Hypotrochoid
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:
where is the angle formed by the horizontal and the center of the rolling circle. When measured in radians, takes values from 0 to .
Special cases include the hypocycloid with and the ellipse with and. The eccentricity of the ellipse is
becoming 1 when .
Image:Ellipse as hypotrochoid.gif|right|400px|thumb|The ellipse may be expressed as a special case of the hypotrochoid, with ; here.
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.