broken path from light to x to camera has length L(x) = |x−light| + |x−camera|. Fixing this length produces an ellipsoid. Camera unwarping removes the second term from the displayed time coordinate, leaving a sphere around the light. It does not remove the camera from the rendering problem.
L = ct − already-travelled path length
camera unwarped: Lᵤ = |x − light|
A regular level set inside a 3D ball has dimension two. A finite gate still has thickness. Dividing its contribution by the gate width and shrinking that width gives a surface integral with a coarea factor.
∫ f(x) δ(L(x)−ℓ) dx
= ∫L=ℓ f(x) / |∇L(x)| dA
This explicit example fixes four unit directions ωᵢ and lets four propagation lengths tᵢ vary independently over [0,1]. Time is T = t₁+t₂+t₃+t₄, using units c=1 and camera-unwarped arrival. A time slice therefore leaves three free coordinates: t₄ = T−t₁−t₂−t₃. Near T=0 the slice is a tetrahedron; interval bounds clip it into other polyhedra as T increases.
The left view uses the remaining path coordinates. The right view maps them to position x = Σ tᵢωᵢ. The translucent outer edges are a projection of all sixteen corners of the unsliced 4D box, not a literal four-dimensional object residing in 3D space.
A = [ ω₁−ω₄ ω₂−ω₄ ω₃−ω₄ ]
d³x = |det A| dt₁ dt₂ dt₃
∫ δ(Σtᵢ−T) f(x(t)) d⁴t = ∫ f(x) / |det A| d³x
The displayed density assumes unit density in the original parameter box. This map is invertible only when the four directions are affinely independent. Coplanar directions make it singular; the slider approaches that case without dividing by zero. Real estimators additionally need the path sampling density, transmittance, scattering factors, visibility and valid segment bounds. A dimension count alone does not supply them.
The ellipsoid uses a finite point camera and a point source, with fixed earlier path length subtracted. There are two travel legs and one intermediate point; calling it a two-bounce scattering path would add a different constraint. The existing orthographic demonstration elsewhere on this page instead subtracts an infinite common camera distance, producing a paraboloid. These are different camera models.
The coarea formula assumes a nonzero arrival gradient. At the minimum full path length, the focal line is degenerate. This example stays above that limit. Occlusion, refraction and spatially varying propagation speed can change the valid support and the time field.
The four-length construction is a bounded explanatory case of the extra analytic dimension used by temporally sliced photon primitives. Their extended space-time formulation tracks position and arrival constraints together through a 4×4 Jacobian; it is not an arbitrary projection of a 4D shape.
Liu, Jiao & Jarosz · Temporally Sliced Photon Primitives (2022) ↗Exact path-length geometry and an independently derived fixed-direction parameter model. These are interactive schematics, not RadianceLab render captures.