The rendering equation

Level SpecializationDifficulty ★★★★★Application⌖ Open in the map

What is it?

Kajiya (1986): the light leaving a point equals the light it emits plus the integral, over all incoming directions, of incoming light times the material's reflectance times a cosine. An integral equation solved by Monte Carlo path tracing — the method behind modern film rendering.

Formulas

L_o(x, \omega_o) = L_e(x, \omega_o) + \int_{\Omega} f_r(x, \omega_i, \omega_o)\,L_i(x, \omega_i)\,(\omega_i\cdot n)\,\dd\omega_i

Why does it matter?

Pixar's RenderMan, Blender Cycles and Arnold are Monte Carlo estimators of this integral; denoisers (often neural) clean up the $1/\sqrt N$ noise.

The mathematics behind it

Definite integral★★★★★fundamental
The colour of a pixel is an integral of incoming light over directions.
Multiple integrals and change of variables★★★★★frequent
Lighting integrals run over the hemisphere of directions and over areas of light sources.
Surface integrals and flux★★★★★frequent
Radiometry (flux, irradiance, radiance) is built on surface and solid-angle integrals.

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.