Editing Cycloid

{{Short description|Curve traced by a point on a rolling circle}}
{{Other uses}}
[[File:Cycloid f.gif|right|frame|A cycloid generated by a rolling circle]]

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

The cycloid, with the [[Cusp (singularity)|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 [[Frequency|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 field|electric]] and [[magnetic field]] perpendicular to one another, the particle’s trajectory draws out a cycloid.

[[File:Brachistochrone_curve.gif|thumb|right| 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 ==
{{quotebox|width=30%|
quote=It was in the left hand [[Try pot|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.
|source=''[[Moby Dick]]'' by [[Herman Melville]], 1851}}

The cycloid has been called "The Helen of Geometers" as, like [[Helen of Troy]], it caused frequent quarrels among 17th-century mathematicians, while [[Sarah B. Hart|Sarah Hart]] sees it named as such  "because the properties of this curve are so beautiful".<ref>{{Cite book | last1=Cajori | first1=Florian | author1-link=Florian Cajori | title=A History of Mathematics | publisher=Chelsea | location=New York | isbn=978-0-8218-2102-2 | year=1999 | page=177 }}</ref><ref>{{Cite web |last=Hart |first=Sarah |date=7 April 2023 |title=The Wondrous Connections Between Mathematics and Literature |url=https://www.nytimes.com/2023/04/07/opinion/the-wondrous-connections-between-mathematics-and-literature.html |access-date=7 April 2023 |website=New York Times}}</ref>

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 [[Greek mathematics|ancients]], citing similar work by [[Carpus of Antioch]] described by [[Iamblichus]].<ref name=Tannery/>  English mathematician [[John Wallis]] writing in 1679 attributed the discovery to [[Nicholas of Cusa]],<ref name=Wallis/> but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost.<ref name=Whitman/>  [[Galileo Galilei]]'s name was put forward at the end of the 19th century<ref name=Cajori/> and at least one author reports credit being given to [[Marin Mersenne]].<ref name=Roidt/> Beginning with the work of [[Moritz Cantor]]<ref name=Cantor/> and [[Siegmund Guenther|Siegmund Günther]],<ref name=Gunther/> scholars now assign priority to French mathematician [[Charles de Bovelles]]<ref name=Phillips/><ref name=Victor/><ref name=Martin/> based on his description of the cycloid in his ''Introductio in geometriam'', published in 1503.<ref name=Bovelles/>  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.<ref name=Whitman/>

Galileo originated the term ''cycloid'' and was the first to make a serious study of the curve.<ref name=Whitman />  According to his student [[Evangelista Torricelli]],<ref name=Torricelli/> in 1599 Galileo attempted the [[Quadrature (geometry)|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.<ref name=Roidt/>  Around 1628, [[Gilles de Roberval|Gilles Persone de Roberval]] likely learned of the quadrature problem from [[Marin Mersenne|Père Marin Mersenne]] and effected the quadrature in 1634 by using [[Cavalieri's principle|Cavalieri's Theorem]].<ref name=Whitman />  However, this work was not published until 1693 (in his ''Traité des Indivisibles'').<ref name=Walker />

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,<ref name=Torricelli/> 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.<ref name=Walker />

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 mass|center of gravity]], area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish [[doubloon]]s.  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.<ref name=Conner />{{rp|198}}  While the contest was ongoing, [[Christopher Wren]] sent Pascal a proposal for a proof of the [[arc length|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.<ref name=Walker />

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 curve|brachistochrone problem]], the solution of which is a cycloid.<ref name=Walker />

== Equations ==
The cycloid through the origin, generated by a circle of radius {{mvar|r}} rolling over the ''{{mvar|x}}-''axis on the positive side ({{math|''y'' ≥ 0}}), consists of the points {{math|(''x'', ''y'')}}, with
<math display="block">\begin{align}
  x &= r(t - \sin t) \\
  y &= r(1 - \cos t),
 \end{align}</math>
where {{mvar|t}} is a real [[parameter]] corresponding to the angle through which the rolling circle has rotated. For given {{mvar|t}}, the circle's centre lies at {{math|1=(''x'', ''y'') = (''rt'', ''r'')}}.

