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Cycloid

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From Wikipedia, the free encyclopedia

Curve traced by a point on a rolling circle

For other uses, seeCycloid (disambiguation).

Ingeometry, acycloid is thecurve traced by a point on acircle as itrolls along astraight line without slipping. A cycloid is a specific form oftrochoid and is an example of aroulette, a curve generated by a curve rolling on another curve.

The cycloid, with thecusps pointing upward, is the curve of fastest descent under uniformgravity (thebrachistochrone curve). It is also the form of a curve for which theperiod of an object insimple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (thetautochrone curve). In physics, when a charged particle at rest is put under a uniformelectric andmagnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid.

Balls rolling under uniform gravity without friction on a cycloid (black) and straight lines with various gradients. It demonstrates that the ball on the curve always beats the balls travelling in a straight line path to the intersection point of the curve and each straight line.

History

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It was in the left handtry-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time.

Moby Dick byHerman Melville, 1851

The cycloid has been called "The Helen of Geometers" as, likeHelen of Troy, it caused frequent quarrels among 17th-century mathematicians, whileSarah 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 historianPaul Tannery speculated that such a simple curve must have been known to theancients, citing similar work byCarpus of Antioch described byIamblichus.^[3] English mathematicianJohn Wallis writing in 1679 attributed the discovery toNicholas 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 toMarin Mersenne.^[7] Beginning with the work ofMoritz Cantor^[8] andSiegmund Günther,^[9] scholars now assign priority to French mathematicianCharles de Bovelles^[10]^[11]^[12] based on his description of the cycloid in hisIntroductio 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 termcycloid and was the first to make a serious study of the curve.^[5] According to his studentEvangelista Torricelli,^[14] in 1599 Galileo attempted thequadrature 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 fromPère Marin Mersenne and effected the quadrature in 1634 by usingCavalieri's Theorem.^[5] However, this work was not published until 1693 (in hisTraité des Indivisibles).^[15]

Constructing thetangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval,Pierre de Fermat andRené 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 thecenter of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanishdoubloons. Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions (byJohn Wallis andAntoine de Lalouvère) was judged to be adequate.^[16]^: 198 While the contest was ongoing,Christopher Wren sent Pascal a proposal for a proof of therectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis'sTractatus 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 thebrachistochrone problem, the solution of which is a cycloid.^[15]

Equations

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

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

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

Wheny is viewed as a function ofx, the cycloid isdifferentiable everywhere except at thecusps on thex-axis, with the derivative tending toward $\infty$ or $-\infty$ near a cusp (wherey=0). The map fromt to(x,y) is differentiable, in fact of classC^∞, with derivative 0 at the cusps.

The slope of thetangent to the cycloid at the point $(x,y)$ 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 $0\leq t\leq 2\pi$ and $0\leq x\leq 2\pi$ .

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

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

If we define $h=2r-y=r(1+\cos t)$ as the height difference from the cycloid's vertex (the point with a horizontal tangent and $\cos t=-1$ ), then we have:

\left({\frac {dx}{dh}}\right)^{2}={\frac {2r}{h}}-1.

Involute

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Theinvolute of the cycloid has exactly thesame 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 alsocycloidal pendulum andarc length).

Demonstration

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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, $P_{1}$ and $P_{2}$ are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially, $P_{1}$ and $P_{2}$ coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, $P_{1}$ and $P_{2}$ traverse two cycloid curves. Considering the red line connecting $P_{1}$ and $P_{2}$ at a given time, one provesthe line is alwaystangent to the lower arc at $P_{2}$ and orthogonal to the upper arc at $P_{1}$ . Let $Q {\displaystyle Q}$ be the point in common between the upper and lower circles at the given time. Then:

