geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...

that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the

cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cu ...

created by rolling a circle on a line.

Properties

If the smaller circle has radius , and the larger circle has radius , then the parametric equations for the curve can be given by either: :

\begin
& x (\theta) = (R - r) \cos \theta + r \cos \left(\frac \theta \right) \\
& y (\theta) = (R - r) \sin \theta - r \sin \left( \frac \theta \right)
\end

or: :

\begin
& 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

If is an integer, then the curve is closed, and has cusps (i.e., sharp corners, where the curve is not differentiable). Specially for the curve is a straight line and the circles are called Cardano circles.

Girolamo Cardano Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...

was the first to describe these hypocycloids and their applications to high-speed

printing Printing is a process for mass reproducing text and images using a master form or template. The earliest non-paper products involving printing include cylinder seals and objects such as the Cyrus Cylinder and the Cylinders of Nabonidus. The ...

. If is a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

, say expressed in simplest terms, then the curve has cusps. If is an

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

, then the curve never closes, and fills the space between the larger circle and a circle of radius . Each hypocycloid (for any value of ) is a brachistochrone for the gravitational potential inside a homogeneous sphere of radius . The area enclosed by a hypocycloid is given by: :

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

The

arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

of a hypocycloid is given by: :

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

Examples

Image:Hypocycloid-3.svg, k=3 — a

deltoid Deltoid (delta-shaped) can refer to: * The deltoid muscle, a muscle in the shoulder * Kite (geometry), also known as a deltoid, a type of quadrilateral * A deltoid curve, a three-cusped hypocycloid * A leaf shape * The deltoid tuberosity, a part o ...

Image:Hypocycloid-4.svg, k=4 — an astroid Image:Hypocycloid-5.svg, k=5 Image:Hypocycloid-6.svg, k=6 Image:Hypocycloid-2-1.svg, k=2.1 = 21/10 Image:Hypocycloid-3-8.svg, k=3.8 = 19/5 Image:Hypocycloid-5-5.svg, k=5.5 = 11/2 Image:Hypocycloid-7-2.svg, k=7.2 = 36/5 The hypocycloid is a special kind of hypotrochoid, which is a particular kind of

roulette Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...

. A hypocycloid with three cusps is known as a

. A hypocycloid curve with four cusps is known as an astroid. The hypocycloid with two "cusps" is a degenerate but still very interesting case, known as the

Tusi couple The Tusi couple is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and fo ...

Relationship to group theory

Any hypocycloid with an integral value of ''k'', and thus ''k'' cusps, can move snugly inside another hypocycloid with ''k''+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 to special unitary groups, denoted SU(''k''), which consist of ''k'' × ''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 a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

to prove that an epicycloid with ''k'' cusps moves snugly inside one with ''k''+1 cusps.

Derived curves

The

evolute In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that cur ...

of a hypocycloid is an enlarged version of the hypocycloid itself, while the

involute In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from o ...

of a hypocycloid is a reduced copy of itself. The

pedal A pedal (from the Latin '' pes'' ''pedis'', "foot") is a lever designed to be operated by foot and may refer to: Computers and other equipment * Footmouse, a foot-operated computer mouse * In medical transcription, a pedal is used to control p ...

of a hypocycloid with pole at the center of the hypocycloid is a

rose curve A rose is either a woody perennial flowering plant of the genus ''Rosa'' (), in the family Rosaceae (), or the flower it bears. There are over three hundred Rose species, species and Garden roses, tens of thousands of cultivars. They form a ...

. The

isoptic In the geometry of curves, an orthoptic is the set of points for which two tangents of a given curve meet at a right angle. Examples: # The orthoptic of a parabola is its directrix (proof: see below), # The orthoptic of an ellipse \tfrac + \ ...

of a hypocycloid is a hypocycloid.

Hypocycloids in popular culture

Curves similar to hypocycloids can be drawn with the

Spirograph 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 ...

toy. Specifically, the Spirograph can draw hypotrochoids and epitrochoids. The

Pittsburgh Steelers The Pittsburgh Steelers are a professional American football team based in Pittsburgh. The Steelers compete in the National Football League (NFL) as a member club of the American Football Conference (AFC) North division. Founded in , the Steel ...

' logo, which is based on the Steelmark, includes three astroids (hypocycloids of four

cusp A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth. Cusp or CUSP may also refer to: Mathematics * Cusp (singularity), a singular point of a curve * Cusp catastrophe, a branch of bifurc ...

s). In his weekly NFL.com column "Tuesday Morning Quarterback," Gregg Easterbrook often refers to the Steelers as the Hypocycloids. Chilean soccer team CD Huachipato based their crest on the Steelers' logo, and as such features hypocycloids. The first Drew Carey season of '' The Price Is Rights 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 to high definition broadcasts starting in 2008, and only the giant price tag prop still features them today.

References

External links

* * *
A free Javascript tool for generating Hypocyloid curves

Plot Hypcycloid — GeoFun
* {{cite web , first=John , last=Snyder , title=Sphere with Tunnel Brachistochrone , work=Wolfram Demonstrations Project , url=http://demonstrations.wolfram.com/SphereWithTunnelBrachistochrone/ Iterative demonstration showing the brachistochrone property of Hypocycloid Roulettes (curve)