The [[Cartesian coordinate system|Cartesian equation]] is obtained by solving the ''{{mvar|y}}''-equation for {{mvar|t}} and substituting into the ''{{mvar|x}}-''equation:<math display="block">x = r \cos^{-1} \left(1 - \frac{y}{r}\right) - \sqrt{y(2r - y)},</math>or, eliminating the multiple-valued inverse cosine:<blockquote><math>r \cos\!\left(\frac{x+\sqrt{y(2r-y)}}{r}\right) + y = r.</math></blockquote>When {{mvar|y}} is viewed as a function of {{mvar|x}}, the cycloid is [[Differentiable function|differentiable]] everywhere except at the [[Cusp (singularity)|cusps]] on the {{mvar|x}}-axis, with the derivative tending toward <math>\infty</math> or <math>-\infty</math> near a cusp (where {{mvar|1=y=0}}). The map from {{mvar|t}} to {{math|(''x'', ''y'')}} is differentiable, in fact of class {{mvar|C}}<sup>∞</sup>, with derivative 0 at the cusps.

The slope of the [[tangent]] to the cycloid at the point <math>(x,y)</math> is given by <math display="inline">\frac{dy}{dx} = \cot(\frac{t}{2})</math>.

A cycloid segment from one cusp to the next is called an arch of the cycloid, for example the points with <math>0 \le t \le 2 \pi</math> and <math>0 \leq x \leq 2\pi</math>.

Considering the cycloid as the graph of a function <math>y = f(x)</math>, it satisfies the [[ordinary differential equation|differential equation]]:<ref>{{cite book |title=Elementary Differential Equations: Applications, Models, and Computing |edition=2nd illustrated |first1=Charles |last1=Roberts |publisher=CRC Press |year=2018 |isbn=978-1-4987-7609-7 |page=141 |url=https://books.google.com/books?id=touADwAAQBAJ}} [https://books.google.com/books?id=touADwAAQBAJ&pg=PA141 Extract of page 141, equation (f) with their ''K''=2''r'']</ref>

:<math>\left(\frac{dy}{dx}\right)^2 = \frac{2r}{y} - 1.</math>
If we define <math>h=2r-y= r(1 + \cos t)</math> as the height difference from the cycloid's vertex (the point with a horizontal tangent and <math>\cos t=-1</math>), then we have:
:<math>\left(\frac{dx}{dh }\right)^2 = \frac{2r}{h} - 1.</math>

==Involute==
[[File:Evolute generation.png|thumb|Generation of the involute of the cycloid unwrapping a tense wire placed on half cycloid arc (red marked)]]
The [[involute]] of the cycloid has exactly the [[Congruence (geometry)|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 [[Cycloid#Cycloidal pendulum|cycloidal pendulum]] and [[Cycloid#Arc length|arc length]]).

===Demonstration===
[[File:Evolute demo.png|thumb|Demonstration of the properties of the involute of a cycloid]]
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, <math>P_1</math> and <math>P_2</math> are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially,  <math>P_1</math> and <math>P_2</math> coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, <math>P_1</math> and <math>P_2</math> traverse two cycloid curves. Considering the red line connecting <math>P_1</math> and <math>P_2</math> at a given time, one proves ''the line is always'' ''tangent to the lower arc at <math>P_2</math> and orthogonal to the upper arc at <math>P_1</math>''. Let <math>Q</math> be the point in common between the upper and lower circles at the given time. Then:
*<math>P_1,Q,P_2</math> are colinear: indeed the equal rolling speed gives equal angles <math>\widehat{P_1O_1Q}=\widehat{P_2O_2Q}</math>, and thus <math>\widehat{O_1 Q P_1} = \widehat{O_2QP_2}</math> . The point <math>Q</math> lies on the line <math>O_1O_2</math> therefore <math>\widehat{P_1 Q O_1} + \widehat{P_1QO_2}=\pi</math> and analogously <math>\widehat{P_2QO_2}+\widehat{P_2QO_1}=\pi</math>. From the equality of <math>\widehat{O_1QP_1}</math> and <math>\widehat{O_2QP_2}</math> one has that also  <math>\widehat{P_1QO_2}=\widehat{P_2QO_1}</math>. It follows <math>\widehat{P_1QO_1}+\widehat{P_2QO_1}=\pi</math> .
*If <math>A</math> is the meeting point between the perpendicular from <math>P_1</math> to the line segment <math>O_1O_2</math> and the tangent to the circle at <math>P_2</math> , then the triangle <math>P_1AP_2</math> is isosceles, as is easily seen from the construction: <math>\widehat{QP_2A}=\tfrac{1}{2}\widehat{P_2O_2Q}</math> and <math>\widehat{QP_1A} = \tfrac{1}{2}\widehat{QO_1R}=</math><math>\tfrac{1}{2}\widehat{QO_1P_1}</math> . For the previous noted equality between <math>\widehat{P_1O_1Q}</math> and <math>\widehat{QO_2P_2}</math> then <math>\widehat{QP_1A}=\widehat{QP_2A}</math> and <math>P_1AP_2</math> is isosceles.
*Drawing from <math>P_2</math> the orthogonal segment to <math>O_1O_2</math>, from <math>P_1</math> the straight line tangent to the upper circle, and calling <math>B</math> the meeting point, one sees that <math>P_1AP_2B</math> is a [[rhombus]] using the theorems on angles between parallel lines
*Now consider the velocity <math>V_2</math> of <math>P_2</math> . It can be seen as the sum of two components, the rolling velocity <math>V_a</math> and the drifting velocity <math>V_d</math>, which are equal in modulus because the circles roll without skidding. <math>V_d</math> is parallel to <math>P_1A</math>, while <math>V_a</math> is tangent to the lower circle at <math>P_2</math> and therefore is parallel to <math>P_2A</math>. The rhombus constituted from the components <math>V_d</math> and <math>V_a</math> is therefore similar (same angles) to the rhombus <math>BP_1AP_2</math> because they have parallel sides. Then <math>V_2</math>, the total velocity of <math>P_2</math>, is parallel to <math>P_2P_1</math> because both are diagonals of two rhombuses with parallel sides and has in common with <math>P_1P_2</math> the contact point <math>P_2</math>. Thus the velocity vector <math>V_2</math> lies on the prolongation of <math>P_1P_2</math> . Because <math>V_2</math> is tangent to the cycloid at <math>P_2</math>, it follows that also <math>P_1P_2</math> coincides with the tangent to the lower cycloid at <math>P_2</math>.
*Analogously, it can be easily demonstrated that <math>P_1P_2</math> is orthogonal to <math>V_1</math> (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 <math>P_1</math> will 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 at <math>P_2</math> to the lower arc because of the tension and the facts demonstrated above. (If it were not tangent there would be a discontinuity at <math>P_2</math> and consequently unbalanced tension forces.)

