In geometry, an epicycloid(also called hypercycloid) is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
An epicycloid with a minor radius (R2) of 0 is a circle. This is a degenerate form.
If the smaller circle has radius r, and the larger circle has radius R = kr, then the
parametric equations for the curve can be given by either:
or:
in a more concise and complex form
where
angle θ is in turns:
smaller circle has radius r
the larger circle has radius kr
(Assuming the initial point lies on the larger circle.) When k is a positive integer, the area of this epicycloid is
It means that the epicycloid is larger than the original stationary circle.
If k is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners).
If k is a rational number, say k = p / q expressed as irreducible fraction, then the curve has p cusps.
Count the animation rotations to see p and q
If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2r.
The distance from (x = 0, y = 0) origin to (the point p on the small circle) varies up and down as
where
R = radius of large circle and
2r = diameter of small circle
File:Epicycloid-1.svg| {{math|1=''k'' = 1 }}; a ''[[cardioid]]''
File:Epicycloid-2.svg| {{math|1=''k'' = 2 }}; a ''[[nephroid]]''
File:Epicycloid-3.svg| {{math|1=''k'' = 3 }}; a ''trefoiloid''
File:Epicycloid-4.svg| {{math|1=''k'' = 4 }}; a ''quatrefoiloid''
File:Epicycloid-2-1.svg| {{math|1=''k'' = 2.1 = 21/10}}
File:Epicycloid-3-8.svg| {{math|1=''k'' = 3.8 = 19/5}}
File:Epicycloid-5-5.svg| {{math|1=''k'' = 5.5 = 11/2}}
File:Epicycloid-7-2.svg| {{math|1=''k'' = 7.2 = 36/5}}
The epicycloid is a special kind of epitrochoid.
An epicycle with one cusp is a cardioid, two cusps is a nephroid.
An epicycloid and its evolute are similar.
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In geometry, an epitrochoid (ɛpᵻˈtrɒkɔɪd or ɛpᵻˈtroʊkɔɪd) is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is at a distance d from the center of the exterior circle. The parametric equations for an epitrochoid are The parameter θ is geometrically the polar angle of the center of the exterior circle. (However, θ is not the polar angle of the point on the epitrochoid.) Special cases include the limaçon with R = r and the epicycloid with d = r.
Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well-known toy version was developed by British engineer Denys Fisher and first sold in 1965. The name has been a registered trademark of Hasbro Inc. since 1998 following purchase of the company that had acquired the Denys Fisher company. The Spirograph brand was relaunched worldwide in 2013, with its original product configurations, by Kahootz Toys.
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid created by rolling a circle on a line. The 2-cusped hypocycloid called Tusi couple was first described by the 13th-century Persian astronomer and mathematician Nasir al-Din al-Tusi in Tahrir al-Majisti (Commentary on the Almagest).