In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle.
The parametric equations for a hypotrochoid are:
where θ is the angle formed by the horizontal and the center of the rolling circle (these are not polar equations because θ is not the polar angle). When measured in radian, θ takes values from 0 to (where LCM is least common multiple).
Special cases include the hypocycloid with d = r and the ellipse with R = 2r and d ≠ r. 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
- ^ J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 165–168. ISBN 0-486-60288-5.
- ^ Gray, Alfred. Modern Differential Geometry of Curves and Surfaces with Mathematica (Second ed.). CRC Press. p. 906. ISBN 9780849371646.
- ^ Aceituno, Pau Vilimelis; Rogers, Tim; Schomerus, Henning (2019-07-16). "Universal hypotrochoidic law for random matrices with cyclic correlations". Physical Review E. 100 (1): 010302. doi:10.1103/PhysRevE.100.010302.
- Weisstein, Eric W. "Hypotrochoid". MathWorld.
- Flash Animation of Hypocycloid
- Hypotrochoid from Visual Dictionary of Special Plane Curves, Xah Lee
- Interactive hypotrochoide animation
- O'Connor, John J.; Robertson, Edmund F., "Hypotrochoid", MacTutor History of Mathematics archive, University of St Andrews