In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radiusr rolling around the inside of a fixed circle of radius R, where the point is a distanced from the center of the interior circle.
The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are R = 5, r = 3, d = 5).
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.[2] The eccentricity of the ellipse is
^J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 165–168. ISBN0-486-60288-5.
^Gray, Alfred. Modern Differential Geometry of Curves and Surfaces with Mathematica (Second ed.). CRC Press. p. 906. ISBN9780849371646.
^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.
hypotrochoid, geometry, hypotrochoid, roulette, traced, point, attached, circle, radius, rolling, around, inside, fixed, circle, radius, where, point, distance, from, center, interior, circle, curve, hypotrochoid, drawn, smaller, black, circle, rolls, around, . 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 red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle parameters are R 5 r 3 d 5 The parametric equations for a hypotrochoid are 1 x 8 R r cos 8 d cos R r r 8 y 8 R r sin 8 d sin R r r 8 displaystyle begin aligned amp x theta R r cos theta d cos left R r over r theta right amp y theta R r sin theta d sin left R r over r theta right end aligned where 8 is the angle formed by the horizontal and the center of the rolling circle these are not polar equations because 8 is not the polar angle When measured in radian 8 takes values from 0 to 2 p LCM r R R displaystyle 2 pi times tfrac operatorname LCM r R R where LCM is least common multiple Special cases include the hypocycloid with d r and the ellipse with R 2r and d r 2 The eccentricity of the ellipse is e 2 d r 1 d r displaystyle e frac 2 sqrt d r 1 d r becoming 1 when d r displaystyle d r see Tusi couple The ellipse drawn in red may be expressed as a special case of the hypotrochoid with R 2r Tusi couple here R 10 r 5 d 1 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 See also EditCycloid Cyclogon Epicycloid Rosetta orbit Apsidal precession SpirographReferences Edit 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 External links EditWeisstein 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 Retrieved from https en wikipedia org w index php title Hypotrochoid amp oldid 1127758477, wikipedia, wiki, book, books, library,
