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\begin{figure}\BoxedEPSF{HypotrochoidDiagram.epsf scaled 600}\end{figure}

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

$\displaystyle x$ $\textstyle =$ $\displaystyle n\cos t+h\cos\left({{n\over b}t}\right)$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle n\sin t-h\sin\left({{n\over b}t}\right),$ (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
$\displaystyle a$ $\textstyle =$ $\displaystyle {2nh\over n+1}$ (3)
$\displaystyle b$ $\textstyle =$ $\displaystyle {(n-1)h\over n+1}.$ (4)

See also Epitrochoid, Hypocycloid, Spirograph


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

Lee, X. ``Hypotrochoid.''

Lee, X. ``Epitrochoid and Hypotrochoid Movie Gallery.''

MacTutor History of Mathematics Archive. ``Hypotrochoid.''

© 1996-9 Eric W. Weisstein