Why is the symbol for solid angle a small omega in the definition of the BRDF?

Question

In many resources the BRDF is defined like $$f_{r}(\omega_{i},\omega_{r}) = \frac{\mathrm{d}L_{r}(\omega_{r})}{L_{i}(\omega_{i})\,\cos{\theta_{i}}\,\mathrm{d}\omega_{i}}.$$ The index $r$ might be replaced by $o$ but overall the equation looks something like this. However, in some other resources I see the definition $$f_{r}(\theta_{i},\phi_{i};\theta_{r},\phi_{r}) \equiv \frac{\mathrm{d}L_{r}(\theta_{i},\phi_{i};\theta_{r},\phi_{r};E_{i})}{L_{i}(\theta_{i},\phi_{i})\,\cos{\theta_{i}}\,\mathrm{d}\omega_{i}}.$$ The second definition seems more precise to me as it – besides other things – clearly states that the BRDF in its purest form is a 4-dimensional function of two directions and clearly differentiates between the direction $(\theta_{i},\phi_{i})$ and the solid angle $\omega_{i}$.

It seems to me as if the first definition mixes up the direction of incidence and the solid angle in the direction of incidence. The symbol $\omega_{i}$ stands for the direction – at least I hope so as the BRDF is not a function of solid angles – to economize on writing, I understand that, but what is $\mathrm{d}\omega_{i}$ then? It must be the solid angle but $\omega_{i}$ is a direction, not a solid angle. Is $\mathrm{d}\omega_{i}$ simply a common way to express the solid angle in a specific direction? May I have misunderstood something?

Another thing I've noticed is that the solid angle in the definitions of radiance, luminance and basically all other radiometric and photometric quantities that deal with a solid angle is symbolized with $\Omega$, i.e. a big omega. Actually, as far as I can tell, solid angles are usually symbolized with $\Omega$. Why is it different in the definition of the BRDF? Am I missing something?

Answer 1

The use of spherical coordinates like $(\theta, \phi)$ is kind of an implementation detail that isn't an essential part of the definition of a BRDF, or other functions defined over a spherical domain generally.

It's a lot like how in vector math, we will often express a point or vector by a single symbol like $v$, understanding that "under the hood" it's going to be expressed as coordinates $(v_x, v_y, v_z)$ in some coordinate system. Writing out all the coordinates in every equation would make notation very verbose and would often obscure the vectorial relationships we really care about.

Similarly, with functions defined over a sphere or hemisphere, we will often express a direction by $\omega$, understanding that when we get to actually writing code this will stand in for some coordinates on the sphere, but the exact choice of representation is not relevant to the math.

The symbol $\mathrm{d}\omega$ represents a differential quantity of solid angle—this is an abuse of notation for sure, but it serves the same purpose of abstracting away the implementation details; in spherical coordinates that would be $\sin(\theta) \, \mathrm{d}\theta \, \mathrm{d}\phi$ for example, but it's kind of assumed as background knowledge that you know how to integrate things over a sphere in whatever coordinate representation you're using.

Differentials as seen in calculus sort of aren't actual values themselves, but are things that can be either integrated over (in which case they specify the measure of the integration), or you can take a ratio of two differentials, which defines a density function (kind of a generalized derivative).

The symbol $\Omega$ is most often used to represent the hemisphere in my experience, not solid angles generically. You'll see something like $\int_\Omega \text{...stuff...}\, \mathrm{d}\omega$ to mean integrate that function over the normal-facing hemisphere with respect to the solid angle measure (i.e. area on the unit sphere).

Answer 2

The $\omega$ is a direction. Whether you parametrise this direction in spherical coordinates $(\phi, \theta)$, in Cartesian coordinates $(x,y,z)$, or some other coordinate system is irrelevant. Thus one uses $\omega$ to represent directions. $dx$ typically refers to a differential. In the current case it is w.r.t. the solid angle measure $\sigma$, so it would be more precise to write $d\sigma(\omega)$. This is still informal and handwavy however. The meaning of the expression $\frac{dL}{d\omega}$ is formalised in the PhD thesis of Eric Veach.