Interactive application Experimental
Interactive
Fit the material response once, then solve direct lighting from an area light analytically.

Analytical Render · LTC

Linearly Transformed Cosines approximate a complex BRDF lobe with a transformed cosine, turning direct lighting from polygonal area lights—especially quads—into closed-form edge integrals. Compare that deterministic result against Monte Carlo in a material garage while the fit and geometry stay inspectable.

Linearly transformed cosinesLight transport & renderingWebGPUCAlgorithm visualizationParticipating mediaRay marchingSIMDWebAssemblyJavaScriptWGSL
Explore connections
A fitted lobe → a polygon integral
FitTransformIntegrate
Open LTC Light Studio Start with the material garage in split view; inspect the integral below the render.
Material garage comparing fitted LTC with BSDF Monte Carlo, including layered finishes and the organic volume cube.
Material garage comparing fitted LTC with BSDF Monte Carlo, including layered finishes and the organic volume cube.
Quick start · live feature guide

From an area emitter to an integral

A distribution fitted in advance makes an expensive lighting question a compact geometric computation.

  1. 01

    Fit the lobe

    Approximate the GGX directional response by a linearly transformed cosine.

  2. 02

    Transform the emitter

    Express the polygon vertices in the fitted cosine distribution’s coordinates.

  3. 03

    Integrate its boundary

    Accumulate oriented edge contributions over the transformed spherical polygon.

  4. 04

    Compare the image

    Place the fitted result beside Monte Carlo direct lighting and inspect approximation error.

Geometry

The integral stays visible

Direction transforms, polygon projection and boundary integration connect the rendered image to its computation.

Materials

Beyond a diffuse test sphere

Procedural material blends, shading normals and an independent clearcoat lobe expose both the usefulness and limits of the fit.

Comparison

Noise versus model error

A stable deterministic image and a sampled reference answer different questions: convergence removes sampling noise, but cannot remove a fit’s bias.

An extended light without a cloud of samples

Soft area lighting normally integrates a response over many incident directions. For a uniform polygonal emitter, the projected solid angle gives an exact diffuse building block under the stated visibility assumptions. LTC extends that geometric route to a fitted glossy distribution.

What the comparison measures

The garage compares deterministic fitted GGX LTC and Monte Carlo direct lighting for the same opaque material model. It is not a full path-tracing comparison. The glass, thin-film and numerical medium terms are shared scene effects; the cube’s single scattering is not analytically integrated by LTC.

JS and WASM expose compatible diffuse studies. The garage’s fitted materials and additional scene effects use the WebGPU implementation, so changing the backend also selects a supported experiment.

Recorded examples · 2Analytical and sampled direct lightingMaterial and emitter studies compare fitted LTC against Monte Carlo under the selected backend.
Image Material garage comparing fitted LTC with BSDF Monte Carlo, including layered finishes and the organic volume cube.
Image WASM diffuse-light study with a split integral/Monte Carlo view and emitter-geometry controls.
More recorded views · 1
Image The working WebGPU material garage: fitted LTC and sampled direct lighting share the same material scene. The organic medium is a separate numerical effect.
Explore this fieldPhoton Primitives Flatland — Light Transport Spectral geometric optics
Page reviewed September 12, 2026. Demonstrations and technical notes reflect the showcased version.
Project overview