== Area ==
Using the above parameterization <math display="inline">   x = r(t - \sin t), \ 
   y = r(1 - \cos t)</math>, the area under one arch, <math>0 \leq t \leq 2\pi,</math> is given by:
<math display="block">  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.
</math>

This is three times the area of the rolling circle.

== Arc length ==
[[File:Cycloid length.png|thumb|The length of the cycloid as consequence of the property of its involute]]
The [[arc length]] {{mvar|S}} of one arch is given by
<math display="block">\begin{align}
  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{align}</math>

Another geometric way to calculate the length of the cycloid is to notice that when a wire describing an [[#Involute|involute]] has been completely unwrapped from half an arch, it extends itself along two diameters, a length of {{math|4''r''}}. This is thus equal to half the length of arch, and that of a complete arch is {{math|8''r''}}.

From the cycloid's vertex (the point with a horizontal tangent and <math>\cos t=-1</math>) to any point within the same arch, the arc length squared is <math>8r^2(1+\cos t)</math>, which is proportional to the height difference <math>r(1+\cos t)</math>; this property is the basis for the cycloid's [[Tautochrone curve#Lagrangian solution|isochronism]]. In fact, the arc length squared is equal to the height difference multiplied by the full arch length {{math|8''r''}}.

== Cycloidal pendulum ==
[[File:CyloidPendulum.png|right|thumb|Schematic of a cycloidal pendulum.]]
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 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 pendulum is [[Tautochrone curve|isochronous]], 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:
<math display="block">\begin{align}
  x &= r[2\theta(t) + \sin 2\theta (t)] \\
  y &= r[-3-\cos2\theta (t)],
\end{align}</math>
where <math>\theta</math> is the angle that the straight part of the string makes with the vertical axis, and is given by
<math display="block">\sin\theta (t) = A \cos(\omega t),\qquad \omega^2 = \frac{g}{L}=\frac{g}{4r},</math>
where {{math|''A'' < 1}} is the "amplitude", <math>\omega</math> is the radian frequency of the pendulum and ''g'' the gravitational acceleration.

[[File:Isochronous cycloidal pendula.gif|thumb|Five isochronous cycloidal pendula with different amplitudes.]]
The 17th-century Dutch mathematician [[Christiaan Huygens#Horology|Christiaan Huygens]] discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to be [[History of longitude|used in navigation]].<ref>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).</ref>

== 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 [[roulette (curve)|roulettes]] with a circle rolled along another curve of uniform [[curvature]]. The cycloid, epicycloids, and hypocycloids have the property that each is [[Similarity (geometry)|similar]] to its [[evolute]]. If ''q'' is the [[product (mathematics)|product]] of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the [[Homothetic transformation|similitude ratio]] of curve to evolute is 1&nbsp;+&nbsp;2''q''.

