Hypocycloid
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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a hypocycloid is a special
plane curve In mathematics, a plane curve is a curve in a plane that may be 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 plane c ...
generated by the trace of a fixed point on a small
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
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 Rolling, rolls along a Line (geometry), straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette (curve), roulette, a curve g ...
created by rolling a circle on a line.


History

The 2-cusped hypocycloid called
Tusi couple The Tusi couple (also known as Tusi's mechanism) 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 ...
was first described by the 13th-century Persian
astronomer An astronomer is a scientist in the field of astronomy who focuses on a specific question or field outside the scope of Earth. Astronomers observe astronomical objects, such as stars, planets, natural satellite, moons, comets and galaxy, galax ...
and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Nasir al-Din al-Tusi Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī (1201 – 1274), also known as Naṣīr al-Dīn al-Ṭūsī (; ) or simply as (al-)Tusi, was a Persians, Persian polymath, architect, Early Islamic philosophy, philosopher, Islamic medicine, phy ...
in ''Tahrir al-Majisti (Commentary on the Almagest)''. German painter and German Renaissance theorist
Albrecht Dürer Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...
described epitrochoids in 1525, and later Roemer and Bernoulli concentrated on some specific hypocycloids, like the astroid, in 1674 and 1691, respectively.


Properties

If the rolling circle has radius , and the fixed 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 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 ...
). Specially for the curve is a straight line and the circles are called
Tusi couple The Tusi couple (also known as Tusi's mechanism) 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 ...
. Nasir al-Din al-Tusi 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 k 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 (for example, The set of all ...
, say k = p/q expressed as
irreducible fraction An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). ...
, then the curve has p cusps. To close the curve and complete the 1st repeating pattern: * \theta=0 to q rotations * \alpha=0 to p rotations * total rotations of rolling circle=p-q rotations If is an
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
, 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 length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
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 Image:Hypocycloid-5.svg, k=5 → a pentoid Image:Hypocycloid-6.svg, k=6 → an exoid 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 (curve), 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 par ...
, which is a particular kind of
roulette Roulette (named after the French language, French word meaning "little wheel") is a casino game which was likely developed from the Italy, Italian game Biribi. In the game, a player may choose to place a bet on a single number, various grouping ...
. A hypocycloid with three cusps is known as a deltoid. 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 (also known as Tusi's mechanism) 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 ...
.


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 matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...
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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
to prove that an epicycloid with ''k'' cusps moves snugly inside one with ''k''+1 cusps.


Derived curves

The evolute 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 (mathematics), locus of a point on a piece of taut string as the string is eith ...
of a hypocycloid is a reduced copy of itself. The pedal 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 toy. Specifically, the Spirograph can draw
hypotrochoid In geometry, a hypotrochoid is a roulette (curve), 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 par ...
s 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 of the American Football Conference (AFC) AFC North, North division. Founded in 1933 P ...
' 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 bifu ...
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 Right ''The Price Is Right'' is an American television game show where contestants compete by guessing the prices of merchandise to win cash and prizes. A 1972 revival by Mark Goodson and Bill Todman of their The Price Is Right (1956 American game ...
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.


See also

* Roulette (curve) * Special cases:
Tusi couple The Tusi couple (also known as Tusi's mechanism) 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 ...
, Astroid, Deltoid * List of periodic functions * Cyclogon * Epicycloid *
Hypotrochoid In geometry, a hypotrochoid is a roulette (curve), 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 par ...
* Epitrochoid * Spirograph * Flag of Portland, Oregon, featuring a hypocycloid * Murray's Hypocycloidal Engine, utilising a
tusi couple The Tusi couple (also known as Tusi's mechanism) 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 ...
as a substitute for a crank


References


Further reading

*


External links

* * *
A free Javascript tool for generating Hypocyloid curves



Plot Hypcycloid — GeoFun
* Iterative demonstration showing the brachistochrone property of Hypocycloid {{Authority control Roulettes (curve)