✨ Math test: the rendering equation, briefly

Note to self: mockup post to test KaTeX (inline + block, fractions, integrals, Greek, matrices). Delete before launch.

Almost everything a technical artist touches in lighting traces back to one equation. It looks intimidating written out, but each piece has a plain-language job. Here it is.

The equation

The outgoing radiance $L_o$ from a point $x$ in direction $\omega_o$ is the emitted light plus everything reflected from the hemisphere $\Omega$ above the surface:

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

Reading it left to right: take the emitted light $L_e$, then add up — that's the integral $\int_\Omega$ — the contribution of every incoming direction $\omega_i$, weighted by how the surface reflects it and how grazing the angle is.

The pieces

• $L_i(x, \omega_i)$ — the light arriving from direction $\omega_i$.
• $f_r(x, \omega_i, \omega_o)$ — the BRDF, the surface's rule for turning incoming light into outgoing light.
• $\omega_i \cdot n$ — the cosine term, $\cos\theta_i$. Light hitting head-on counts fully; light at a grazing angle barely counts at all.

That cosine is why a floor looks bright underfoot and dim toward the horizon.

A concrete BRDF

For a simple diffuse surface with albedo $\rho$, the BRDF is just a constant:

$$f_r = \frac{\rho}{\pi}$$

The $\pi$ in the denominator is the part everyone forgets — it's what keeps energy conserved so a surface never reflects more light than it receives.

Specular is where it gets interesting. The Fresnel term, using Schlick's approximation, is:

$$F(\theta) = F_0 + (1 - F_0)\,(1 - \cos\theta)^5$$

where $F_0$ is the reflectance at normal incidence. For most dielectrics $F_0 \approx 0.04$, which is why even matte plastic gets a bright rim at grazing angles.

Why the integral is hard

We can't evaluate $\int_\Omega$ exactly in real time, so we approximate it. Monte Carlo integration samples $N$ random directions and averages:

$$\int_{\Omega} f(\omega)\, d\omega \approx \frac{1}{N} \sum_{k=1}^{N} \frac{f(\omega_k)}{p(\omega_k)}$$

The $p(\omega_k)$ is the probability of picking each sample — importance sampling just means choosing $p$ to match where $f$ is large, so you spend samples where they matter.

A tiny bit of linear algebra

Transforming a normal isn't the same as transforming a position. If $M$ is your model matrix, normals must use the inverse-transpose:

$$n' = (M^{-1})^{\mathsf{T}}\, n$$

For a rotation like

$$R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$$

the inverse-transpose equals $R$ itself, which is why pure rotations never distort your normals — but non-uniform scale does, and that is the usual culprit behind broken lighting.

Wrapping up

You rarely type these equations into a shader directly, but knowing what each term does turns "the lighting looks wrong" into a diagnosis. The cosine, the $\pi$, the Fresnel rim — each one is a knob with a physical meaning.

If the math above rendered as crisp symbols rather than raw dollar signs, KaTeX is working.