# Dictionary Definition

cycloid adj : resembling a circle [syn: cycloidal] n : a line
generated by a point on a circle rolling along a straight
line

# User Contributed Dictionary

## English

### Noun

cycloid (plural cycloids)#### Translations

zoology: fish with cycloid scales

### Adjective

cycloid#### Translations

resembling a circle

- French: cycloïdal (pl: cycloïdaux), cycloïdale

fish scales: thin and rounded

- French: cycloïde

### Related terms

### See also

# Extensive Definition

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.

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 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 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, created by a circle of radius r, consists of the points (x, y) with

- x = r(t - \sin t)\,

- y = r(1 - \cos t)\,

where t is a real parameter; rt is the
x-coordinate of the center of the rolling circle.

Using degrees, a more modifiable form of the
equation can be found: x = a(cos(270-t)) + 2πrt/360

y = a(sin(270-t)) + r

This curve is differentiable everywhere
except at the cusps
where it hits the x-axis, with the derivative tending toward \infty
or -\infty as one approaches a cusp. It satisfies the
differential equation

- \left(\frac\right)^2 = \frac.

## Area

One arch of a cycloid genereated by a circle of radius r can be parametrized by- x = r(t - \sin t),\,

- y = r(1 - \cos t),\,

with

- 0 \le t \le 2 \pi.\,

Since

- \frac = r(1- \cos t),

we find the area under the arch to be

- A=\int_^ y \, dx = \int_^ r^2(1-\cos t)^2 \, dt

## Arc length

The arc length S of one arch is given by- \int_^ \left(\left(\frac\right)^2+\left(\frac\right)^2\right)^\, dt=\int_^ 2r \sin(t/2) \, dt = 8r.

## 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.An up-side-down cycloid is called a tautochrone
which is a path of a cycloidal pendulum.

## 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 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 (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).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.

## See also

## 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

- Retrieved April 27, 2007.

## External links

- 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, The Wolfram Demonstrations Project.

cycloid in Bulgarian: Циклоида

cycloid in Catalan: Cicloide

cycloid in Czech: Cykloida

cycloid in German: Zykloide

cycloid in Spanish: Cicloide

cycloid in French: Cycloïde

cycloid in Italian: Cicloide

cycloid in Hebrew: ציקלואידה

cycloid in Hungarian: Ciklois

cycloid in Dutch: Cycloïde

cycloid in Japanese: サイクロイド

cycloid in Polish: Cykloida

cycloid in Portuguese: Ciclóide

cycloid in Romanian: Cicloidă

cycloid in Russian: Циклоида

cycloid in Slovenian: Cikloida

cycloid in Swedish: Cykloid

cycloid in Chinese:
摆线