Epitrochoid

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 (1686), L'Hospital (1690), Jakob Bernoulli (1690), la Hire (1694), Johann Bernoulli (1695), Daniel Bernoulli (1725), Euler (1745, 1781). An epitrochoid appears in Dürer's work Instruction in Measurement with Compasses and Straight Edge (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

$$x = (a+b)\cos t-h\cos\left({{a+b\over b} t}\right)$$

$$y = (a+b)\sin t-h\sin\left({{a+b\over b} t}\right),$$

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, Spirograph, Trochoid

References

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

Lee, X. ``Epitrochoid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Epitrochoid_dir/epitrochoid.html.

Lee, X. ``Epitrochoid and Hypotrochoid Movie Gallery.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/EpiHypoTMovieGallery_dir/epiHypoTMovieGallery.html.