Rendering, with the hidden steps exposed
A scene reduced to two dimensions. Every sampled direction, failed connection, and weighted contribution remains visible. Three path estimators and a surface photon map reveal two ways to reconstruct the same light field.
The scene is a diagram, not a 3D beauty render. Path colors identify construction strategy; the sensor strips below show computed radiance.
Each strip contains 64 angular bins. Every estimator has the same bounce cutoff. The blue photon-map curve adds finite-radius smoothing and eight fixed camera quadrature directions per bin, so close agreement is approximate. Equal launches are not equal work: explicit connections add visibility rays. Changing exposure only changes the display.
Angular radiance profile · left to right follows the camera field of view
Blocked and missed paths remain in the denominator. Light tracing sums every splat into its own bin and divides by all emitted paths.
± The displayed uncertainty is one estimated standard error of the image-wide mean, computed over independent launches—not a per-pixel guarantee. Photon mapping reuses a correlated map, so no independent-launch standard error is shown for it.
A small model with explicit rules
Directions live on a circle. A diffuse surface uses f = ρ / 2 and p(θ) = cos θ / 2, since the hemisphere cosine integral is 2. The angular-to-length conversion is dθ = |cos θₗ| ds / r. Three-dimensional factors such as π and r² do not carry over unchanged.
Path tracing may reach the emitter by chance. Next-event estimation samples a point on it explicitly. The MIS mode uses the power heuristic with both densities expressed per radian; an emitter reached by continuation receives the complementary weight. This is direct-light MIS, not full bidirectional path tracing.
Light tracing begins at a line emitter, reflects, then connects each diffuse vertex to the camera. The camera reports mean radiance over each angular bin. It is an ideal point receiver with a 1D angular film, not the original finite-aperture camera or a complete camera radiometry model.
Photons store incoming flux before the receiving surface scatters it. Gather only on the same object and side; divide by all launched photons and the actual surface support length, then apply ρ / 2. A circle uses arc length and a line uses length: this 2D kernel is not a 3D disk of area πr². Larger radii hide noise but blur lighting boundaries. This is fixed-radius surface photon mapping, not VCM or progressive photon mapping.
Companion to RadianceLab
RadianceLab's photon-primitive studies examine how sampling densities, missed intersections, and small Jacobians affect variance. Flatland isolates the earlier mechanics: constructing a path, testing a connection, accounting for its density, and either accumulating weighted path contributions or reconstructing illumination from a photon density.
This corrected Ruby engine is a new surface-transport edition of the original Raylib visualizer. The Raylib desktop interface has draggable camera controls, material and phase-function inspectors, glass, rough reflection and medium experiments. Those use older tracer classes with known numerical inconsistencies; they are not the core running here. This browser app runs the corrected Ruby source directly in WebAssembly, including the new photon map. It retains diffuse surfaces and an angular pinhole camera; full BDPT, VCM and MLT are not implemented in either edition.
Inspect the actual Ruby source ↗ Read the numerical checks ↗