In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius, where the point is a distance from the center of the interior circle.
The parametric equations for a hypotrochoid are:[1]
\begin{align} &x(\theta)=(R-r)\cos\theta+d\cos\left({R-r\overr}\theta\right)\\ &y(\theta)=(R-r)\sin\theta-d\sin\left({R-r\overr}\theta\right) \end{align}
where is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because is not the polar angle). When measured in radian, takes values from 0 to
2\pi x \tfrac{\operatorname{LCM}(r,R)}{R}
Special cases include the hypocycloid with and the ellipse with and .[2] The eccentricity of the ellipse is
e= | 2\sqrt{d/r |
becoming 1 when
d=r
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.[3]
. Modern Differential Geometry of Curves and Surfaces with Mathematica. Second. 29 December 1997 . Alfred Gray (mathematician). CRC Press. 9780849371646. en. 906.