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Hypocycloid

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Curve traced by a point on a circle rolling within another circle

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The red path is a hypocycloid traced as the smaller black circle rolls around inside the larger black circle (parameters are R=4.0, r=1.0, and so k=4, giving anastroid).

Ingeometry, ahypocycloid is a specialplane curve generated by the trace of a fixed point on a smallcircle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like thecycloid created by rolling a circle on a line.

History

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The 2-cusped hypocycloid calledTusi couple was first described by the 13th-centuryPersian astronomer andmathematician Nasir al-Din al-Tusi inTahrir al-Majisti (Commentary on the Almagest).^[1]^[2] German painter and German Renaissance theoristAlbrecht Dürer describedepitrochoids in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively.^[3]

Properties

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If the rolling circle has radiusr, and the fixed circle has radiusR =kr, then theparametric equations for the curve can be given by either: ${\begin{aligned}&x(\theta )=(R-r)\cos \theta +r\cos \left({\frac {R-r}{r}}\theta \right)\\&y(\theta )=(R-r)\sin \theta -r\sin \left({\frac {R-r}{r}}\theta \right)\end{aligned}}$ or: ${\begin{aligned}&x(\theta )=r(k-1)\cos \theta +r\cos \left((k-1)\theta \right)\\&y(\theta )=r(k-1)\sin \theta -r\sin \left((k-1)\theta \right)\end{aligned}}$

Ifk is an integer, then the curve is closed, and haskcusps (i.e., sharp corners, where the curve is notdifferentiable). Specially fork = 2 the curve is a straight line and the circles are calledTusi couple. Nasir al-Din al-Tusi was the first to describe these hypocycloids and their applications to high-speedprinting.^[4]^[5]

If $k {\displaystyle k}$ is arational number, say $k=p/q$ expressed asirreducible fraction, then the curve has $p {\displaystyle p}$ cusps.

To close the curve and

complete the 1st repeating pattern :

θ = 0 toq rotations

α = 0 top rotations

total rotations of rolling circle =p -q rotations

Ifk is anirrational number, then the curve never closes, and fills the space between the larger circle and a circle of radiusR − 2r.

Each hypocycloid (for any value ofr) is abrachistochrone for the gravitational potential inside a homogeneous sphere of radiusR.^[6]

The area enclosed by a hypocycloid is given by:^[3]^[7]

$A={\frac {(k-1)(k-2)}{k^{2}}}\pi R^{2}=(k-1)(k-2)\pi r^{2}$

Thearc length of a hypocycloid is given by:^[7]

$s={\frac {8(k-1)}{k}}R=8(k-1)r$

Examples

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Hypocycloid Examples
k=3 → adeltoid
k=4 → anastroid
k=5 → a pentoid
k=6 → an exoid
k=2.1 = 21/10
k=3.8 = 19/5
k=5.5 = 11/2
k=7.2 = 36/5

The hypocycloid is a special kind ofhypotrochoid, which is a particular kind ofroulette.

A hypocycloid with three cusps is known as adeltoid.

A hypocycloid curve with four cusps is known as anastroid.

The hypocycloid with two "cusps" is a degenerate but still very interesting case, known as theTusi couple.

Relationship to group theory

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Hypocycloids "rolling" inside one another. The cusps of each of the smaller curves maintain continuous contact with the next-larger hypocycloid.

Any hypocycloid with an integral value ofk, and thusk cusps, can move snugly inside another hypocycloid withk+1 cusps, such that the points of the smaller hypocycloid will always be in contact with the larger. This motion looks like 'rolling', though it is not technically rolling in the sense of classical mechanics, since it involves slipping.

Hypocycloid shapes can be related tospecial unitary groups, denoted SU(k), which consist ofk ×k unitary matrices with determinant 1. For example, the allowed values of the sum of diagonal entries for a matrix in SU(3), are precisely the points in the complex plane lying inside a hypocycloid of three cusps (a deltoid). Likewise, summing the diagonal entries of SU(4) matrices gives points inside an astroid, and so on.

Thanks to this result, one can use the fact that SU(k) fits inside SU(k+1) as asubgroup to prove that anepicycloid withk cusps moves snugly inside one withk+1 cusps.^[8]^[9]

Derived curves

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Theevolute of a hypocycloid is an enlarged version of the hypocycloid itself, whiletheinvolute of a hypocycloid is a reduced copy of itself.^[10]

Thepedal of a hypocycloid with pole at the center of the hypocycloid is arose curve.

Theisoptic of a hypocycloid is a hypocycloid.

Hypocycloids in popular culture

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Curves similar to hypocycloids can be drawn with theSpirograph toy. Specifically, the Spirograph can drawhypotrochoids andepitrochoids.

ThePittsburgh Steelers' logo, which is based on theSteelmark, includes threeastroids (hypocycloids of fourcusps). In his weekly NFL.com column "Tuesday Morning Quarterback,"Gregg Easterbrook often refers to the Steelers as the Hypocycloids. Chilean soccer teamCD Huachipato based their crest on the Steelers' logo, and as such features hypocycloids.

The first Drew Carey season ofThe Price Is Right's set features astroids on the three main doors, giant price tag, and the turntable area. The astroids on the doors and turntable were removed when the show switched tohigh definition broadcasts starting in 2008, and only the giant price tag prop still features them today.^[11]

References

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^Weisstein, Eric W."Tusi Couple".mathworld.wolfram.com. Retrieved2023-02-27.
^Blake, Stephen P. (2016-04-08).Astronomy and Astrology in the Islamic World. Edinburgh University Press.ISBN 978-0-7486-4911-2.
^^a ^b"Area Enclosed by a General Hypocycloid"(PDF).Geometry Expressions. RetrievedJan 12, 2019.
^White, G. (1988), "Epicyclic gears applied to early steam engines",Mechanism and Machine Theory,23 (1):25–37,doi:10.1016/0094-114X(88)90006-7,Early experience demonstrated that the hypocycloidal mechanism was structurally unsuited to transmitting the large forces developed by the piston of a steam engine. But the mechanism had shown its ability to convert linear motion to rotary motion and so found alternative low-load applications such as the drive for printing machines and sewing machines.
^Šír, Zbyněk; Bastl, Bohumír; Lávička, Miroslav (2010), "Hermite interpolation by hypocycloids and epicycloids with rational offsets",Computer Aided Geometric Design,27 (5):405–417,doi:10.1016/j.cagd.2010.02.001,G. Cardano was the first to describe applications of hypocycloids in the technology of high-speed printing press (1570).
^Rana, Narayan Chandra; Joag, Pramod Sharadchandra (2001),"7.5 Barchistochrones and tautochrones inside a gravitating homogeneous sphere",Classical Mechanics, Tata McGraw-Hill, pp. 230–2,ISBN 0-07-460315-9
^^a ^b"Hypocycloid".Wolfram Mathworld. RetrievedJan 16, 2019.
^Baez, John."Deltoid Rolling Inside Astroid".AMS Blogs. American Mathematical Society. Retrieved22 December 2013.
^Baez, John (3 December 2013)."Rolling hypocycloids".Azimuth blog. Retrieved22 December 2013.
^Weisstein, Eric W."Hypocycloid Evolute".MathWorld. Wolfram Research.
^Keller, Joel (21 August 2007)."A glimpse at Drew Carey's Price is Right".TV Squad. Archived fromthe original on 27 May 2010.
^Trombold, John; Donahue, Peter, eds. (2006),Reading Portland: The City in Prose, Oregon Historical Society Press, p. xvi,ISBN 9780295986777,At the center of the flag lies a star — technically, a hypocycloid — which represents the city at the confluence of the two rivers.