$P_{1},Q,P_{2}$ are colinear: indeed the equal rolling speed gives equal angles ${\widehat {P_{1}O_{1}Q}}={\widehat {P_{2}O_{2}Q}}$ , and thus ${\widehat {O_{1}QP_{1}}}={\widehat {O_{2}QP_{2}}}$ . The point $Q {\displaystyle Q}$ lies on the line $O_{1}O_{2}$ therefore ${\widehat {P_{1}QO_{1}}}+{\widehat {P_{1}QO_{2}}}=\pi$ and analogously ${\widehat {P_{2}QO_{2}}}+{\widehat {P_{2}QO_{1}}}=\pi$ . From the equality of ${\widehat {O_{1}QP_{1}}}$ and ${\widehat {O_{2}QP_{2}}}$ one has that also ${\widehat {P_{1}QO_{2}}}={\widehat {P_{2}QO_{1}}}$ . It follows ${\widehat {P_{1}QO_{1}}}+{\widehat {P_{2}QO_{1}}}=\pi$ .
If $A {\displaystyle A}$ is the meeting point between the perpendicular from $P_{1}$ to the line segment $O_{1}O_{2}$ and the tangent to the circle at $P_{2}$ , then the triangle $P_{1}AP_{2}$ is isosceles, as is easily seen from the construction: ${\widehat {QP_{2}A}}={\tfrac {1}{2}}{\widehat {P_{2}O_{2}Q}}$ and ${\widehat {QP_{1}A}}={\tfrac {1}{2}}{\widehat {QO_{1}R}}=$ ${\tfrac {1}{2}}{\widehat {QO_{1}P_{1}}}$ . For the previous noted equality between ${\widehat {P_{1}O_{1}Q}}$ and ${\widehat {QO_{2}P_{2}}}$ then ${\widehat {QP_{1}A}}={\widehat {QP_{2}A}}$ and $P_{1}AP_{2}$ is isosceles.
Drawing from $P_{2}$ the orthogonal segment to $O_{1}O_{2}$ , from $P_{1}$ the straight line tangent to the upper circle, and calling $B {\displaystyle B}$ the meeting point, one sees that $P_{1}AP_{2}B$ is arhombus using the theorems on angles between parallel lines
Now consider the velocity $V_{2}$ of $P_{2}$ . It can be seen as the sum of two components, the rolling velocity $V_{a}$ and the drifting velocity $V_{d}$ , which are equal in modulus because the circles roll without skidding. $V_{d}$ is parallel to $P_{1}A$ , while $V_{a}$ is tangent to the lower circle at $P_{2}$ and therefore is parallel to $P_{2}A$ . The rhombus constituted from the components $V_{d}$ and $V_{a}$ is therefore similar (same angles) to the rhombus $BP_{1}AP_{2}$ because they have parallel sides. Then $V_{2}$ , the total velocity of $P_{2}$ , is parallel to $P_{2}P_{1}$ because both are diagonals of two rhombuses with parallel sides and has in common with $P_{1}P_{2}$ the contact point $P_{2}$ . Thus the velocity vector $V_{2}$ lies on the prolongation of $P_{1}P_{2}$ . Because $V_{2}$ is tangent to the cycloid at $P_{2}$ , it follows that also $P_{1}P_{2}$ coincides with the tangent to the lower cycloid at $P_{2}$ .
Analogously, it can be easily demonstrated that $P_{1}P_{2}$ is orthogonal to $V_{1}$ (the other diagonal of the rhombus).
This proves that the tip of a wire initially stretched on a half arch of the lower cycloid and fixed to the upper circle at $P_{1}$ will follow the point along its pathwithout changing its length because the speed of the tip is at each moment orthogonal to the wire (no stretching or compression). The wire will be at the same time tangent at $P_{2}$ to the lower arc because of the tension and the facts demonstrated above. (If it were not tangent there would be a discontinuity at $P_{2}$ and consequently unbalanced tension forces.)

Area

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Using the above parameterization ${\textstyle x=r(t-\sin t),\ y=r(1-\cos t)}$ , the area under one arch, $0\leq t\leq 2\pi ,$ is given by: $A=\int _{x=0}^{2\pi r}y\,dx=\int _{t=0}^{2\pi }r^{2}(1-\cos t)^{2}dt=3\pi r^{2}.$

This is three times the area of the rolling circle.