The classic [[Spirograph]] toy traces out hypotrochoid and [[epitrochoid]] curves.

== Other uses ==
[[File:Kimbell Art Museum.jpg|thumb|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 for the Arts|Hopkins Center]] at [[Dartmouth College]] in [[Hanover, New Hampshire]].<ref name=DartmouthAlumniMagazine/>

Early research indicated that some transverse arching curves of the plates of golden age violins are closely modeled by curtate cycloid curves.<ref>{{cite journal|last=Playfair|first=Q|title=Curtate Cycloid Arching in Golden Age Cremonese Violin Family Instruments|journal=Catgut Acoustical Society Journal|volume=4|series=II|issue=7|pages=48–58}}</ref> Later work indicates that curtate cycloids do not serve as general models for these curves,<ref>{{cite journal|last=Mottola|first=RM|title=Comparison of Arching Profiles of Golden Age Cremonese Violins and Some Mathematically Generated Curves|journal=Savart Journal|year=2011|volume=1|issue=1|url=http://savartjournal.org/index.php/sj/article/view/12|access-date=2012-08-13|archive-date=2017-12-11|archive-url=https://web.archive.org/web/20171211053817/http://www.savartjournal.org/index.php/sj/article/view/12|url-status=dead}}</ref> which vary considerably.



== See also ==
* [[Cyclogon]]
* [[Cycloid gear]]
* [[List of periodic functions]]
* [[Tautochrone curve]]
* [[Trochoid]] (for points located outside the circle)

== References ==
{{Reflist | refs=

<ref name=Conner>
{{citation
| last=Conner
| first=James A.
| title=Pascal's Wager: The Man Who Played Dice with God
| pages=[https://archive.org/details/pascalswagermanw00conn/page/224 224]
| isbn=9780060766917
| edition=1st
| year=2006
| publisher=HarperCollins
| url-access=registration
| url=https://archive.org/details/pascalswagermanw00conn/page/224
}}</ref>

<ref name=Torricelli>
{{citation 
| last=Torricelli
| first=Evangelista
| title=Opera geometrica
| year=1644
| oclc=55541940
}}
</ref>

<ref name=Phillips>
{{citation
| title=Brachistochrone, Tautochrone, Cycloid&mdash;Apple of Discord
| last=Phillips
| first=J. P.
| journal=The Mathematics Teacher
| volume=60
| number=5
|date=May 1967
| pages=506&ndash;508
| doi=10.5951/MT.60.5.0506
| jstor=27957609
}}{{subscription required}}
</ref>

<ref name=Gunther>
{{citation 
| title=Vermischte untersuchungen zur geschichte der mathematischen wissenschaften
| last=Günther
| first=Siegmund
| year=1876
| location=Leipzig
| publisher=Druck und Verlag Von B. G. Teubner
| oclc=2060559
| pages=352
}}
</ref>

<ref name=Wallis>
{{Cite journal | last1 = Wallis | first1 = D. | title = 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 | doi = 10.1098/rstl.1695.0098 | journal = Philosophical Transactions of the Royal Society of London | volume = 19 | issue = 215–235 | pages = 561–566 | year = 1695 | url = https://zenodo.org/record/1432146 | doi-access = free }} (Cited in Günther, p. 5)
</ref>

<ref name=Cajori>
{{citation 
| last=Cajori
| first=Florian
| title=A History of Mathematics
| url=https://books.google.com/books?id=mGJRjIC9fZgC
| edition=5th
| isbn=0-8218-2102-4
| page=162
| year=1999
| publisher=American Mathematical Soc.
}}(Note:  The [https://archive.org/details/ahistorymathema00cajogoog 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.)
</ref>

<ref name=Bovelles>
{{citation 
| title=Introductio in geometriam ... Liber de quadratura circuli. Liber de cubicatione sphere. Perspectiva introductio.
| last=de Bouelles
| first=Charles
| year=1503
| oclc=660960655
}}
</ref>

<ref name=Victor>
{{citation
| title=Charles de Bovelles, 1479-1553: An Intellectual Biography
| last=Victor
| first=Joseph M.
| url=https://books.google.com/books?id=bw4lM9wF1lgC
| page=42
| year=1978
| publisher=Librairie Droz
| isbn=978-2-600-03073-1
}}
</ref>

<ref name=Martin>
{{Cite journal | last1 = Martin | first1 = J. | doi = 10.4169/074683410X475083 | title = The Helen of Geometry | journal = The College Mathematics Journal | volume = 41 | pages = 17–28 | year = 2010 | s2cid = 55099463 }}
</ref>

