random-points-in-disk#

How many random points does it take before a disk looks like a disk?

Draw N points uniformly at random from a disk, bin them into a grid of voxels, and compare the resulting density map against the exact one. Each voxel's count is essentially Poisson with mean μ = points per voxel, so its relative noise is 1/√μ — which depends only on how many points land in a voxel, not on the size of the disk or the resolution per se. The app plots three panels side by side:

Ideal — exact density, each voxel shaded by the fraction of it inside the disk
Sampled — counts from N random points, on the same color scale
Noise — the error in units of its expected standard deviation, (count − ideal)/√ideal

Sliding N and the resolution shows the tradeoff directly: quadrupling the resolution quadruples the number of voxels, so it takes 4× the points to hold the same noise level. The stats row reports the measured coefficient of variation next to the 1/√μ prediction, and how many points a target noise level would require.

Statistics are computed only over voxels lying entirely inside the disk; boundary voxels have a smaller expected count and would otherwise inflate the measured spread.

Motivation#

This is the discretization question behind isochromat-based MRI simulation: a voxel's signal is a sum over the isochromats that landed in it, so randomly placed isochromats inject noise of order 1/√μ on top of the physics being modeled.

Running locally#

No build step — it is a single static index.html. Open it directly, or serve the directory:

python3 -m http.server 8000