In my last article, “Physically-Based Rendering in Game, Part 1“,  I introduce reflectance equation and Lambert. In this article, we mainly focus on physically-based lighting.

## Punctual Light Sources

These are classic point, directional, and spot lights in compute games. These local light source can be abstracted to a concept named “punctual light sources“. They are infinitely small light with centain direction. Since what we want is the lighting reach a surface point, we don’t need to consider the attenuation. A punctual light source are parameterized by the light color $\mathbf{c}_{light}$ and the light direction vector $\mathbf{l_c}$. The light color $\mathbf{c}_{light}$ is specified as the color a white Lambertian surface would have when illuminated by the light from a direction parallel to the surface normal ($\mathbf{l_c}=\mathbf{n}$).

How to calculate the contribution from this light source to a point? Here we will start by defining a tiny area light source centered on $\mathbf{l_c}$, with a small angular extent $\varepsilon$. This tiny area light illuminates a shaded surface point with the incoming radiance function $L_{tiny}(\mathbf{l})$. The incoming radiance function has the following two properties:

$\forall\mathbf{l}|\angle(\mathbf{l}, \mathbf{l_c})>\varepsilon, L_{tiny}(\mathbf{l})=0$

$if \mathbf{l_c}=\mathbf{n}, \mathbf{c}_{light}=\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i$

The first property says that no light is incoming for any light directions which form an angle greater than $\varepsilon$ with $\mathbf{l_c}$. The second property follows from the definition of $\mathbf{c}_{light}$. Since the surface is white, $\mathbf{c}_{diff}=1$. Applying reflectance equation and Lambert from last article, there’s the second property. Because $\mathbf{c}_{light}$ requires $\mathbf{l_c}=\mathbf{n}$, it is the limit as $\varepsilon$ goes to 0:

$if \mathbf{l_c}=\mathbf{n}, \mathbf{c}_{light}=\lim_{\varepsilon \to 0}(\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i)$

Since $\mathbf{l_c}=\mathbf{n}$ and $\varepsilon \to 0$, we can assume $\mathbf{n} \cdot \mathbf{l}=1$, which gives us:

$\mathbf{c}_{light}=\lim_{\varepsilon \to 0}(\frac{1}{\pi}\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i)$

That is:

$\lim_{\varepsilon \to 0}(\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i) = \pi \mathbf{c}_{light}$

Now we shall apply our tiny area light to a general BRDF, and look at its behavior in the limit as goes to 0:

$L_0(\mathbf{v})=\lim_{\varepsilon \to 0}(\int_{\Omega} \rho(\mathbf{l},\mathbf{v}) \otimes L_{tiny}(\mathbf{l}) (\mathbf{n} \cdot \mathbf{l}) d\omega_i)=\rho(\mathbf{l_c}, \mathbf{v}) \otimes \lim_{\varepsilon \to 0}(\int_{\Omega} L_{tiny}(\mathbf{l}) d\omega_i)(\mathbf{n} \cdot \mathbf{l_c})$

So

$L_0(\mathbf{v})=\pi \rho(\mathbf{l_c}, \mathbf{v}) \otimes \mathbf{c}_{light} (\mathbf{n} \cdot \mathbf{l_c})$

As you see, the tiny area light source term disappears. The remaining of our equation is what we familar with.

The second article in this serial discusses the equations in physically-based lighting. The next one is about the BRDF, i.e. $\rho(\mathbf{l_c}, \mathbf{v})$.