Spirograph Generator

How the Spirograph Generator works

Spirograph generator: what it does and why

A spirograph curve is traced by a pen fixed at some distance from the centre of a gear that rolls around the inside or outside of a ring. Rolling inside gives a hypotrochoid, the familiar looping flower; rolling outside gives an epitrochoid, which bulges outward and looks more like a crown. The three numbers that define the curve are the ring size R, the gear size r and the pen offset d, all in the same units. The toy expressed them as tooth counts, so the sliders here do too.

How many loops. The pen returns to its starting point after the gear has gone around the ring r ÷ gcd(R, r) times, where gcd is the greatest common divisor. The stats line shows this number. R = 96 and r = 56 share a factor of 8, so the curve closes after 7 loops and has 12 lobes. Change r to 55, which shares no factor with 96, and it needs 55 loops to close and fills the ring almost solid. Pick tooth counts with a large common factor for a simple shape and coprime counts for a dense one.

Pen offset. At d = 0 the pen is at the gear's centre and draws a plain circle. As d grows the lobes get pointier, then at d = r the curve has cusps, and past that it develops loops at each lobe. The classic toy only allowed d less than r because the holes were inside the gear; this page allows more.

Colour. Rainbow shifts the hue along the length of the curve, so you can see where the pen was early and late. Single picks one hue from the seed. Regenerate changes only the colour; the shape is entirely determined by the sliders, which is what you want when tuning it.

Export. The curve is scaled to fit the shorter side of the export, so a 4K desktop export and a phone export both contain the whole figure, centred.

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