Arc length

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The length of the cycloid as consequence of the property of its involute

Thearc lengthS of one arch is given by ${\begin{aligned}S&=\int _{0}^{2\pi }{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt\\&=\int _{0}^{2\pi }r{\sqrt {2-2\cos t}}\,dt\\&=2r\int _{0}^{2\pi }\sin {\frac {t}{2}}\,dt\\&=8r.\end{aligned}}$

Another geometric way to calculate the length of the cycloid is to notice that when a wire describing aninvolute has been completely unwrapped from half an arch, it extends itself along two diameters, a length of4r. This is thus equal to half the length of arch, and that of a complete arch is8r.

From the cycloid's vertex (the point with a horizontal tangent and $\cos t=-1$ ) to any point within the same arch, the arc length squared is $8r^{2}(1+\cos t)$ , which is proportional to the height difference $r(1+\cos t)$ ; this property is the basis for the cycloid'sisochronism. In fact, the arc length squared is equal to the height difference multiplied by the full arch length8r.

Cycloidal pendulum

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If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the string is constrained to be tangent to one of its arches, and the pendulum's lengthL is equal to that of half the arc length of the cycloid (i.e., twice the diameter of the generating circle,L = 4r), the bob of thependulum also traces a cycloid path. Such a pendulum isisochronous, with equal-time swings regardless of amplitude. Introducing a coordinate system centred in the position of the cusp, the equation of motion is given by: ${\begin{aligned}x&=r[2\theta (t)+\sin 2\theta (t)]\\y&=r[-3-\cos 2\theta (t)],\end{aligned}}$ where $\theta$ is the angle that the straight part of the string makes with the vertical axis, and is given by $\sin \theta (t)=A\cos(\omega t),\qquad \omega ^{2}={\frac {g}{L}}={\frac {g}{4r}},$ whereA < 1 is the "amplitude", $\omega$ is the radian frequency of the pendulum andg the gravitational acceleration.

Five isochronous cycloidal pendula with different amplitudes.

The 17th-century Dutch mathematicianChristiaan Huygens discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to beused in navigation.^[18]

Related curves

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Several curves are related to the cycloid.

Trochoid: generalization of a cycloid in which the point tracing the curve may be inside the rolling circle (curtate) or outside (prolate).
Hypocycloid: variant of a cycloid in which a circle rolls on the inside of another circle instead of a line.
Epicycloid: variant of a cycloid in which a circle rolls on the outside of another circle instead of a line.
Hypotrochoid: generalization of a hypocycloid where the generating point may not be on the edge of the rolling circle.
Epitrochoid: generalization of an epicycloid where the generating point may not be on the edge of the rolling circle.

All these curves areroulettes with a circle rolled along another curve of uniformcurvature. The cycloid, epicycloids, and hypocycloids have the property that each issimilar to itsevolute. Ifq is theproduct of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then thesimilitude ratio of curve to evolute is 1 + 2q.

The classicSpirograph toy traces out hypotrochoid andepitrochoid curves.

Other uses

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Cycloidal arches at theKimbell Art Museum

The cycloidal arch was used by architectLouis Kahn in his design for theKimbell Art Museum inFort Worth, Texas. It was also used byWallace K. Harrison in the design of theHopkins Center atDartmouth College inHanover, New Hampshire.^[19]

Early research indicated that some transverse arching curves of the plates of golden age violins are closely modeled by curtate cycloid curves.^[20] Later work indicates that curtate cycloids do not serve as general models for these curves,^[21] which vary considerably.

