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

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line.

He demonstrated that for an object to descend down a curve under gravity in the same time interval, regardless of the starting point, it must follow a cycloid curve rather than the circular arc of a pendulum.

* 1658-Christian Huygens experimentally discovers that balls placed anywhere inside an inverted cycloid reach the lowest point of the cycloid in the same time and thereby experimentally shows that the cycloid is the tautochrone

: On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after having departed from any point on the cycloid, are equal to each other ...

Accepting the challenge, Johann proposed the cycloid, the path of a point on a moving wheel, pointing out at the same time the relation this curve bears to the path described by a ray of light passing through strata of variable density.

If, in this case, the point lies on the circle then the roulette is a cycloid.

If the orbital velocity and the tether rotation rate are synchronized, in the rotovator concept the tether tip moves in a cycloid, and at the lowest point is momentarily stationary with respect to the ground.

* Isochrone curve, the curve ( a cycloid ) for which objects starting at different points finish at the same time and point when released

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 ).

By geometrical methods which were an early use of calculus, he showed that this curve is a cycloid, not the circular arc of a pendulum's bob, so pendulums are not isochronous.

The curve is a cycloid, and the time is equal to π times the square root of the radius over the acceleration of gravity.

He proved geometrically in his Horologium oscillatorium, originally published in 1673, that the curve was a cycloid.

This particular inverted cycloid is a brachistochrone curve.

After deriving the differential equation for the curve by the method given above, he went on to show that it does yield a cycloid.

He announced that the curve required in the first problem must be a cycloid, and he gave a method of determining it.

Further research by Marshall Meyers, Kahn's project architect for the Kimbell museum, revealed that using a cycloid curve for the gallery vaults would reduce the ceiling height and provide other benefits as well.

The relatively flat cycloid curve would produce elegant galleries that were wide in proportion to their height, allowing the ceiling to be lowered to 20 feet ( 6 m ).

Richard Kelly, lighting consultant, determined that a reflecting screen made of perforated anodized aluminum with a specific curve could be used to distribute natural light evenly across the cycloid curve of the ceiling.

In 1658 Christopher Wren showed that the length of one arch of a cycloid is four times the diameter of its generating circle.

When y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps where it hits the x-axis, with the derivative tending toward or as one approaches a cusp.

If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the " string " is constrained between the adjacent arcs of the cycloid, and the pendulum's length is equal to that of half the arc length of the cycloid ( i. e. twice the diameter of the generating circle ), the bob of the pendulum also traces a cycloid path.

The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute.

A cycloid generated by a rolling circle

A cycloid generated by a circle of radius r = 2

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

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

Also his contribution to mathematics should be noted ; in 1658, he found the length of an arc of the cycloid using an exhaustion proof based on dissections to reduce the problem to summing segments of chords of a circle which are in geometric progression.

Huygens also proved that the time of descent is equal to the time a body takes to fall vertically the same distance as the diameter of the circle which generates the cycloid, multiplied by < sup > π </ sup >⁄< sub > 2 </ sub >.

In modern terms, this means that the time of descent is π √, where r is the radius of the circle which generates the cycloid and g is the gravity of Earth.

Substituting and, we see that these equations for and are those of a circle rolling along a horizontal line — a cycloid:

which is the differential equation of an inverted cycloid generated by a circle of diameter D.

* Gilles de Roberval shows that the area under a cycloid is three times the area of its generating circle.

It is comparable to the cycloid but instead of the circle rolling along a line, it rolls within a circle.

A cycloid ( a common trochoid ) generated by a rolling circle

Reptile scale types include: cycloid, granular ( which appear bumpy ), and keeled ( which have a center ridge ).

The Jewish law of kashrut, which only permits the consumption of fish with scales, forbids sturgeon, as they have ganoid scales instead of the permitted ctenoid and cycloid scales.

In fact, this concept can be traced further back, via Ernest Crocker in 1922, to Henry Edward Armstrong, who in 1890, in an article entitled The structure of cycloid hydrocarbons, wrote the ( six ) centric affinities act within a cycle ... benzene may be represented by a double ring ( sic ) ... and when an additive compound is formed, the inner cycle of affinity suffers disruption, the contiguous carbon-atoms to which nothing has been attached of necessity acquire the ethylenic condition.

Unlike the standard controller, the Max has a small button-shaped item called a cycloid, which can be moved in directions to control on-screen movement, somewhat like the newer Nintendo 3DS's Circle Pad.

Antoine de Laloubère ( 1600 – 1664 ), a Jesuit, born in Languedoc, is chiefly known for an incorrect solution of Pascal's problems on the cycloid, which he gave in 1660, but he has a better claim to distinction in having been the first mathematician to study the properties of the helix.

Instead, the brachistochrone is a half cycloid, which was only proved much later with the development of calculus.

The Geren firm, which had been asked to look for ways to keep costs low, objected that the cycloid vaults would be too expensive and urged a flat roof instead.

His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola, which requires approximation of the natural logarithm function by infinite series.

The tip of the tether moves in approximately a cycloid, in which it is momentarily stationary with respect to the ground.