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Cycloid

## Summary

A cycloid generated by a rolling circle

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 constant 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).

## History

It was in the left hand try-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 by Herman Melville, 1851

The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians.[1]

Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was known in antiquity.[2] English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa,[3] but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost.[4] Galileo Galilei's name was put forward at the end of the 19th century[5] and at least one author reports credit being given to Marin Mersenne.[6] Beginning with the work of Moritz Cantor[7] and Siegmund Günther,[8] scholars now assign priority to French mathematician Charles de Bovelles[9][10][11] based on his description of the cycloid in his Introductio in geometriam, published in 1503.[12] 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.[4]

Galileo originated the term cycloid and was the first to make a serious study of the curve.[4] According to his student Evangelista Torricelli,[13] 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 but incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible.[6] 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.[4] However, this work was not published until 1693 (in his Traité des Indivisibles).[14]

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 Viviana, who were able to produce a quadrature. This result and others were published by Torricelli in 1644,[13] 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.[14]

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.[15]: 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 Tractus Duo, giving Wren priority for the first published proof.[14]

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.[14]

## Equations

The cycloid through the origin, with a horizontal base given by the x-axis, generated by a circle of radius r rolling over the "positive" side of the base (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).

Solving for t and replacing, the Cartesian equation is found to be:

${\displaystyle x=r\cos ^{-1}\left(1-{\frac {y}{r}}\right)-{\sqrt {y(2r-y)}}.}$

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 ${\displaystyle \infty }$ or ${\displaystyle -\infty }$ as one approaches a cusp. The map from t to (x, y) is a differentiable curve or parametric curve of class C, and the singularity where the derivative is 0 is an ordinary cusp.

The slope of the tangent to the cycloid at the point ${\displaystyle (x,y)}$ is given by:

${\displaystyle {\frac {dy}{dx}}=\cot {\frac {t}{2}}}$

A cycloid segment from one cusp to the next is called an arch of the cycloid. The first arch of the cycloid consists of points such that ${\displaystyle 0\leq t\leq 2\pi .}$

The equation of the cycloid satisfies the differential equation:[16]

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

## Involute

Generation of the involute of the cycloid unwrapping a tense wire placed on half cycloid arc (red marked)

The involute of the cycloid has the property of being exactly the same cycloid it originates from. This can otherwise be seen from the tip of a wire initially lying on a half arc of cycloid describing a cycloid arc equal to the one it was lying on once unwrapped (see also cycloidal pendulum and arc length).

### Demonstration

Demonstration of the properties of the involute of a cycloid

There are several demonstrations of the assertion. The one presented here uses the physical definition of cycloid and the kinematic property that the instantaneous velocity of a point is tangent to its trajectory. Referring to the adjacent picture, ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ are two tangent points belonging to two rolling circles. The two circles start to roll with same speed and same direction without skidding. ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ start to draw two cycloid arcs as in the picture. Considering the line connecting ${\displaystyle P_{1}}$ and ${\displaystyle P_{2}}$ at an arbitrary instant (red line), it is possible to prove that the line is anytime tangent in ${\displaystyle P_{2}}$ to the lower arc and orthogonal to the tangent in ${\displaystyle P_{1}}$ of the upper arc. One sees that calling ${\displaystyle Q}$ the point in common between the upper circle and the lower circle:

