Hypotrochoid

The Roulette traced by a point $P$ attached to a Circle of radius $b$ rolling around the inside of a fixed Circle of radius $a$. The parametric equations for a hypotrochoid are

$$x = n\cos t+h\cos\left({{n\over b}t}\right)$$ (1)

$$y = n\sin t-h\sin\left({{n\over b}t}\right),$$ (2)

where $n\equiv a-b$ and $h$ is the distance from $P$ to the center of the rolling Circle. Special cases include the Hypocycloid with $h=b$, the Ellipse with $a=2b$, and the Rose with

$$a = {2nh\over n+1}$$ (3)

$$b = {(n-1)h\over n+1}.$$ (4)

See also Epitrochoid, Hypocycloid, Spirograph

References

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 165-168, 1972.

Lee, X. ``Hypotrochoid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Hypotrochoid_dir/hypotrochoid.html

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

MacTutor History of Mathematics Archive. ``Hypotrochoid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Hypotrochoid.html.