Path Tracer

The rendering equation recurses forever because light keeps bouncing. Truncating paths at a fixed depth silently deletes all the energy those deeper bounces would have carried, which darkens the image: that's bias. Russian roulette terminates paths probabilistically instead. At each bounce, kill the path with probability \(q\); if it survives, divide its contribution by \(1-q\).

The survivors are boosted by exactly the factor needed to make up for the killed paths, so the expected value is untouched, so paths become finite while the estimator stays unbiased. The price is variance: the occasional long-surviving path carries a large weight. I keep \(q\) modest (around 0.3) and only start roulette after a few guaranteed bounces, so short paths (which carry most of the energy) are never gambled away.

A naive path tracer only finds light when a random bounce happens to hit an emitter. For small lights that almost never happens, so most paths contribute nothing and the image stays noisy. Event splitting fixes this by splitting the reflected radiance at every bounce into two estimators: a direct term, sampled by picking a point on a light source deliberately, and an indirect term, sampled by bouncing the path onward.

For the direct term, I sample a point \(y\) uniformly on an area light (\(p_A(y) = 1/A\)) and cast a shadow ray to test visibility. Because the sample is drawn over the light's surface rather than over directions, the integrand picks up a geometry term that converts between the two measures:

\(\cos\theta_x\) is the angle at the shaded point, \(\cos\theta_y\) the angle at the light (a light facing away contributes nothing), the squared distance gives the natural falloff, and \(V\) is the shadow ray's 0-or-1 visibility. To avoid counting the same energy twice, emission is ignored when an indirect bounce ray happens to land on a light, since that light was already accounted for by the direct term at the previous vertex. The exception is mirror and refractive bounces: their BRDF is a delta function that shadow rays can't sample, so emission is kept when following them.

Monte Carlo variance comes from the mismatch between the integrand and the sampling distribution, and the ideal distribution is proportional to the integrand itself. Uniform hemisphere sampling wastes samples on grazing directions whose \(\cos\theta\) factor nearly zeroes them out.

Lambertian. The BRDF is constant (\(\rho/\pi\)), so the integrand is proportional to \(\cos\theta\) alone. I sample a cosine-weighted hemisphere (uniform point on the unit disk, projected up, which is Malley's method), whose pdf is exactly \(\cos\theta/\pi\). The weight then collapses:

Every bounce just multiplies the path throughput by the albedo, and the direction-dependent variance vanishes entirely.

Phong. The glossy lobe \(f_r \propto \cos^n\alpha\) concentrates around the mirror direction \(\omega_r\), and gets narrower as the exponent \(n\) grows. I sample directions around \(\omega_r\) with \(\cos\alpha = u_1^{1/(n+1)},\ \phi = 2\pi u_2\), giving \(p(\omega) = \frac{n+1}{2\pi}\cos^n\alpha\). The \(\cos^n\alpha\) in the pdf cancels the one in the BRDF, so a rough and a near-mirror surface converge equally fast instead of the tight lobe being missed by uniform samples.

At a dielectric boundary the transmitted direction follows Snell's law, \(\eta_1 \sin\theta_1 = \eta_2 \sin\theta_2\). Solving for the transmitted ray requires \(\sin^2\theta_t \le 1\); past the critical angle (glass-to-air at roughly 42°) there is no solution and all light reflects. This is total internal reflection, which is what makes the bottom of a glass object glitter.

Real dielectrics both reflect and refract, with the split governed by the Fresnel equations. I use Schlick's approximation:

Glass reflects ~4% head-on but almost 100% at grazing angles, which is why windows become mirrors at night when viewed obliquely. Rather than splitting every hit into two rays (which would branch the path tree exponentially), the path chooses: reflect with probability \(R\), refract with probability \(1-R\), each weighted by one over its probability. The Fresnel factor cancels against the weight, one ray continues, and the expected value is exactly the full split.

A pinhole camera renders everything in perfect focus because every pixel sees through a single point. Real lenses have area, and that's where focus blur comes from. The thin-lens model adds two parameters: an aperture radius and a focal distance.

For each camera ray, first find the point \(P\) where the original pinhole ray crosses the focal plane. Then jitter the ray origin to a uniformly-sampled point on the aperture disk and aim the new ray at \(P\):

Every lens sample converges on \(P\), so geometry on the focal plane stays sharp no matter where on the aperture the ray started. Geometry off the plane is hit by rays from different lens points at different locations, and averaging those samples smears it into the circle of confusion, which is the bokeh, produced by the same Monte Carlo machinery. The aperture radius directly controls blur strength: shrink it to zero and the pinhole camera returns.