In
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 ...
, an epicycloid is a
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 pla ...
produced by tracing the path of a chosen point on the circumference of a
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 const ...
—called an ''
epicycle
In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun ...
''—which rolls without slipping around a fixed circle. It 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 ...
.
Equations
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 distribu ...
for the curve can be given by either:
:
or:
:
in a more concise and complex form
:
where
* angle is in turns:
* smaller circle has radius
* the larger circle has radius
Area
(Assuming the initial point lies on the larger circle.) When is a positive integer, the area of this epicycloid is
:
If is a positive integer, then the curve is closed, and has
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 (i.e., sharp corners).
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 ration ...
, say 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 cusps.
Count the animation rotations to see and
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 integ ...
, then the curve never closes, and forms a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of the space between the larger circle and a circle of radius .
The distance from origin to (the point on the small circle) varies up and down as
:
where
* = radius of large circle and
* = diameter of small circle
File:Epicycloid-1.svg, ; a ''cardioid
In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spi ...
''
File:Epicycloid-2.svg, ; a ''nephroid
In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger by a factor of one-half.
Name
Although the term ''nephroid'' was used to describe other curves, it was ...
''
File:Epicycloid-3.svg, ; a ''trefoiloid''
File:Epicycloid-4.svg, ; a ''quatrefoiloid''
File:Epicycloid-2-1.svg,
File:Epicycloid-3-8.svg,
File:Epicycloid-5-5.svg,
File:Epicycloid-7-2.svg,
The epicycloid is a special kind of
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 ...
.
An epicycle with one cusp is a
cardioid
In geometry, a cardioid () is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spi ...
, two cusps is a
nephroid
In geometry, a nephroid () is a specific plane curve. It is a type of epicycloid in which the smaller circle's radius differs from the larger by a factor of one-half.
Name
Although the term ''nephroid'' was used to describe other curves, it was ...
.
An epicycloid and its
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 curv ...
are
similar.
Epicycloid Evolute - from Wolfram MathWorld
/ref>
Proof
We assume that the position of is what we want to solve, is the angle from the tangential point to the moving point , and is the angle from the starting point to the tangential point.
Since there is no sliding between the two cycles, then we have that
:
By the definition of angle (which is the rate arc over radius), then we have that
:
and
:.
From these two conditions, we get the identity
:.
By calculating, we get the relation between and , which is
:.
From the figure, we see the position of the point on the small circle clearly.
:
:
See also
* List of periodic functions
This is a list of some well-known periodic functions. The constant function , where is independent of , is periodic with any period, but lacks a ''fundamental period''. A definition is given for some of the following functions, though each funct ...
* 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 curve ...
* Cyclogon
In geometry, a cyclogon is the curve traced by a vertex of a polygon that rolls without slipping along a straight line. There are no restrictions on the nature of the polygon. It can be a regular polygon like an equilateral triangle or a square. ...
* Deferent and epicycle
In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun ...
* Epicyclic gearing
An epicyclic gear train (also known as a planetary gearset) consists of two gears mounted so that the center of one gear revolves around the center of the other. A carrier connects the centers of the two gears and rotates the planet and sun gea ...
* 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 ...
* Hypocycloid
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 crea ...
* 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 ...
* Multibrot set
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a port ...
* Roulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes.
Definition
Informal definition
Roughly speaking, a roulette i ...
* 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 ...
References
*
External links
*
Epicycloid
by Michael Ford, The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
, 2007
*{{MacTutor, class=Curves, id=Epicycloid, title=Epicycloid
Animation of Epicycloids, Pericycloids and Hypocycloids
Spirograph -- GeoFun
Historical note on the application of the epicycloid to the form of Gear Teeth
Algebraic curves
Roulettes (curve)