Cycloid (Gr. a circle, and form), the curve described by a point on the circumference of a circle when the circle rolls along a straight line. A nail in the tire of a wagon wheel, as the wheel rolls along the street, describes a cycloid. It is usually described upon paper as follows: Fasten a bit of black lead to the edge of a coin, for example a cent, so that when the coin is laid flat upon the paper and moved the lead may trace a mark; lay a ruler upon the paper and place the edge of the coin where the lead is attached against the edge of the ruler; roll the coin along the edge of the ruler, and the lead will describe a cycloid. "When the coin has rolled a distance equal to one half its circumference, the lead will be at a distance from the ruler equal to the diameter of the coin; when it has rolled a distance equal to the circumference of the coin, the lead will again be next to the ruler. The lead will thus describe an arch which to the eye appears like the half of an ellipse or oval, but is in reality an entirely different curve. If the coin continue to roll along the ruler, another arch exactly like the first will be described, and so on indefinitely.
As they are all merely repetitions of the first one, the investigation of the properties of one is sufficient; and unless the contrary is stated, the word "cycloid" is generally understood as meaning a single branch. The straight line along which the circle rolls is called the base of the cycloid, and its length equals the circumference of the circle. The length of the cycloid is four times the diameter of the circle; and the area of the surface included between the cycloid and its base is three times the area of the circle. If a cycloid be inverted so that its concave side is upward, then a body rolling down the curve will reach the lowest point in the same time from whatever point of the curve it starts. The cycloid is also called the brachystochrone (Gr. shortest, and time), or the curve of swiftest descent, because a body starting from any point of the curve, when in its last mentioned position, will reach its lowest point in less time than it would had it rolled from the same starting point along any other curve or straight line. The cycloid is one of the most important curves in the theory of mechanics, and the investigation of its properties was among the earliest applications of the differential calculus.
Cycloids, an order of bony fishes, established by Agassiz. See Comparative Anatomy.