Spirograph curve

Parametric equations

This spirograph curve is a hypotrochoid, the path of a small circle rolling inside a fixed circle:

• R is the radius of the fixed circle, the large circle.
• r is the radius of the rolling circle, the small circle.
• d is the distance from the pen to the center of the rolling circle.

Interaction

• Large circle R: change the fixed radius. Switching R or r changes the corner structure, and the curve grows in again
• Small circle r: change the rolling radius. The corner structure follows R and r
• Pen d: change how far the pen sits from the center of the small circle. The sharpness and the crossings change continuously

What to notice

• How long the orbit takes to close depends on $\gcd (R,r)$. A coprime ratio makes a richer pattern.
• When $d$ is near $r$, sharp points appear. A smaller $d$ keeps the curve more inward.
• A hypotrochoid is the curve drawn by a circle rolling inside a fixed circle.

Related

• Rose curve
• Lissajous curve
• Fourier series

Further reading

• Hypotrochoid (Wikipedia)