Cycloid

A cycloid is the curve traced by a point on the rim of a 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.

The 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 an object in descent without friction inside this curve does not depend on the object's starting position).

History

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. The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th century mathematicians.

Equations

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

:$x\; =\; r(t\; -\; \backslash sin\; t)\backslash ,$

:$y\; =\; r(1\; -\; \backslash cos\; t)\backslash ,$

where t is a real parameter, corresponding to the angle through which the rolling circle has rotated, measured in radians. For given t, the circle's centre lies at x = rt, y = r.

Solving for t and replacing, the Cartesian equation would be

:$x\; =\; r\; \backslash cos^\{-1\}\; \backslash left(1-\backslash frac\{y\}\{r\}\backslash right)-\backslash sqrt\{y(2r-y)\}$

The first arch of the cycloid consists of points such that

:$0\; \backslash le\; t\; \backslash le\; 2\; \backslash pi.\backslash ,$

The cycloid is differentiable everywhere except at the cusps where it hits the x-axis, with the derivative tending toward $\backslash infty$ or $-\backslash infty$ as one approaches a cusp. The map from t to (x, y) is a differentiable curve or parametric curve of class C^∞ and the singularity where the derivative is 0 is an ordinary cusp.

The cycloid satisfies the differential equation:

:$\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)^2\; =\; \backslash frac\{2r-y\}\{y\}.$

Area

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

:$x\; =\; r(t\; -\; \backslash sin\; t),\backslash ,$

:$y\; =\; r(1\; -\; \backslash cos\; t),\backslash ,$

with

:$0\; \backslash le\; t\; \backslash le\; 2\; \backslash pi.\backslash ,$

Since

:$\backslash frac\{dx\}\{dt\}\; =\; r(1-\; \backslash cos\; t),$

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 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, 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 curve, and the cusp of an inverted cycloid forces the pendulum bob to move in a cycloidal path.

The 17th Century Dutch mathematician Christiaan Huygens discovered this property of the cycloid and applied it to the design of more accurate clocks for use in navigation.

Related curves

Several curves are related to the cycloid.

Curtate cycloid: Here the point tracing out the curve is inside the circle, which rolls on a line.

Prolate cycloid: Here the point tracing out the curve is outside the circle, which rolls on a line.

Trochoid: refers to any of the cycloid, the curtate cycloid and the prolate cycloid.

Hypocycloid: The point is on the edge of the circle, which rolls not on a line but on the inside of another circle.

Epicycloid: The point is on the edge of the circle, which rolls not on a line but on the outside of another circle.

Hypotrochoid: As hypocycloid but the point need not be on the edge of its circle.

Epitrochoid: As epicycloid but the point need not be on the edge of its 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.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

Use in architecture

The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas. It was also used in the design of the Hopkins Center in Hanover, New Hampshire.

See also

Spirograph

Epicycloid

Hypocycloid

Epitrochoid

Hypotrochoid

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

Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 196–200, Simon & Schuster.

External links

Retrieved April 27, 2007.

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.

Cicloides y trocoides

Cycloid Curves by Sean Madsen with contributions by David von Seggern, Wolfram Demonstrations Project.

Category:Curves