Hypotrochoid Explorations

hypotrochoid spirograph geometry interactive

A hypotrochoid is the curve traced by a point attached to a circle rolling inside a larger fixed circle. It’s the math behind every Spirograph kit.

Interactive Demo

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The Math

Given fixed circle radius R, rolling circle radius r, and pen distance d from the rolling center:

  • x = (R − r) cos(θ) + d cos(((R − r) / r) θ)
  • y = (R − r) sin(θ) − d sin(((R − r) / r) θ)

Varying the ratios produces an endless family of hypnotic patterns.