Cycloid (red) generated by a rolling circle

A cycloid is the curve defined by a fixed point on a wheel as it rolls, or, more precisely, the locus of a point on the rim of a circle rolling along a straight line. animated cycloid, from Polish Wikipedias Cykloida. ... animated cycloid, from Polish Wikipedias Cykloida. ... In mathematics, a locus (plural loci) is a collection of points which share a common property. ... In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...

The cycloid was first studied by Nicholas of Cusa and later by Mersenne. It was named by Galileo 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 showed that the length of a cycloid is four times the diameter of its generating circle. Nicholas of Cusa Nicholas of Cusa (1401 – August 11, 1464) was a German cardinal of the Catholic Church, a philosopher, jurist, mathematician, and an astronomer. ... Marin Mersenne, Marin Mersennus or le Père Mersenne (September 8, 1588 – September 1, 1648) was a French theologian, philosopher, mathematician and music theorist. ... Portrait of Galileo Galilei by Giusto Sustermans. ... Events The Jesuit educational plan known as the Ratio Studiorum is issued (January 8). ... Events Moses Amyrauts Traite de la predestination is published Curaçao captured by the Dutch Treaty of Polianovska First meeting of the Académie française The witchcraft affair at Loudun Jean Nicolet lands at Green Bay, Wisconsin Opening of Covent Garden Market in London English establish a settlement... Gilles Personne de Roberval (August 8, 1602 - October 27, 1675), French mathematician, was born at Roberval, near Beauvais, France. ... Events January 13 - Edward Sexby, who had plotted against Oliver Cromwell, dies in Tower of London February 6 - Swedish troops of Charles X Gustav of Sweden cross The Great Belt (Storebælt) in Denmark over frozen sea May 1 - Publication of Hydriotaphia, Urn Burial and The Garden of Cyrus by... Christopher Wren by Godfrey Kneller, 1711. ...

The upside down cycloid is the solution to the brachistochrone problem (i.e. it is the curve of fastest descent under gravity) and the related tautochrone problem (i.e. the period of a ball rolling back and forth inside it does not depend on the ball's starting position). The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century mathematicians. 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 passes down along the curve to the second point, under the action of constant gravity and... A tautochrone curve is the curve for which the time taken by a particle sliding down it under uniform gravity to its lowest point is independent of its starting point. ... A mathematician is a person whose primary area of study and research is mathematics. ...

Graph of cycloid generated by a circle of radius r=2

The cycloid through the origin, created by a circle of radius r, consists of the points (x,y) with Image File history File links Cycloid, from Chinese Wikipedias Mc_Cycloid. ... Image File history File links Cycloid, from Chinese Wikipedias Mc_Cycloid. ...

x = r(t - sin t)

y = r(1 - cos t)

where t is a real parameter, equal to the center of the rolling circle. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... A parameter is a measurement or value on which something else depends. ...

If seen as a function y(x), it is arbitrary often differentiable everywhere except at the cusps where it hits the x-axis; the slope at the cusps is infinite. It satisfies the differential equation Partial plot of a function f. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In common parlance, a cusp is an important moment usually regarded as a decision point upon which consequent events are determined. ... Look up Slope in Wiktionary, the free dictionary The slope or the gradient is commonly used to describe the measurement of the steepness, incline or grade of a straight line. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

Related curves

Several curves are related to the cycloid. When we relax the requirement that the fixed point be on the rim 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 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 (a straight line is a circle of infinite radius) then we get the epicycloid (circle rolling on outside of another circle, point on the rim of the rolling circle), the hypocycloid (circle on the inside, point on the rim), the epitrochoid (circle on the outside, point anywhere on circle), and the hypotrochoid (circle on the inside, point anywhere on circle).. A curve described by a fixed point as a circle rolls along a straight line. ... In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls around without slipping around a fixed circle. ... In geometry, a hypocycloid is a special plane curve, a roulette, generated by the trace of a fixed point on a small circle that rolls within a larger circle. ... An epitrochoid is a roulette traced by a point attached to a circle of radius b rolling around the outside of a fixed circle of radius a, where the point is a distance h from the center of the exterior circle. ... 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. ...

All these curves are roulettes with a circle rolled along a uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product 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. In the differential geometry of curves, a roulette is the general concept behind cycloids, epicycloids, hypocycloids, and involutes. ... Curvature is the amount by which a geometric object deviates from being flat. ... Several equivalence relations in mathematics are called similarity. ... In the differential geometry of curves, the evolute of a curve is the set of all its centers of curvature. ... In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin. ...

References

• An application from physics: Ghatak, A. & Mahadevan, L. Crack street: the cycloidal wake of a cylinder tearing through a sheet. Physical Review Letters, 91, (2003). http://link.aps.org/abstract/PRL/v91/e215507

A right circular cylinder In mathematics, a cylinder is a quadric, i. ...

External links

• http://mathworld.wolfram.com/Cycloid.html
• Cycloids at cut-the-knot
• A Treatise on The Cycloid and all forms of Cycloidal Curves, monograph by Richard A. Proctor, B.A. posted by Cornell University Library.