This evening is more preparing of tutorial materials. In particular, for understanding the rendering equation and radiometry, the three most frequently used terms (radiant flux, radiance and irradiance), understanding what an integral is, why we need a surface integral over the hemisphere, why the integral over the unit hemisphere is 2π
, why that is also the solid angle subtended by that hemisphere, why the irradiance with constant radiance L
is π·L
and not 2π·L
, why a lambertian BRDF function is defined as c/π
, why a BRDF can evaluate to infinity, what Monte Carlo integration is, what a probability density function (pdf) is, why the pdf of a randomly chosen vector over the hemisphere is 1/(2π)
and why the hell we should need all of this when building a physically correct path tracer.
And I constantly lose grasp of certain things and have to re-read them.
Also, there is a nice explanation video about the rendering equation https://www.youtube.com/watch?v=eo_MTI-d28s , though I pm’ed Eric Arnebäck on Twitter about a slight error: BRDF evaluating to 1 does not equate to all light from omega_i reflecting towards omega_o - if that were the case, the BRDF would essentially represent a perfect mirror and would need to be the dirac delta function evaluating to infinity for that pair of directions, since the cosine-weighted BRDF still needs to integrate to <= 1 over the hemisphere).
EDIT: Just saw that the original James T. Kajiya paper “The Rendering Equation” from 1986 gives the answer to why it’s possible for basically all path tracers today to approximate the recursively defined rendering equation using an iterative process by following a light path over multiple bounces and accumulating the received light: It has got to do with reformulating the so called “Fredholm integral equation of the second kind” (of which the rendering equation is an example) into an infinite series (the Liouville–Neumann series), of which one can simply evaluate only the first n
summands, where the first summand is just the direct light from a light source to the eye, the second summand is light from a source to a point on a surface and then to the eye, the third summand is light hitting two surfaces before reaching the eye, and so on.
I never understood why it works this way.