experiment 036 · geometry · kinematics

Spirograph

Simulate hypotrochoids and epitrochoids by rolling one circle inside or outside another fixed circle. A pen attached to the rolling circle traces out beautiful, intricate, continuous mathematical curves.

The Mathematics of Spirographs

A Spirograph produces curves known mathematically as hypotrochoids and epitrochoids.

Hypotrochoids

A hypotrochoid is produced when a smaller circle of radius $r$ rolls around the inside of a fixed larger circle of radius $R$. The pen is placed at a distance $d$ from the center of the interior rolling circle. The parametric equations are:

$$ x(\theta) = (R - r) \cos \theta + d \cos\left(\frac{R - r}{r} \theta\right) $$ $$ y(\theta) = (R - r) \sin \theta - d \sin\left(\frac{R - r}{r} \theta\right) $$

Epitrochoids

An epitrochoid is produced when the rolling circle rolls along the outside of the fixed circle. Its parametric equations are:

$$ x(\theta) = (R + r) \cos \theta - d \cos\left(\frac{R + r}{r} \theta\right) $$ $$ y(\theta) = (R + r) \sin \theta - d \sin\left(\frac{R + r}{r} \theta\right) $$

Special Cases

Hypocycloid: When the pen is exactly on the edge of the rolling circle ($d = r$), the hypotrochoid becomes a hypocycloid.
Epicycloid: Similarly, an epitrochoid with $d = r$ is an epicycloid.
Ellipse: A hypotrochoid with $R = 2r$ traces an ellipse (or a straight line if $d = r$).

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