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Trochoid

curve, cycloid, time, circle, generating, body, line and path

TROCHOID, in mathematics is the path that any point of a circle moving along a plane, and around its centre, traces in the air ; so that a nail in the rim of a cartwheel moves in a trochoid, as the cart goes along, and the wheel itself both turns around its axle and is carried along the ground. The word trochoid is de rived from the Greek rpftxos, from rpcx‘i run, and e'en shape. In mathematical parlance, the word trochoid conveys the same idea as the word cycloid.

Few curves have afforded finer scope for exercise of modern geometry than the trochoid or cycloid (q.v.). Its properties successively engaged the attention of Roberval, Femat, Descartes, Pascal, Slusius, Wren, Wallace and Huygens. Huygens rectified this curve as early as 1657; and having afterward discovered the isochronism, or its remarkable property that all bodies along it will descend from any point in the same time and likewise that it produces a similar trochoid by its development, he ap plied the discoveries to the improvement of the pendulum,. and showed how perfectly synchron ous vibrations could be procured, theoretically at least, by causing a flexible rod to vibrate be tween the trochoidal cheeks. Upon this geo metrical curve, then, depends the better part of the whole doctrine of pendulums. Huygens demonstrated that from whatever point or height a heavy body oscillating on a fixed centre begins to descend, while it continues to move in a trochoid (or cycloid) the times of its falls or oscillations will be equal to each other. It is likewise the curve of quickest de scent — that is, a body falling in it from any given point above to another not exactly beneath it will come to this point in less time than in any other curve passing through those two points. This very singular property of the trochoid with respect to motion was first discovered by one of the Bernouillis in 1697. From this it is easily seen that if any body whatever move in a trochoid by its own weight or swing together with some other force act ing upon it all the while, it will go through all the distances of the same curve in exactly the same time, and accordingly pendulums have sometimes been contrived to swing in such a manner that they shall describe trochoids, or near trochoids, and thus move in equal times whether they go through a longer or shorter path of the same curve.

As the curve of quickest descent, the trochoid is understood to assume its inverted position. It is to be noticed that although the trochoid is longer than many curves and the straight line that may connect two points not in the same verticle line, yet it is, of all lines which can be drawn, the one through which a body will fall in the shortest time. John Bernouilli made

the enquiry after the curve which possesses the property: ((That a body setting out from any point of it, as A, and impelled solely by the force of gravity, will reach another point of it, as B, in a shorter space of time than it could reach the same point by following any other path" To this curve he gave the name bra chystochrone, from flPaxicrros shortest, and xPovos time. It is plain from what has been said about the curve of quickest descent alone, that the brachystochrone and the trochoid (or cycloid) are one and the same curve.

This curve possesses an additional histori cal interest entirely independent of the par ticular nature of the curve, for the determina tion of the nature of this line suggested to Lagrange the idea of an entirely new branch of mathematics, the calculus of variations.

The trochoid (or cycloid, under which name the brachystochrone is best known) may be made to assume an endless variety of forms by placing the tracing point not in the circum ference of the generating circle but without or within it. The meaning in this article of a curve is the totality of points, whose co-ordinates are functions of a parameter which may be dif ferentiated as often as may be required. Con sult Klein, H., (Elementar Mathematik vom boheren Standpunkte aus' (Leipzig 1909, Vol. II, p. 354).

If the generating point lies upon the cir cumference of the generating circle, the curve is known as a common trochoid or a cycloid. When the point is without the circumference of the generating circle, the resulting curve is called curtate or contracted. If the point of generation lies within the circumference of the generating circle, the curve is called a prolate or inflected trochoid or cycloid. In all three in stances the wheel or generating circle is thought of as rolling along a straight line in a single plane. For trochoids otherwise conditioned by the surface over which the wheel can roll see ROULETTE; HYPOCYCLOID ; CYCLOID.

One more fact in connection with this in teresting curve may be noted: It is believed by some that birds, such as the eagle, which build in the rocks, drop or fly down from height to height in this curve. It is impossible to make very accurate observations of their flight and path, but there is a general resemblance be tween it and the trochoid or cycloid, which has led several ingenious men to adopt this opinion. With the further development of photographic observation, this hypothesis may be confirmed or abandoned.