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:

$x\; =\; (R\; -\; r)costheta\; +\; dcosleft(\{R\; -\; r\; over\; r\}thetaright)$

$y\; =\; (R\; -\; r)sintheta\; -\; dsinleft(\{R\; -\; r\; over\; r\}thetaright)$

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

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

See also

• Epitrochoid
• Cycloid
• Hypocycloid
• Epicycloid
• Spirograph

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications.

External links

• Flash Animation of Hypocycloid
• Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee.