In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve.

The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). In physics, when a charged particle at rest is put under a uniform electric and magnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid.

History

The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th-century mathematicians, while Sarah Hart sees it named as such "because the properties of this curve are so beautiful".^[1][2]

Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery speculated that such a simple curve must have been known to the ancients, citing similar work by Carpus of Antioch described by Iamblichus.^[3] English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa,^[4] but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost.^[5] Galileo Galilei's name was put forward at the end of the 19th century^[6] and at least one author reports credit being given to Marin Mersenne.^[7] Beginning with the work of Moritz Cantor^[8] and Siegmund Günther,^[9] scholars now assign priority to French mathematician Charles de Bovelles^[10][11][12] based on his description of the cycloid in his Introductio in geometriam, published in 1503.^[13] In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel.^[5]

Galileo originated the term cycloid and was the first to make a serious study of the curve.^[5] According to his student Evangelista Torricelli,^[14] in 1599 Galileo attempted the quadrature of the cycloid (determining the area under the cycloid) with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them. He discovered the ratio was roughly 3:1, which is the true value, but he incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible.^[7] Around 1628, Gilles Persone de Roberval likely learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem.^[5] However, this work was not published until 1693 (in his Traité des Indivisibles).^[15]

Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviani, who were able to produce a quadrature. This result and others were published by Torricelli in 1644,^[14] which is also the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the controversy cut short by Torricelli's early death in 1647.^[15]

In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons. Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions (by John Wallis and Antoine de Lalouvère) was judged to be adequate.^[16]: 198 While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's Tractatus Duo, giving Wren priority for the first published proof.^[15]

Fifteen years later, Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of an inverted cycloidal arch in the same amount of time, regardless of its starting point. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation. In 1696, Johann Bernoulli posed the brachistochrone problem, the solution of which is a cycloid.^[15]

Equations

The cycloid through the origin, generated by a circle of radius r rolling over the x-axis on the positive side (y ≥ 0), consists of the points (x, y), with $${\displaystyle {\begin{aligned}x&=r(t-\sin t)\\y&=r(1-\cos t),\end{aligned}}}$$ where t is a real parameter corresponding to the angle through which the rolling circle has rotated. For given t, the circle's centre lies at (x, y) = (rt, r).

The Cartesian equation is obtained by solving the y-equation for t and substituting into the x-equation:$${\displaystyle x=r\cos ^{-1}\left(1-{\frac {y}{r}}\right)-{\sqrt {y(2r-y)}},}$$or, eliminating the multiple-valued inverse cosine:

${\displaystyle r\cos \!\left({\frac {x+{\sqrt {y(2r-y)}}}{r}}\right)+y=r.}$

When y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps on the x-axis, with the derivative tending toward ${\displaystyle \infty }$ or ${\displaystyle -\infty }$ near a cusp. The map from t to (x, y) is differentiable, in fact of class C^∞, with derivative 0 at the cusps.

The slope of the tangent to the cycloid at the point ${\displaystyle (x,y)}$ is given by ${\textstyle {\frac {dy}{dx}}=\cot({\frac {t}{2}})}$.

A cycloid segment from one cusp to the next is called an arch of the cycloid, for example the points with ${\displaystyle 0\leq t\leq 2\pi }$ and ${\displaystyle 0\leq x\leq 2\pi }$.

Considering the cycloid as the graph of a function ${\displaystyle y=f(x)}$, it satisfies the differential equation:^[17]

${\displaystyle \left({\frac {dy}{dx}}\right)^{2}={\frac {2r}{y}}-1.}$

Involute

The involute of the cycloid has exactly the same shape as the cycloid it originates from. This can be visualized as the path traced by the tip of a wire initially lying on a half arch of the cycloid: as it unrolls while remaining tangent to the original cycloid, it describes a new cycloid (see also cycloidal pendulum and arc length).

Demonstration

This demonstration uses the rolling-wheel definition of cycloid, as well as the instantaneous velocity vector of a moving point, tangent to its trajectory. In the adjacent picture, ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially, ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ traverse two cycloid curves. Considering the red line connecting ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ at a given time, one proves the line is always tangent to the lower arc at ${\displaystyle P_{2}}$ and orthogonal to the upper arc at ${\displaystyle P_{1}}$. Let ${\displaystyle Q}$ be the point in common between the upper and lower circles at the given time. Then:

• ${\displaystyle P_{1},Q,P_{2}}$ are colinear: indeed the equal rolling speed gives equal angles ${\displaystyle {\widehat {P_{1}O_{1}Q}}={\widehat {P_{2}O_{2}Q}}}$, and thus ${\displaystyle {\widehat {O_{1}QP_{1}}}={\widehat {O_{2}QP_{2}}}}$ . The point ${\displaystyle Q}$ lies on the line ${\displaystyle O_{1}O_{2}}$ therefore ${\displaystyle {\widehat {P_{1}QO_{1}}}+{\widehat {P_{1}QO_{2}}}=\pi }$ and analogously ${\displaystyle \stackrel{}{{}_{}}}$Zdroj:https://en.wikipedia.org?pojem=Cycloid
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