Spirograph

A red five-lobed curve traced inside a grey ring by a pen on a smaller rolling wheel

A wheel rolling inside a ring, with a pen that traces a hypotrochoid; the wheel animates, and the sizes of the ring and wheel and the pen's distance from the wheel's centre can be edited.

A Spirograph is a toothed wheel that rolls inside a toothed ring, with a pen in one of its holes. The pen traces a curve called a hypotrochoid. This document rolls the wheel: the ring has radius R, the wheel radius w, and the pen sits at distance d from the wheel's centre.

When the wheel's centre has moved through the angle s around the ring, the wheel itself has turned the other way by \frac{R - w}{w}s, because it rolls without slipping: the arc it has covered on the ring equals the arc that has touched it. The pen is therefore at a(s) = (R - w)\cos s + d\cos\left(\frac{R - w}{w}s\right) and b(s) = (R - w)\sin s - d\sin\left(\frac{R - w}{w}s\right). With R = 5 and w = 3 the wheel's centre goes round three times before the pen returns to its start, and the curve has five lobes: in general the curve closes after w/\gcd(R, w) turns and has R/\gcd(R, w) lobes.

T animates the wheel over three turns of its centre, which is the whole path for these sizes; the faint curve is that path, the red curve is the part traced so far, and the spoke joins the wheel's centre to the pen. Edit R, w and d to change the pattern (w must stay between 0 and R; other sizes may need more or fewer turns to close): with d = w the pen is on the rim and traces a hypocycloid with cusps; with R = 2w the curve is an ellipse, and a diameter of the ring when d = w as well.

After a JSXGraph example by The Center of Mobile Learning with Digital Technology.

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