What is a spirograph and what equations describe its curves?

A spirograph is a drawing toy that uses interlocking gears to trace roulette curves. When a small gear of radius r rolls inside a fixed outer gear of radius R, the pen traces a hypotrochoid: x(t)=(R−r)cos(t)+d·cos((R−r)/r·t), y(t)=(R−r)sin(t)−d·sin((R−r)/r·t), where d is the pen distance from the rolling gear's centre and t is the rotation angle. If the gear rolls outside, the result is an epitrochoid.

How many petals or points does a spirograph pattern have?

For a hypotrochoid with outer radius R and inner radius r, when d=r (pen on gear rim) the number of cusps equals R/gcd(R,r). For general pen positions, the pattern has R/gcd(R,r) lobes or petals. The curve closes after r/gcd(R,r) full rotations of the inner gear. For example, R=5, r=3 gives 5/gcd(5,3)=5 lobes and closes after 3/1=3 gear rotations.

What is the difference between a hypotrochoid and an epitrochoid?

A hypotrochoid is traced by a point attached to a circle rolling inside another circle (the inner gear case, used by the classic Spirograph toy). An epitrochoid is traced by a point on a circle rolling outside another circle. Famous special cases: hypotrochoid with d=r gives a hypocycloid (e.g., deltoid for R=3r, astroid for R=4r); epitrochoid with d=r gives an epicycloid (cardioid for R=r, nephroid for R=2r).

How do roulette curves appear in nature and engineering?

Spirograph curves appear across science and engineering: the astroid describes the envelope of lines in optics and the shape of some crystal cross-sections; cardioids appear in microphone polar patterns and as the boundary of bifurcation in the Mandelbrot set; epitrochoids model planetary gear systems used in Wankel rotary engines; and Lissajous figures (a limiting case when gear radii approach infinity) appear on oscilloscope screens representing phase relationships between signals.

Home▸Geometry▸Spirograph Simulator

🧪 Spirograph Simulator

Interactive spirograph simulator. Trace hypotrochoids and epitrochoids — the mathematical curves behind the classic Spirograph toy. Adjust outer radius, inner radius, and pen offset. 6 presets, rainbow trail, animated gear overlay. Free.

Geometry2DModerate60 FPS

⚙ Under the hood

This simulation explores the geometry of spirographs, specifically hypotrochoids and epitrochoids. It demonstrates how circular motion can generate intricate curves and patterns.

SpirographHypotrochoidEpitrochoid

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install