🎡 Spirograph Generator

A spirograph is a drawing toy — patented by Denys Fisher in 1965 — that traces hypotrochoid and epitrochoid curves. A small toothed gear rolls inside (or around) a fixed ring, and a pen placed in an offset hole of the gear draws the curve as the gear turns. The interplay of the ring, the gear and the pen position produces the looping, star-like rosettes the toy is famous for.

When the gear rolls inside the ring, the pen traces a hypotrochoid given by the parametric equations x = (R − r)·cos θ + d·cos(((R − r)/r)·θ) and y = (R − r)·sin θ − d·sin(((R − r)/r)·θ), where R is the ring radius, r the gear radius, d the pen offset and θ the rolling angle. The pattern closes after r / gcd(R, r) full turns, when the pen finally returns to its starting point.

The ratio R : r sets how many lobes the pattern has — a larger ring relative to the gear gives more petals. The pen offset d controls the shape: small d stays rounded and near-circular, while d close to r makes sharp, pointed cusps. Try coprime values of R and r (their gcd is 1) to get dense, many-pointed stars that only close after many revolutions.