Spirograph
A spirograph traces the path of a point on a small gear as it rolls around a big one. Roll it inside and you get hypotrochoids; roll it outside and you get epitrochoids — the looping, petalled, star-like curves from the classic toy. The shape depends on the ratio of the two radii and how far the pen sits from the rolling gear’s centre: a rational ratio closes into a neat figure, while a nearly irrational one nearly never closes and fills a dense band. Drag the ring size, wheel size and pen offset and watch the curve draw live in colour, or randomise for a fresh pattern, then export it as a PNG.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- The pen position is a sum of two rotations, one for the gear rolling around the ring and one for the pen around the gear, giving the parametric hypotrochoid and epitrochoid equations.
- The pattern closes after the rolling gear completes a whole number of trips, set by the ratio of the radii in lowest terms: 5/3 closes quickly, 97/89 takes far longer and looks almost solid.
- Moving the pen offset from the gear’s centre out past its rim turns gentle rounded curves into sharp cusped loops — the same knob as the hole you picked on the plastic wheel.
- Runs 100% in your browser — simulations are computed locally on your device.
Related tools
3D Geometry Board
Points, lines, planes, spheres and polyhedra on an orbiting camera — slice solids with planes, unfold nets, drop the Dandelin spheres into a cone and drag everything.
Bézier Curve Playground
Drag control points and watch De Casteljau’s algorithm construct a Bézier curve step by step.
Central Limit Theorem Demo
Sample from any lopsided distribution and watch the averages pile up into a bell curve.
Conway’s Game of Life
Paint cells and watch Conway’s cellular automaton breed gliders, guns and oscillators.
Dynamic Geometry Board
Construct with points, lines, circles and transformations, then drag anything — the construction stays true, and a checker tells you which relations are theorems.
Fantasy Map Generator
Roll a seed and get a continent: coastlines, mountains, rivers, biomes and named towns, ready for your campaign.