Hypotrochoid
Rolling curves (roulettes)The spirograph curve: a pen fixed at distance d from the center of a circle rolling inside another.
Adjust parameters
Equation
Open the Creator with a ready-made animation brief for this curve — review the scene plan and Manim code before rendering.
What is the Hypotrochoid?
The hypotrochoid generalizes the hypocycloid: the drawing point sits at any distance d from the rolling circle’s center, not just on the rim. With d < r the cusps soften into waves; with d > r they blossom into loops. This is exactly the geometry of the Spirograph toy — the pen hole you choose is the parameter d.
Closure is pure number theory: the pattern completes after r/gcd(R, r) trips around the ring, so the lobe count and the density of the lacework are read directly off the integers R and r. The classic Spirograph look, R = 5, r = 3, d = 5, closes after three laps with a rich three-fold symmetry.
Key properties
- d = r recovers the hypocycloid; d = 0 gives a circle.
- The pattern closes after r/gcd(R, r) revolutions around the ring.
- R = 2r produces an ellipse for any d — a rolling-circle ellipsograph.
- Cusps (d = r), waves (d < r), or loops (d > r) — one parameter decides.
- All hypotrochoids stay within radius R − r + d of the center.
Where you'll see it
The Spirograph toy, guilloché engraving on banknotes and watch faces, and harmonic drawing machines.