Cone, disk and sphere each release two of (polar angle, azimuth, distance). The colored surfaces share one test event. Changing the camera direction reveals how responsibility shifts between them.
The sphere holds distance; the cone holds polar angle; the disk holds azimuth. Their different tangent directions make their poorly conditioned regions complementary. The disk is a meridian half-plane; its finite circular rim here only bounds the drawing.
A sphere and cone use quadratic crossings; a disk uses a plane crossing and domain test. Their local densities are inexpensive. Larger area lights change the sampled source position and variance, while each conditioned source still gives these simple shapes.
pₛ = qₜ |ω·d|; p𝒸 = |cosθ (ω·d) − a·d| / t; p𝒹 = |(a × ω)·d| / (2πt sin²θ)
wᵢ = pᵢ / Σⱼ pⱼ
Densities are per unit camera-ray arclength, conditional on one source point with hemispherical support. qₜ = σₜ exp(−σₜt), σₜ = 0.5 here. Balance weights use equal technique sample counts. This is a local density probe, not a radiance or variance measurement.
Computed geometry and local density ratios, derived from RadianceLab’s transport implementation. Line density, tube shading and glow are diagram styling. No native film, convergence curve or frame-time benchmark runs here.