• ${\displaystyle P_{1},Q,P_{2}}$ are aligned because ${\displaystyle {\widehat {P_{1}O_{1}Q}}={\widehat {P_{2}O_{2}Q}}}$ (equal rolling speed) and therefore ${\displaystyle {\widehat {O_{1}QP_{1}}}={\widehat {O_{2}QP_{2}}}}$ . The point ${\displaystyle Q}$ lies on the line ${\displaystyle O_{1}O_{2}}$ therefore ${\displaystyle {\widehat {P_{1}QO_{1}}}+{\widehat {P_{1}QO_{2}}}=\pi }$ ad analogously ${\displaystyle {\widehat {P_{2}QO_{2}}}+{\widehat {P_{2}QO_{1}}}=\pi }$. From the equality of ${\displaystyle {\widehat {O_{1}QP_{1}}}}$ and ${\displaystyle {\widehat {O_{2}QP_{2}}}}$ one has that also ${\displaystyle {\widehat {P_{1}QO_{2}}}={\widehat {P_{2}QO_{1}}}}$. It follows ${\displaystyle {\widehat {P_{1}QO_{1}}}+{\widehat {P_{2}QO_{1}}}=\pi }$ .
• If ${\displaystyle A}$ is the meeting point between the perpendicular from ${\displaystyle P_{1}}$ to the straight of ${\displaystyle O_{1}O_{2}}$ and the tangent to the circle in ${\displaystyle P2}$ , then the triangle ${\displaystyle P_{1}AP_{2}}$ is isosceles because ${\displaystyle {\widehat {QP_{2}A}}={\frac {1}{2}}{\widehat {P_{2}O_{2}Q}}}$ and ${\displaystyle {\widehat {QP_{1}A}}={\frac {1}{2}}{\widehat {QO_{1}R}}=}$(easy to prove seen the construction)${\displaystyle ={\frac {1}{2}}{\widehat {QO_{1}P_{1}}}}$ . For the previous noted equality between ${\displaystyle {\widehat {P_{1}O_{1}Q}}}$ and ${\displaystyle {\widehat {QO_{2}P_{2}}}}$ then ${\displaystyle {\widehat {QP_{1}A}}={\widehat {QP_{2}A}}}$ and ${\displaystyle P_{1}AP_{2}}$ is isosceles.
• Conducting from ${\displaystyle P_{2}}$ the orthogonal straight to ${\displaystyle O_{1}O_{2}}$ , from ${\displaystyle P_{1}}$ the straight line tangent to the upper circle and calling ${\displaystyle B}$ the meeting point is now easy to see that ${\displaystyle P_{1}AP_{2}B}$ is a rhombus, using the theorems concerning the angles between parallel lines
• Now consider the speed ${\displaystyle V_{2}}$ of ${\displaystyle P_{2}}$ . It can be seen as the sum of two components, the rolling speed ${\displaystyle V_{a}}$ and the drifting speed ${\displaystyle V_{d}}$ . Both speeds are equal in modulus because the circles roll without skidding. ${\displaystyle V_{d}}$ is parallel to ${\displaystyle P_{1}A}$ and ${\displaystyle V_{a}}$ is tangent to the lower circle in ${\displaystyle P_{2}}$ therefore is parallel to ${\displaystyle P_{2}A}$ . The rhombus constituted from the components ${\displaystyle V_{d}}$ and ${\displaystyle V_{a}}$ is therefore similar (same angles) to the rhombus ${\displaystyle BP_{1}AP_{2}}$ because they have parallel sides. The total speed of ${\displaystyle P_{2},V_{2}}$ is then parallel to ${\displaystyle P_{2}P_{1}}$ because both are diagonals of two rhombuses with parallel sides and has in common with ${\displaystyle P_{1}P_{2}}$ the contact point ${\displaystyle P_{2}}$ . It follows that the speed vector ${\displaystyle V_{2}}$ lies on the prolongation of ${\displaystyle P_{1}P_{2}}$ . Because ${\displaystyle V_{2}}$ is tangent to the arc of cycloid in ${\displaystyle P_{2}}$ (property of velocity of a trajectory), it follows that also ${\displaystyle P_{1}P_{2}}$ is coinciding with the tangent to the lower cycloid arc in ${\displaystyle P_{2}}$.
• Analogously, it can be easily demonstrated that ${\displaystyle P_{1}P_{2}}$ is orthogonal to ${\displaystyle V_{1}}$ (other diagonal of the rhombus).
• The tip of an inextensible wire initially stretched on half arc of lower cycloid and bounded to the upper circle in ${\displaystyle P_{1}}$ will then follow the point along its path without 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 in ${\displaystyle P_{2}}$ to the lower arc because the tension and the demonstrated items. If it would not be tangent then there would be a discontinuity in ${\displaystyle P_{2}}$ and consequently there would be unbalanced tension forces.

## Area

Using the above parameterization for one arch of a cycloid generated by a circle of radius r,

{\displaystyle {\begin{aligned}x&=r(t-\sin t)\\y&=r(1-\cos t)\end{aligned}}}

for ${\displaystyle 0\leq t\leq 2\pi ,}$ the area under the arch is given by

{\displaystyle {\begin{aligned}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}.\end{aligned}}}

This result, and some generalizations, can be obtained without calculation by Mamikon's visual calculus.

