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 (these are not polar equations because is not the polar angle). When measured in radians, takes values from 0 to (where is least common multiple).
Special cases include the hypocycloid with and the ellipse with and . The eccentricity of the ellipse is
becoming 1 when (see Tusi couple).
The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.
Hypotrochoids describe the support of the eigenvalues of some random matrices with cyclic correlations.
See also Cycloid Cyclogon Epicycloid Rosetta (orbit) Apsidal precession Spirograph
References
External links
Flash Animation of Hypocycloid Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee Interactive hypotrochoide animation
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