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Epitrochoid


EpitrochoidDiagram1Epitrochoid1
EpitrochoidDiagram2Epitrochoid2

The roulette traced by a point P attached to a circle of radius b rolling around the outside of a fixed circle of radius a. These curves were studied by Dürer (1525), Desargues (1640), Huygens (1679), Leibniz, Newton in 1686, L'Hospital in 1690, Jakob Bernoulli in 1690, la Hire in 1694, Johann Bernoulli in 1695, Daniel Bernoulli in 1725, and Euler in 1745 and 1781. An epitrochoid appears in Dürer's work Instruction in Measurement with Compasses and Straight Edge in 1525. He called epitrochoids spider lines because the lines he used to construct the curves looked like a spider.

The parametric equations for an epitrochoid are

x=(a+b)cost-hcos((a+b)/bt)
(1)
y=(a+b)sint-hsin((a+b)/bt),
(2)

where h is the distance from P to the center of the rolling circle. Special cases include the limaçon with a=b, the circle with a=0, and the epicycloid with h=b.


See also

Epicycloid, Hypotrochoid, Rose Curve, Spirograph, Trochoid

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References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 168-170, 1972.

Cite this as:

Weisstein, Eric W. "Epitrochoid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Epitrochoid.html

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