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 ...

, a hypocycloid is a special

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...

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 con ...

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 Parametric may refer to: Mathematics *Parametric equation, a representation of a curve through equations, as functions of a variable *Parametric statistics, a branch of statistics that assumes data has come from a type of probability distribut ...

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 In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

). 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 rat ...

, 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 In physics and mathematics, a brachistochrone curve (), or curve of fastest descent, is the one lying on the plane between a point ''A'' and a lower point ''B'', where ''B'' is not directly below ''A'', on which a bead slides frictionlessly under ...

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 Image:Hypocycloid-4.svg, k=4 — an

astroid In mathematics, an astroid is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius. By double generation, it ...

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 In geometry, a hypotrochoid is a roulette traced by a point attached to a circle of radius rolling around the inside of a fixed circle of radius , where the point is a distance from the center of the interior circle. The parametric equations f ...

, which is a particular kind of roulette. A hypocycloid with three cusps is known as a deltoid. A hypocycloid curve with four cusps is known as an

. 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 group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...

s, 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 In geometry, an epicycloid 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. Equati ...

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 or ...

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. The isoptic 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

s and

epitrochoid In geometry, an epitrochoid ( or ) is a roulette traced by a point attached to a circle of radius rolling around the outside of a fixed circle of radius , where the point is at a distance from the center of the exterior circle. The parametric ...

s. The Pittsburgh Steelers' logo, which is based on the

Steelmark The Steelmark is a logo representing steel and the steel industry owned by the American Iron and Steel Institute, and used by it to promote the product and its manufacturers. The logo was incorporated as the emblem of the Pittsburgh Steelers and ...

, includes three

s (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 Club Deportivo Huachipato is a Chilean football club based in Talcahuano that is a current member of the Chilean Primera División. The club was founded 7 June 1947 and plays its home games at the Estadio CAP, which has a capacity of 10,500 p ...

based their crest on the Steelers' logo, and as such features hypocycloids. The first Drew Carey season of ''

The Price Is Right ''The Price Is Right'' is a television game show franchise created by Bob Stewart, originally produced by Mark Goodson and Bill Todman; currently it is produced and owned by Fremantle. The franchise centers on television game shows, but also inc ...

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 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)