The rendering equation for lambertian BRDF is (see Spherical integrals for info on how to solve):
Assume single directional light with direction (a, b) in spherical coordinates:

Where is a delta distribution with value in the direction of the light and everywhere else. This is written as (see Dirac delta for unit sphere):
Now we can solve the integral:
Note now that is precisely "N dot L" as you typically know it, so this matches typical realtime shading.

Note, I used Mathematica to solve the last integral. Wolfram isn't good at delta distributions.

#math #light-transport #pbr