<ref name=Cantor>
{{citation
| title=Vorlesungen über Geschichte der Mathematik, Bd. 2
| last=Cantor
| first=Moritz
| publisher=B. G. Teubner
| location=Leipzig
| year=1892
| url=https://archive.org/details/vorlesungenuberg00cant
| oclc=25376971
}}
</ref>

<ref name=Roidt>
{{cite thesis
| title=Cycloids and Paths
| last=Roidt
| first=Tom
| year=2011
| type=MS
| page=4
| url=http://www.web.pdx.edu/~caughman/Cycloids%20and%20Paths.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.web.pdx.edu/~caughman/Cycloids%20and%20Paths.pdf |archive-date=2022-10-09 |url-status=live
| publisher=Portland State University
}}
</ref>

<ref name=Whitman>
{{citation 
| title=Some historical notes on the cycloid
| journal=The American Mathematical Monthly
| volume=50
| number=5
| date=May 1943
| pages=309&ndash;315
| jstor=2302830
| last=Whitman
| first=E. A.
| doi=10.2307/2302830
}} {{subscription required}}</ref>

<ref name=Tannery>
{{citation
| last=Tannery
| first=Paul
| author-link=Paul Tannery
| year=1883
| title=Pour l'histoire des lignes et surfaces courbes dans l'antiquité 
| journal=Bulletin des sciences mathématiques et astronomiques
| series=Ser. 2
| volume=7
| department=Mélanges
| pages=278–291
| quote-page=284
| url=https://archive.org/details/bulletindesscie21publgoog/page/284/mode/1up
| quote=Avant de quitter la citation de Jamblique, j'ajouterai que, dans la courbe de ''double mouvement'' de Carpos, il est difficile de ne pas reconnaître la cycloïde dont la génération si simple n'a pas dû échapper aux anciens.
| trans-quote=Before leaving the citation of Iamblichus, I will add that, in the curve of ''double movement'' of [[Carpus of Antioch|Carpus]], it is difficult not to recognize the cycloid, whose so-simple generation couldn't have escaped the ancients.
}} (cited in Whitman 1943);  
</ref>

<ref name=Walker>
{{citation
| last=Walker
| first=Evelyn
| title=A Study of Roberval's Traité des Indivisibles
| publisher=Columbia University
| year=1932
}} (cited in Whitman 1943);
</ref>

<ref name=DartmouthAlumniMagazine>
{{citation
| title= 101 Reasons to Love Dartmouth
| url=https://dartmouthalumnimagazine.com/articles/101-reasons-love-dartmouth
| publisher=Dartmouth Alumni Magazine
| year=2016
}}
</ref>

}}

== Further reading ==
* ''An application from physics'': Ghatak, A. & Mahadevan, L. Crack street: the cycloidal wake of a [[cylinder (geometry)|cylinder]] tearing through a sheet. Physical Review Letters, 91, (2003). [http://link.aps.org/abstract/PRL/v91/e215507 link.aps.org]
* Edward Kasner & James Newman (1940) [[Mathematics and the Imagination]], pp 196&ndash;200, [[Simon & Schuster]].
* {{cite book | author = Wells D | author-link=David G. Wells | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 445–47 | url = https://archive.org/details/penguindictionar0000well | url-access = registration }}

== External links ==
* {{MacTutor | class=Curves | id=Cycloid | title=Cycloid}}
* {{MathWorld | urlname=Cycloid | title=Cycloid}} Retrieved April 27, 2007.
* [http://www.cut-the-knot.org/pythagoras/cycloids.shtml Cycloids] at [[cut-the-knot]]
* [http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=02260001&seq=9 A Treatise on The Cycloid and all forms of Cycloidal Curves], monograph by Richard A. Proctor, B.A. posted by [https://web.archive.org/web/20060127232450/http://historical.library.cornell.edu/math/index.html Cornell University Library].
* ''[http://demonstrations.wolfram.com/CycloidCurves/ Cycloid Curves]'' by Sean Madsen with contributions by David von Seggern, [[Wolfram Demonstrations Project]].
* [https://web.archive.org/web/20120604030020/http://communities.ptc.com/videos/2077 Cycloid on PlanetPTC (Mathcad)]
* [https://web.archive.org/web/20160616201751/http://www.its.caltech.edu/~mamikon/VisualCalc.html A VISUAL Approach to CALCULUS problems] by Tom Apostol

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[[Category:Roulettes (curve)]]