How to use it
Press Play and a small toothed wheel rolls round the inside of a fixed ring while a pen in one of its holes draws a curve. A finished curve stays on the paper as a layer, so you can change the gears, pen hole, colour or rotation and draw the next one on top. If you want pictures straight away, open a preset, press Random or let Auto pattern draw a whole rosette.
Controls
- Play, Step, Finish instantly: Play rolls the wheel, Step advances it by one tooth, Finish instantly draws the rest of the curve at once.
- Undo and Clear: Undo removes the last finished curve and restores its settings. Clear empties the paper, and Undo brings it back once.
- Ring teeth N and wheel teeth n: use the sliders or the real-toy tooth counts. Rolling inside needs a wheel smaller than the ring.
- Pen hole: a fraction of the wheel radius, from 0.1 near the centre to 1.5 beyond the rim. Hole 1 is on the rim, hole 10 near the centre.
- Rotation and Inside/Outside: rotation turns the whole drawing, and Inside or Outside chooses whether the wheel rolls in the ring or around it.
- Pen and paper: colour, width, opacity, rainbow, dark or light paper.
- Mouse, touch, keyboard: drag on the canvas to roll the wheel by hand, even backwards. With the canvas focused, Space plays or pauses, the period key and arrow keys move the wheel, F finishes the curve.
- Save PNG, Save SVG, Copy link: PNG can be transparent, SVG keeps every curve as a vector path, and the link rebuilds your layers anywhere.
How it works
This is a roulette: the curve traced by a point fixed to one curve while it rolls without slipping along another. Let the fixed ring have radius R and the wheel radius r, both proportional to their tooth counts, and let the pen sit at distance d from the wheel centre. Rolling inside gives a hypotrochoid, rolling outside an epitrochoid:
Outside: x = (R + r) cos t − d cos((R + r)t / r), y = (R + r) sin t − d sin((R + r)t / r)
Here t is the angle of the line from the ring centre to the wheel centre. With N ring teeth, n wheel teeth and g = gcd(N, n), the curve closes when the same teeth meet again. That takes the least common multiple of N and n teeth, so the wheel goes round the ring n / g times (t runs from 0 to 2πn / g). In that time the pen reaches its outermost points N / g times, which is the number of petals. For N = 96 and n = 36, g = 12, so you get 8 petals and 3 laps. If N and n share no factor, you get N petals and n laps.
A pen on the wheel rim gives a hypocycloid inside and an epicycloid outside. If N = 2n and the pen is on the rim, the curve collapses to a straight line, the Tusi couple. N = 3n gives the three-cusped deltoid and N = 4n the four-cusped astroid. With N = 2n and the pen off the rim you get an ellipse, and a pen at the centre draws a circle.
Such curves are everywhere: the housing of a rotary piston engine is an epitrochoid, and the guilloche patterns on banknotes and watch dials are engraved by machines that combine circular motions.
Things to try
- Set N = 96, n = 36, hole 0.7, count the petals and compare with the readout (8).
- Put the pen on the rim (hole 1) with wheel 32 in ring 96 for a deltoid, or wheel 24 for the astroid.
- Set N = 96, n = 48 and drag the hole from 1.0 towards 0.5: the straight line opens into an ellipse.
- Pick coprime sizes such as 105 and 52: the wheel makes 52 laps before the curve closes.
- Press Auto pattern with rotation stepping for a classic rosette, then try hole stepping.
- Move the pen hole above 1.0 for loops, or try Outside mode for flowers.
Frequently asked questions
Why does my curve take so many laps to close?
When the tooth counts share a large factor the curve closes quickly. When they are nearly coprime the wheel laps the ring up to n times, because the same teeth only meet again after the least common multiple of N and n teeth.
Why is the drawing almost a circle?
A pen near the wheel centre barely moves relative to the wheel, so it follows the centre, which travels on a circle of radius R − r. Move the hole outwards to see petals.
Why must the wheel be smaller than the ring?
A wheel rolling inside has to fit inside, so n is smaller than N. Outside mode has no such limit.
Can I draw with a pen hole beyond the rim?
Yes, up to 1.5 times the wheel radius. The maths is the same with d larger than r, and the curve gains loops. A real wheel has no holes there.
Is it exactly like a real gear toy?
It is the ideal version, with perfect rolling and no play between teeth. Real drawings differ slightly, but the shapes follow the same equations.