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Epicycloid

 
Epicycloid

 
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Epicycloid
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an epicycloid is a plane curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 produced by tracing the path of a chosen point of a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 — called epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette
Roulette (curve)

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
.

If the smaller circle has radius r, and the larger circle has radius R = kr, then the






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an epicycloid is a plane curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
 produced by tracing the path of a chosen point of a circle
Circle

A circle is a simple shape of Euclidean geometry consisting of those point in a plane which are the same distance from a given point called the center....
 — called epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette
Roulette (curve)

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
.

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

If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable).

If k is a rational number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
, say k=p/q expressed in simplest terms, then the curve has p cusps.

If k is an irrational number
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
, then the curve never closes, and fills the space between the larger circle and a circle of radius R + 2r.

Image:Epicycloid-1.svg| k = 1 Image:Epicycloid-2.svg| k = 2 Image:Epicycloid-3.svg| k = 3 Image:Epicycloid-4.svg| k = 4 Image:Epicycloid-2-1.svg| k = 2.1 = 21/10 Image:Epicycloid-3-8.svg| k = 3.8 = 19/5 Image:Epicycloid-5-5.svg| k = 5.5 = 11/2 Image:Epicycloid-7-2.svg| k = 7.2 = 36/5

The epicycloid is a special kind of epitrochoid
Epitrochoid

An epitrochoid is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle....
.

An epicycle with one cusp is a cardioid
Cardioid

A cardioid is closed curve with one Cusp ....
.

An epicycloid and its evolute
Evolute

In the differential geometry of curves, the evolute of a curve is the locus of all its Osculating circle. Equivalently, it is the envelope of the perpendicular to a curve....
 are similar.

Proof

We assume that the position of is what we want to solve, is the radian from the tangential point to the moving point , and is the radian 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 radian (which is the rate arc over radius), then we have that From the two condition, we get the identity By calculating, we get the relation between and , which is

From the figure, we see the position of the point clearly.

See also

  • Special cases: Cardioid
    Cardioid

    A cardioid is closed curve with one Cusp ....
    , Nephroid
    Nephroid

    The nephroid is a plane curve whose name means kidney-shaped Although the term nephroid was used to describe other curves, it was applied to the curve in this article by Proctor in 1878....
  • Cycloid
    Cycloid

    A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line.It is an example of a roulette , a curve generated by a curve rolling on another curve....
  • Hypocycloid
    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....
  • Epitrochoid
    Epitrochoid

    An epitrochoid is a roulette traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle....
  • Hypotrochoid
    Hypotrochoid

    A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle....
  • Spirograph
    Spirograph

    Spirograph is a geometric drawing toy that produces mathematical curves of the variety technically known as hypotrochoids and epitrochoids. The term has also been used to describe a variety of software applications that display similar curves, and applied to the class of curves that can be produced with the drawing equipment ....
  • Deferent and epicycle
    Deferent and epicycle

    In the Ptolemaic system of astronomy, the epicycle was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets....
  • Epicyclic gearing
    Epicyclic gearing

    Epicyclic gearing or planetary gearing is a gear system that consists of one or more outer gears, or planet gears, revolving about a central, or sun gear....


External links

  • , MathWorld
    MathWorld

    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....
  • "" by Michael Ford, The Wolfram Demonstrations Project, 2007