References

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^Cajori, Florian (1999).A History of Mathematics. New York: Chelsea. p. 177.ISBN 978-0-8218-2102-2.
^Hart, Sarah (7 April 2023)."The Wondrous Connections Between Mathematics and Literature".New York Times. Retrieved7 April 2023.
^Tannery, Paul (1883),"Pour l'histoire des lignes et surfaces courbes dans l'antiquité", Mélanges,Bulletin des sciences mathématiques et astronomiques, Ser. 2,7:278–291, p. 284:Avant de quitter la citation de Jamblique, j'ajouterai que, dans la courbe dedouble mouvement de Carpos, il est difficile de ne pas reconnaître la cycloïde dont la génération si simple n'a pas dû échapper aux anciens. [Before leaving the citation of Iamblichus, I will add that, in the curve ofdouble movement ofCarpus, it is difficult not to recognize the cycloid, whose so-simple generation couldn't have escaped the ancients.] (cited in Whitman 1943);
^Wallis, D. (1695)."An Extract of a Letter from Dr. Wallis, of May 4. 1697, Concerning the Cycloeid Known to Cardinal Cusanus, about the Year 1450; and to Carolus Bovillus about the Year 1500".Philosophical Transactions of the Royal Society of London.19 (215–235):561–566.doi:10.1098/rstl.1695.0098. (Cited in Günther, p. 5)
^^a ^b ^c ^dWhitman, E. A. (May 1943), "Some historical notes on the cycloid",The American Mathematical Monthly,50 (5):309–315,doi:10.2307/2302830,JSTOR 2302830(subscription required)
^Cajori, Florian (1999),A History of Mathematics (5th ed.), American Mathematical Soc., p. 162,ISBN 0-8218-2102-4(Note: Thefirst (1893) edition and its reprints state that Galileo invented the cycloid. According to Phillips, this was corrected in the second (1919) edition and has remained through the most recent (fifth) edition.)
^^a ^bRoidt, Tom (2011).Cycloids and Paths(PDF) (MS). Portland State University. p. 4.Archived(PDF) from the original on 2022-10-09.
^Cantor, Moritz (1892),Vorlesungen über Geschichte der Mathematik, Bd. 2, Leipzig: B. G. Teubner,OCLC 25376971
^Günther, Siegmund (1876),Vermischte untersuchungen zur geschichte der mathematischen wissenschaften, Leipzig: Druck und Verlag Von B. G. Teubner, p. 352,OCLC 2060559
^Phillips, J. P. (May 1967), "Brachistochrone, Tautochrone, Cycloid—Apple of Discord",The Mathematics Teacher,60 (5):506–508,doi:10.5951/MT.60.5.0506,JSTOR 27957609(subscription required)
^Victor, Joseph M. (1978),Charles de Bovelles, 1479-1553: An Intellectual Biography, Librairie Droz, p. 42,ISBN 978-2-600-03073-1
^Martin, J. (2010). "The Helen of Geometry".The College Mathematics Journal.41:17–28.doi:10.4169/074683410X475083.S2CID 55099463.
^de Bouelles, Charles (1503),Introductio in geometriam ... Liber de quadratura circuli. Liber de cubicatione sphere. Perspectiva introductio.,OCLC 660960655
^^a ^bTorricelli, Evangelista (1644),Opera geometrica,OCLC 55541940
^^a ^b ^c ^dWalker, Evelyn (1932),A Study of Roberval's Traité des Indivisibles, Columbia University (cited in Whitman 1943);
^Conner, James A. (2006),Pascal's Wager: The Man Who Played Dice with God (1st ed.), HarperCollins, pp. 224,ISBN 9780060766917
^Roberts, Charles (2018).Elementary Differential Equations: Applications, Models, and Computing (2nd illustrated ed.). CRC Press. p. 141.ISBN 978-1-4987-7609-7.Extract of page 141, equation (f) with theirK=2r
^C. Huygens, "The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula (sic) as Applied to Clocks," Translated by R. J. Blackwell, Iowa State University Press (Ames, Iowa, USA, 1986).
^101 Reasons to Love Dartmouth, Dartmouth Alumni Magazine, 2016
^Playfair, Q. "Curtate Cycloid Arching in Golden Age Cremonese Violin Family Instruments".Catgut Acoustical Society Journal. II.4 (7):48–58.
^Mottola, RM (2011)."Comparison of Arching Profiles of Golden Age Cremonese Violins and Some Mathematically Generated Curves".Savart Journal.1 (1). Archived fromthe original on 2017-12-11. Retrieved2012-08-13.