Hypotrochoid - Alchetron, The Free Social Encyclopedia

A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.

The parametric equations for a hypotrochoid are:

x ( θ ) = ( R − r ) cos ⁡ θ + d cos ⁡ ( R − r r θ ) y ( θ ) = ( R − r ) sin ⁡ θ − d sin ⁡ ( R − r r θ ) .

Where θ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ is not the polar angle).

Special cases include the hypocycloid with d = r and the ellipse with R = 2r.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

References

Hypotrochoid Wikipedia

(Text) CC BY-SA