Rendering

When you shoot rays into a scene or sample directions on a hemisphere, you are choosing among a finite set of direction bins. That is a circle. This page takes the one rendering-adjacent fact the project actually proves — that the bin schedule is safe and closes — and is careful to claim nothing about image quality.

Lesson Goal

Model a direction sampler as a finite circle of bins, and separate the proved part (safe, closing indexing) from the parts that need pictures and metrics (coherence, noise, frame time).

The Idea

Partition directions into n bins, 0 through n-1. A sampler steps through them by a stride, exactly like a coil on C_n:

bin(ray) = ray_index mod n

The proved seed is the direction-bin schedule: indices stay in range and the schedule closes after one full pass. That rules out a sampler walking off the end of its bin table. It says nothing about whether the resulting image is less noisy or renders faster — those are empirical questions that need rendered frames and baselines, not theorems.

Worked Example

A sampler with 8 direction bins:

ray 3  -> bin 3 mod 8 = 3
ray 11 -> bin 11 mod 8 = 3   (same bin, one lap later)
ray 8  -> bin 8 mod 8 = 0    (the schedule closes and repeats)

The indexing is just C_8 rotation. The closure fact (a full pass returns to the start) is the certified part.

Common Mistake

A safe, closing bin schedule is not a rendering-quality result. The proved seed covers bin indexing and closure only; ray coherence, lower noise, and better frame time would need images, metrics, and baselines. Rendering demos are executable references, not proof artifacts and not performance claims. See What “Proved” Means Here.

Checkpoint

What extra evidence — beyond the closure theorem — would you need before claiming a direction-bin schedule actually improves a rendered image? (Answer: rendered frames, a noise/quality metric, and a fair baseline.)

Source Trail

Paper source: Coil Ray And Sampling