Hypotrochoid
The spirograph curve: a pen fixed at distance d from the center of a circle rolling inside another.
Adjust parameters
Export
Formulas and key values
- Parametric equations
- Period · whole-number R, r
- Farthest from the center
R is the fixed ring, r the rolling circle, and d the pen’s distance from the rolling circle’s center.
Explainer video
Make a video of the Hypotrochoid
Opens a video template with the parameters on screen. Every number in the video is computed before it renders; choose landscape or vertical.
Start the video01 · Definition
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.
02 · Properties
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.
03 · Questions
Hypotrochoid: common questions
- What is the equation of the Hypotrochoid?
- Parametric equations: x = (R-r) cos t + d cos(((R-r)/r)t), y = (R-r) sin t - d sin(((R-r)/r)t). R is the fixed ring, r the rolling circle, and d the pen’s distance from the rolling circle’s center.
04 · Wonders
See it in a math wonder
05 · Keep exploring