The diffuse walk — Parallax Sketchbook

SKETCHBOOK / 03 The Lantern Room · 2-week chapter

Page 1 of 6: A scattering sample is a contract

BSDF / PDF pairing

A scattering sample is a contract

A Lambertian BRDF is reflectance divided by pi. Its reflected-energy fraction is the reflectance, not the BRDF value by itself. With cosine-weighted hemisphere sampling, the cosine-over-pi density cancels the corresponding BRDF and cosine factors in the throughput update.

That cancellation is specific to the paired model and sampler. It is not permission to delete PDF handling from the integrator. Return a scattering sample containing direction, density, and the information needed to distinguish an ordinary density from a delta event later.

The runnable bridge in diffuse.cpp keeps the formula visible even though the matched terms cancel. Its validated scene owns sphere/material values; reflectance channels must lie in [0,1]. It builds the world-space frame, uses a documented small-scene ray offset, and propagates invalid sampling inputs as ordinary errors before any division.

LOOK CLOSER

#include "lantern.hpp"
#include <cmath>
#include <expected>
#include <numbers>

std::expected<double, lantern::Error>
diffuse_weight(double reflectance, const lantern::DirectionSample& s) {
    if (!std::isfinite(reflectance) || reflectance < 0 || reflectance > 1 ||
        !std::isfinite(s.pdf) || s.pdf <= 0 || !std::isfinite(s.direction.z) ||
        s.direction.z < 0 || s.direction.z > 1)
        return std::unexpected(lantern::Error{"Invalid scattering input."});
    const double weight = (reflectance / std::numbers::pi) * s.direction.z / s.pdf;
    if (!std::isfinite(weight))
        return std::unexpected(lantern::Error{"Non-finite weight."});
    return weight;
}

Isolated scalar throughput calculation, not a complete BSDF or integrator.

From the page to your own work

Your studio assignment.

Add the first indirect diffuse bounce to your own Lantern Room.

A finish line you can see

A bounded diffuse path loop, a constant-environment test, and a trace of one sample.

Check a matched sampler/BRDF cancellation numerically.
Name the local/world frame conversion.
Test zero and maximum bounce behavior.
Describe the remaining truncation bias.

Keep the evidence in your weekly notebook ↗

Explain it like an engineer

Try answering aloud before opening the review notes. Use your own implementation as evidence.

Why does the same albedo update fail for a different directional sampler?

The cancellation depends on a specific PDF.
Write f times cosine divided by the new density.
Keep directional measure and local hemisphere conventions consistent.

These prompts support practice; they are not an automatic assessment of your work.

A tip for this exercise

Debug a single sample under a constant environment before adding the colored wall. A short trace of direction, PDF, throughput, and termination reason makes the first indirect bounce much easier to inspect.

Hints, when you need them.

Try a first step, then reveal as much help as you need. Each hint gets a little more specific.

1Hint 1 · Find a starting point

List the state that persists between hits: the ray, accumulated radiance, throughput, and depth. Decide which value is added and which is multiplied at a scattering event.

2Hint 2 · Choose an approach

Keep the sampled direction in its local frame while checking the cosine and PDF. Then transform it through an orthonormal frame and add just one bounded indirect step.

3Hint 3 · Check your reasoning

For reflectance 0.6 and a valid positive-cosine sample from the matched sampler, the scalar throughput factor should be 0.6. Check this before testing your documented zero-depth and maximum-depth behavior.

A small creative variation ↗

Render one colored wall twice, once with one indirect bounce and once with a larger bounded depth.

Optional. Save it for later if this chapter already feels full.

Coming back after a break? ↩

Run the constant-environment check and fixed-ray trace unchanged before editing the throughput update.

Books & principles behind this study +

Original Parallax explanations and assignments, with these works for further reading.

YOUR MARGIN NOTES

Leave yourself a good question.

What changed your mind? What would you try next?

Notes for The diffuse walk

Notes are private to this browser and saved as you write.

A CHECKPOINT, NOT A SCORE

Can you explain the idea and try the exercise yourself?

Completion records your own assessment. It does not compile or test your C++.

Companion source 10 files