## Frequent Links

# Hypotrochoid

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:^{[1]}

- <math>x (\theta) = (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right)</math>
- <math>y (\theta) = (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right).</math>

Where <math>\theta</math> is the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because <math>\theta</math> is not the polar angle).

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

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

## See also

## References

**^**J. Dennis Lawrence (1972).*A catalog of special plane curves*. Dover Publications. pp. 165–168. ISBN 0-486-60288-5.

## External links

- Flash Animation of Hypocycloid
- Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
- Interactive hypotrochoide animation
- O'Connor, John J.; Robertson, Edmund F., "Hypotrochoid",
*MacTutor History of Mathematics archive*, University of St Andrews.

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