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Cycloid

 
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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
Roulette (curve)

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

The cycloid is the solution to the brachistochrone problem
Brachistochrone curve

A Brachistochrone curve , or curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and is constrained to move along the curve to the second point, under the action of constant gravity and assuming no friction....
 (i.e. it is the curve of fastest descent under gravity) and the related tautochrone problem
Tautochrone curve

A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point....
 (i.e. the period of a ball rolling back and forth inside it does not depend on the ball's starting position).

cycloid was first studied by Nicholas of Cusa
Nicholas of Cusa

Nicholas of Kues was a Roman Catholic cardinal from Germany , a Philosophy, jurist, Mathematics, and an Astronomy. He is widely considered as one of the greatest geniuses and polymaths of the 15th century....
 and later by Mersenne
Marin Mersenne

Marin Mersenne, Marin Mersennus or le P?re Mersenne was a France theology, philosopher, mathematician and Music theory, often referred to as the "father of acoustics" ....
.






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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
Roulette (curve)

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

The cycloid is the solution to the brachistochrone problem
Brachistochrone curve

A Brachistochrone curve , or curve of fastest descent, is the curve between two points that is covered in the least time by a body that starts at the first point with zero speed and is constrained to move along the curve to the second point, under the action of constant gravity and assuming no friction....
 (i.e. it is the curve of fastest descent under gravity) and the related tautochrone problem
Tautochrone curve

A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction in uniform gravity to its lowest point is independent of its starting point....
 (i.e. the period of a ball rolling back and forth inside it does not depend on the ball's starting position).

History

The cycloid was first studied by Nicholas of Cusa
Nicholas of Cusa

Nicholas of Kues was a Roman Catholic cardinal from Germany , a Philosophy, jurist, Mathematics, and an Astronomy. He is widely considered as one of the greatest geniuses and polymaths of the 15th century....
 and later by Mersenne
Marin Mersenne

Marin Mersenne, Marin Mersennus or le P?re Mersenne was a France theology, philosopher, mathematician and Music theory, often referred to as the "father of acoustics" ....
. It was named by Galileo
Galileo Galilei

Galileo Galilei was a Grand Duchy of Tuscany physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution....
 in 1599. In 1634 G.P. de Roberval showed that the area under a cycloid is three times the area of its generating circle. In 1658 Christopher Wren
Christopher Wren

Sir Christopher Wren was a 17th century England designer, astronomer, geometer, and one of the greatest English architects in history. Wren designed 53 London churches, including St Paul's Cathedral, as well as many secular buildings of note....
 showed that the length of a cycloid is four times the diameter of its generating circle. The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century mathematician
Mathematician

A mathematician is a person whose primary area of study and/or research is the field of mathematics....
s.

Equations

The cycloid through the origin, generated by a circle of radius r, consists of the points (x, y) with


where t is a real parameter
Parameter

In mathematics, statistics, and the mathematical sciences, a parameter is a quantity that defines certain characteristics of systems or function s....
, corresponding to the angle through which the rolling circle has rotated, measured in radian
Radian

The radian is a unit of plane angle, equal to 180/pi Degree , or about 57.2958 degrees, or about 57?17'45?. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s (not degrees). For given t, the circle's centre lies at x = rt, y = r.

This curve is differentiable everywhere except at the cusp
Cusp (singularity)

In singularity theory a cusp is a Singular point of a curve. Spinode is an alternative name, but this is less commonly used today.For a curve defined as the zero set of a function of two variables , the cusps on the curve will have the following properties:...
s where it hits the x-axis, with the derivative tending toward or as one approaches a cusp. It satisfies the differential equation
Ordinary differential equation

In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable....

Area

One arch of a cycloid generated by a circle of radius r can be parametrized by

with

Since

we find the area under the arch to be

Arc length

The arc length S of one arch is given by

Cycloidal pendulum

If its length is equal to that of half the cycloid, the bob of a pendulum
Pendulum

A pendulum is a weight suspended from a pivot so it can swing freely.When a pendulum is displaced from its resting Mechanical equilibrium, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position....
 suspended from the cusp of an inverted cycloid, such that the "string" is constrained between the adjacent arcs of the cycloid, also traces a cycloid path. Such a cycloidal pendulum is isochronous
Isochronous

Isochronous : From Greek iso, equal + chronos, time. It literally means to occur at the same time or at equal time intervals. The term is used in different technical contexts....
, regardless of amplitude. This is because the path of the pendulum bob traces out a cycloidal path (presuming the bob is suspended from a supple rope or chain); a cycloid is its own involute
Involute

In the differential geometry of curves, an involute of a smooth curve is another curve, obtained by attaching an imaginary taut string to the given curve and tracing its free end as it is wound onto that given curve; or in reverse, unwound....
 curve, and the cusp of an inverted cycloid forces the pendulum bob to move in a cycloidal path.

Related curves

Several curves are related to the cycloid. When we relax the requirement that the fixed point be on the edge of the circle, we get the curtate cycloid and the prolate cycloid. In the former case, the point tracing out the curve is inside the circle, and, in the latter case, it is outside. A trochoid
Trochoid

Trochoid is the word created by Gilles de Roberval for the curve described by a fixed point as a circle rolls along a straight line. As a circle of radius a rolls without slipping along a line L, the center C moves parallel to L, and every other point P in the rotating plane rigidly attached to the circle traces the curve...
 refers to any of the cycloid, the curtate cycloid and the prolate cycloid. If we further allow the line on which the circle rolls to be an arbitrary circle then we get the epicycloid
Epicycloid

In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle ? called epicycle ? which rolls without slipping around a fixed circle....
 (circle rolling on outside of another circle, point on the rim of the rolling circle), the 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....
 (circle on the inside, point on the rim), the 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....
 (circle on the outside, point anywhere on circle), and the 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....
 (circle on the inside, point anywhere on circle).

All these curves are roulettes
Roulette (curve)

In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, and involutes....
 with a circle rolled along a uniform curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
. The cycloid, epicycloids, and hypocycloids have the property that each is similar to 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....
. If q is the product
Product (mathematics)

In the a mathematics, a product is the result of Multiplication, or an expression that identifies divisors to be multiplied. The order in real number or complex number numbers are multiplied has no bearing on the product; this is known as the Commutativity of multiplication....
 of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the curve:evolute similitude ratio is 1+2q.

The classic 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 ....
 toy traces out hypotrochoid and 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....
 curves.

Use in architecture


Kimbell Art Museum
The cycloidal arch was used by architect Louis Khan in his design for the Kimbell Art Museum
Kimbell Art Museum

The Kimbell Art Museum is situated in the Cultural District of Fort Worth, Texas, USA. It houses a small collection of European, Asian and Pre-Columbian works, as well as hosting travelling art exhibitions....
 in Fort Worth, Texas. It was also used in the design of the Hopkins Center in Hanover, New Hampshire.

See also


  • 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 ....
  • Epicycloid
    Epicycloid

    In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle ? called epicycle ? which rolls without slipping around a fixed circle....
  • 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....


External links

  • at cut-the-knot
    Cut-the-knot

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • , monograph by Richard A. Proctor, B.A. posted by .
  • by Sean Madsen with contributions by David von Seggern, Wolfram Demonstrations Project
    Wolfram Demonstrations Project

    The Wolfram Demonstrations Project is a website developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience....
    .