## Arc length

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

The arc length S of one arch is given by

{\displaystyle {\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 immediate way to calculate the length of the cycloid given the properties of the involute is to notice that when a wire describing an involute has been completely unwrapped it extends itself along two diameters, a length of 4r. Because the wire does not change length during the unwrapping, it follows that the length of half an arc of cycloid is 4r and that of a complete arc is 8r.

## Cycloidal pendulum

Schematic of a cycloidal pendulum.

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 L 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 the pendulum also traces a cycloid path. Such a cycloidal pendulum is isochronous, regardless of amplitude. Introducing a coordinate system centred in the position of the cusp, the equation of motion is given by:

{\displaystyle {\begin{aligned}x&=r[2\theta (t)+\sin(2\theta (t))]\\y&=r[-3-\cos(2\theta (t))],\end{aligned}}}

where ${\displaystyle \theta }$ is the angle of the straight part of the string with respect to the vertical axis, and is given by

${\displaystyle \sin \theta (t)=A\cos(\omega t),\qquad \omega ^{2}={\frac {g}{L}}={\frac {g}{4r}},}$

where A<1 is the "amplitude", ${\displaystyle \omega }$ is the radian frequency of the pendulum and g the gravitational acceleration.

Five isochronous cycloidal pendula with different amplitudes.

The 17th-century Dutch mathematician Christiaan Huygens discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to be used in navigation.[17]

## Related curves

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 are roulettes with a circle rolled along another curve of uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the curve:evolute similitude ratio is 1 + 2q.

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

## Other uses

Cycloidal arches at the Kimbell Art Museum

The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas. It was also used by Wallace K. Harrison in the design of the Hopkins Center at Dartmouth College in Hanover, New Hampshire.[18]

Early research indicated that some transverse arching curves of the plates of golden age violins are closely modeled by curtate cycloid curves.[19] Later work indicates that curtate cycloids do not serve as general models for these curves,[20] which vary considerably.

## References

1. ^ Cajori, Florian (1999). A History of Mathematics. New York: Chelsea. p. 177. ISBN 978-0-8218-2102-2.
2. ^ Tannery, Paul (1883), "Pour l'histoire des lignes et surfaces courbes dans l'antiquité", Bulletin des sciences mathèmatique, Paris: 284 (cited in Whitman 1943);
3. ^ 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)
4. ^ a b c d Whitman, 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)
5. ^ Cajori, Florian (1999), A History of Mathematics (5th ed.), p. 162, ISBN 0-8218-2102-4(Note: The first (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.)
6. ^ a b Roidt, Tom (2011). Cycloids and Paths (PDF) (MS). Portland State University. p. 4.
7. ^ Cantor, Moritz (1892), Vorlesungen über Geschichte der Mathematik, Bd. 2, Leipzig: B. G. Teubner, OCLC 25376971
8. ^ Günther, Siegmund (1876), Vermischte untersuchungen zur geschichte der mathematischen wissenschaften, Leipzig: Druck und Verlag Von B. G. Teubner, p. 352, OCLC 2060559
9. ^ 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)
10. ^ Victor, Joseph M. (1978), Charles de Bovelles, 1479-1553: An Intellectual Biography, p. 42, ISBN 978-2-600-03073-1
11. ^ Martin, J. (2010). "The Helen of Geometry". The College Mathematics Journal. 41: 17–28. doi:10.4169/074683410X475083. S2CID 55099463.
12. ^ de Bouelles, Charles (1503), Introductio in geometriam ... Liber de quadratura circuli. Liber de cubicatione sphere. Perspectiva introductio., OCLC 660960655
13. ^ a b Torricelli, Evangelista (1644), Opera geometrica, OCLC 55541940
14. ^ a b c d Walker, Evelyn (1932), A Study of Roberval's Traité des Indivisibles, Columbia University (cited in Whitman 1943);
15. ^ Conner, James A. (2006), Pascal's Wager: The Man Who Played Dice with God (1st ed.), HarperCollins, pp. 224, ISBN 9780060766917
16. ^ 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 their K=2r
17. ^ 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).
18. ^ 101 Reasons to Love Dartmouth, Dartmouth Alumni Magazine, 2016
19. ^ Playfair, Q. "Curtate Cycloid Arching in Golden Age Cremonese Violin Family Instruments". Catgut Acoustical Society Journal. II. 4 (7): 48–58.
20. ^ Mottola, RM (2011). "Comparison of Arching Profiles of Golden Age Cremonese Violins and Some Mathematically Generated Curves". Savart Journal. 1 (1).