The Rendering Equation

Building on Radiometry

In the previous lecture, we established the fundamental radiometric quantities. Now we’ll use them to derive the rendering equation.

Recall from fig.radiometry-diagram the geometric setup. The irradiance equation eq.irradiance gave us:

E = \frac{d\Phi}{dA}

And radiance eq.radiance extended this to directional quantities:

L = \frac{d^2\Phi}{dA \, d\omega \, \cos\theta}

The Rendering Equation

The rendering equation describes how light bounces through a scene:

L_o(\mathbf{p}, \omega_o) = L_e(\mathbf{p}, \omega_o) + \int_{\Omega} f_r(\mathbf{p}, \omega_i, \omega_o) L_i(\mathbf{p}, \omega_i) (\omega_i \cdot \mathbf{n}) \, d\omega_i

eq.2.1

This equation (eq.rendering) states that outgoing radiance equals emitted radiance plus reflected radiance from all incoming directions.

Note: For a complete mathematical derivation of the rendering equation from first principles, see Derivation D1: Rendering Equation Derivation.

Rendering equation diagram — **Figure 2.1**: The rendering equation geometry showing incoming and outgoing light directions

As shown in fig.rendering-diagram, we integrate over the hemisphere of incoming directions. Each incoming ray’s contribution depends on:

The BRDF $f_r$ (bidirectional reflectance distribution function)
The incoming radiance $L_i$ (which we learned about in eq.radiance)
The cosine term $\omega_i \cdot \mathbf{n}$

Path Tracing

Path tracing solves eq.rendering by Monte Carlo integration. We randomly sample directions from the hemisphere and average their contributions.

Try hovering over the cross-page references above to see previews from lecture 1! The geometric foundations from fig.radiometry-diagram directly inform how we set up the rendering equation shown in fig.rendering-diagram.

We can always refer fig.lol for the same.