Spirograph & Lissajous Curve Generator - Geometric Art Visualizer

How to use this tool

1. Choose Spirograph or Lissajous mode.
2. Adjust the sliders to change the parameters.
3. Watch the pattern update in real time.

What is a spirograph?

A spirograph traces the path of a point on a small circle rolling inside (or outside) a larger fixed circle. Curves traced by a point inside a rolling circle are called hypotrochoids; curves traced by a point outside a rolling circle are epitrochoids. The physical toy Spirograph® (introduced in 1965) uses toothed rings and wheels to draw these curves on paper.

The formula

For a hypotrochoid, the parametric equations are:

x(t) = (R − r) cos(t) + d · cos((R − r)t / r)

y(t) = (R − r) sin(t) − d · sin((R − r)t / r)

where R is the outer circle radius, r is the rolling circle radius, and d is the distance from the center of the rolling circle to the pen point.

Parameter guide

• Outer radius (R): the size of the fixed ring. Larger values scale the overall pattern.
• Inner radius (r): the size of the rolling wheel. The ratio R/r determines how many lobes or petals the curve has.
• Pen offset (d): the distance from the wheel’s center to the pen. When d equals r, the pen is on the edge of the wheel and traces a specific classic curve. Moving d inside the wheel produces rounded petals; outside produces overlapping loops.
• Speed ratio: when R/r is rational (a fraction), the curve eventually closes. When it’s irrational, the curve never closes and the pen will fill the ring given enough time.

Famous examples

• Rose curves (rhodonea): R/r = whole number; produces petal shapes.
• Epicycloids: d = r (pen on wheel edge) produces classic looped outer curves like cardioids and nephoids.
• Hypocycloids: d = r rolling inside; 3-petal = deltoid, 4-petal = astroid.
• Lissajous-like patterns: certain ratios produce interlocking grid patterns resembling Lissajous figures.

Why the curve closes

A spirograph curve closes into a finite pattern only when the ratio R/r is a rational number (expressible as a fraction p/q in lowest terms). In that case, the small circle makes exactly r / gcd(R, r) rotations before returning to its starting position. If R/r is irrational (such as π or √2), the pen traces a path that never exactly repeats and would eventually fill the ring densely given infinite time.

Roulette curve gallery

Special values of d and r relative to R produce named classic curves:

• Astroid (hypocycloid): rolling circle inside, r = R/4, d = r. Produces a 4-pointed star with concave curved sides.
• Straight line: rolling circle inside, r = R/2, d = r. The pen traces a perfect straight line (diameter of the outer circle). This is Tusi's couple.
• Cardioid (epicycloid): rolling circle outside, r = R, d = r. Produces a heart-shaped curve with one cusp.
• Deltoid (hypotrochoid): r = R/3, d = r. Produces a 3-pointed curved triangle used in rotary engine design (Wankel engine).