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

CC BY-SA 4.0
2021-05-21T08:14:58.737Z
2021-05-21T08:14:58.737Z
lightxbulb
<p>The use of spherical coordinates like <span class="math-container">$(\theta, \phi)$</span> 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.</p>
<p>It's a lot like how in vector math, we will often express a point or vector by a single symbol like <span class="math-container">$v$</span>, understanding that &quot;under the hood&quot; it's going to be expressed as coordinates <span class="math-container">$(v_x, v_y, v_z)$</span> 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.</p>
<p>Similarly, with functions defined over a sphere or hemisphere, we will often express a direction by <span class="math-container">$\omega$</span>, 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.</p>
<p>The symbol <span class="math-container">$\mathrm{d}\omega$</span> 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 <span class="math-container">$\sin(\theta) \, \mathrm{d}\theta \, \mathrm{d}\phi$</span> 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.</p>
<p>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).</p>
<p>The symbol <span class="math-container">$\Omega$</span> is most often used to represent the hemisphere in my experience, not solid angles generically. You'll see something like <span class="math-container">$\int_\Omega \text{...stuff...}\, \mathrm{d}\omega$</span> to mean integrate that function over the normal-facing hemisphere with respect to the solid angle measure (i.e. area on the unit sphere).</p>

CC BY-SA 4.0
2021-05-24T12:10:51.687Z
In other fields $\Omega$ is used to represent any solid angle, while in graphics it's most often the hemisphere centered around the normal at a specific point (also $\Omega_{\pmb{x}}$ for a point $\pmb{x}$ , the normal being implicit; or also $\Omega_{\pmb{n}}$ for a normal $\pmb{n}$).
lightxbulb
CC BY-SA 4.0
2021-05-23T19:16:35.920Z
2021-05-23T19:16:35.920Z
Nathan Reed
<p>In many resources the BRDF is defined like <span class="math-container">$$f_{r}(\omega_{i},\omega_{r}) = \frac{\mathrm{d}L_{r}(\omega_{r})}{L_{i}(\omega_{i})\,\cos{\theta_{i}}\,\mathrm{d}\omega_{i}}.$$</span> The index <span class="math-container">$r$</span> might be replaced by <span class="math-container">$o$</span> but overall the equation looks something like this. However, in some other resources I see the definition <span class="math-container">$$f_{r}(\theta_{i},\phi_{i};\theta_{r},\phi_{r}) \equiv \frac{\mathrm{d}L_{r}(\theta_{i},\phi_{i};\theta_{r},\phi_{r};E_{i})}{L_{i}(\theta_{i},\phi_{i})\,\cos{\theta_{i}}\,\mathrm{d}\omega_{i}}.$$</span> 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 <span class="math-container">$(\theta_{i},\phi_{i})$</span> and the solid angle <span class="math-container">$\omega_{i}$</span>.</p>
<p>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 <span class="math-container">$\omega_{i}$</span> 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 <span class="math-container">$\mathrm{d}\omega_{i}$</span> then? It must be the solid angle but <span class="math-container">$\omega_{i}$</span> is a direction, not a solid angle. Is <span class="math-container">$\mathrm{d}\omega_{i}$</span> simply a common way to express the solid angle in a specific direction? May I have misunderstood something?</p>
<p>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 <span class="math-container">$\Omega$</span>, i.e. a big omega. Actually, as far as I can tell, solid angles are usually symbolized with <span class="math-container">$\Omega$</span>. Why is it different in the definition of the BRDF? Am I missing something?</p>

<brdf>
Why is the symbol for solid angle a small omega in the definition of the BRDF?
akuzminykh
# Why is the symbol for solid angle a small omega in the definition of the BRDF?

(Asked by: username_0 on 2021-05-21)

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?

**Tags:** brdf


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## 2 answers

### Answer by username_2 on 2021-05-23. Score: 6 | Accepted answer

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).

> **Comments:**
> **username_1**: In other fields $\Omega$ is used to represent any solid angle, while in graphics it's most often the hemisphere centered around the normal at a specific point (also $\Omega_{\pmb{x}}$ for a point $\pmb{x}$ , the normal being implicit; or also $\Omega_{\pmb{n}}$ for a normal $\pmb{n}$).  

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### Answer by username_1 on 2021-05-21. Score: 3

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.


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