Cycloid

Rolling circle draws a cycloid

The cycloid (from the Greek. Κυκλοειδής "round") is a flat transcendental curve .

A cycloid is defined kinematically as the trajectory of a fixed point of a generating circle (radius $r$ ${\ displaystyle r}$ $r$ ) rolling without sliding in a straight line .

Content

Equations

We take the horizontal coordinate axis as the straight line along which the generating circle of radius $r$ ${\ displaystyle r}$ . The cycloid is described:

parametrically
$x=rt-r\sin t$ ${\ displaystyle x = rt-r \ sin t}$
$y=r-r\cos t$ ${\ displaystyle y = rr \ cos t}$ .
equation in cartesian coordinates
$x=r\arccos {\frac {r-y}{r}}-{\sqrt {2ry-y^{2}}}$ ${\ displaystyle x = r \ arccos {\ frac {ry} {r}} - {\ sqrt {2ry-y ^ {2}}}}$ .
as a solution to a differential equation
$\left({\frac {dy}{dx}}\right)^{2}={\frac {2r-y}{y}}$ ${\ displaystyle \ left ({\ frac {dy} {dx}} \ right) ^ {2} = {\ frac {2r-y} {y}}}$ .

Properties

Tautochronism of cycloids

Oscillations with a cycloidal regulator.

Cycloid - a periodic function along the abscissa, with a period $2\pi r$ ${\ displaystyle 2 \ pi r}$ . It is convenient to take singular points ( return points ) of the form beyond the boundaries of the period $t=2\pi k$ ${\ displaystyle t = 2 \ pi k}$ where $k$ ${\ displaystyle k}$ Is an arbitrary integer.
To draw a tangent to the cycloid at its arbitrary point A, it is enough to connect this point with the top point of the generating circle. Combining A with the bottom point of the generating circle, we get the normal .
The length of the cycloid arch is $8r$ ${\ displaystyle 8r}$ . This property was discovered by Christopher Wren ( 1658 ). The dependence of the length of the cycloid arc (s) on the parameter t is as follows ^[1] : $s(t)=4r(1-\cos {t \over 2})$ ${\ displaystyle s (t) = 4r (1- \ cos {t \ over 2})}$ .
The area under each arch of the cycloid is three times larger than the area of the generating circle. Torricelli said that Galileo discovered this fact experimentally: he compared the weight of the plates with a circle and with an arch of cycloids. ^[2] Mathematically, this fact was first proved by Roberval around 1634 using the indivisible method .
The radius of curvature of the first arch of the cycloid is $4r\sin {\frac {t}{2}}$ ${\ displaystyle 4r \ sin {\ frac {t} {2}}}$ .
The “inverted” cycloid is a speedy descent curve ( brachistochrone ). Moreover, it also has the property of tautochronism : a heavy body placed anywhere in the arch of the cycloid reaches the horizontal in the same time.
The oscillation period of a material point moving along an inverted cycloid does not depend on the amplitude . (A direct consequence of tautochronism).
The evolute of a cycloid is a cycloid congruent to the original one and parallelly shifted from the original one so that the vertices become “ points ”.
- The last two properties discovered by Huygens were used by him to create precise mechanical watches .
The details of machines that simultaneously perform uniform rotational and translational motion describe cycloidal curves : cycloid, epicycloid , hypocycloid , trochoid , astroid ( compare the construction of Bernoulli lemniscates ).

Historical Review

The first of the scientists drew attention to the cycloid of Nikolai Kuzansky in the 15th century and Charles de Bovel in his work in 1501. But a serious study of this curve began only in the 17th century .

The name of the cycloid was invented by Galileo (in France this curve was first called roulette ). A substantial study of cycloids was conducted by a contemporary of Galileo Mersenne . Among transcendental curves , i.e., curves whose equation cannot be written as a polynomial in $x, y$ ${\ displaystyle x, y}$ , cycloid is the first of those studied.

Pascal wrote about the cycloid ^[3] ^[4] :

Roulette is a line so ordinary that after a straight line and a circle there is no more common line; it is so often drawn in front of everyone’s eyes that one should be surprised at how the ancients did not consider it ... for it is nothing more than a path described in the air with a nail of a wheel ...
Original text (Fr.)
La Roulette est une ligne si commune, qu'apres la droitte, & la circulaire, il n'y en a point de si frequente; Et elle se décrit si fouuent aux yeux de tout le monde, qu'il ya lieu de s'estonner qu'elle n'ait point esté considerée par les anciens, dans lesquels on n'en trouue rien: Car ce n'est autre chose que le chemin que fait en l'air, le clou d'une rouë ...

The new curve quickly gained popularity and underwent a deep analysis, in which Descartes , Fermat , Newton , Leibniz , the Bernoulli brothers and other luminaries of science of the XVII-XVIII centuries participated. The methods of mathematical analysis that appeared in those years were actively honed on the cycloid.

The fact that the analytical study of the cycloid turned out to be as successful as the analysis of algebraic curves made a great impression and became an important argument in favor of the “equation of rights” of algebraic and transcendental curves.

Notes

↑ Arkhipov G.I. , Sadovnichy V.A. , Chubarikov V.N. Lectures on Mathematical Analysis / Ed. V. A. Sadovnichogo. - 2nd ed. - M .: Higher school , 2000 .-- S. 261. - 695 p. - 8000 copies. - ISBN 5-06-003955-2 .
↑ Aleksandrova N.V. History of mathematical terms, concepts, designations: Dictionary Dictionary, ed. 3rd - St. Petersburg: LCI, 2008 .-- S. 213. - 248 p. - ISBN 978-5-382-00839-4 .
↑ Klyaus E.M., Pogrebysky I.B. , Frankfurt W.I. Pascal. - M .: Nauka , 1971. - S. 191. - ( Scientific and biographical literature ). - 10,000 copies.
↑ Pascal, Blaise. Histoire de la roulette, appellée autrement la trochoïde, ou la cycloïde, où l'on rapporte par quels degrez on est arrivé à la connoissance de la nature de cette ligne . 10 octobre 1658. P.1.

Literature

Berman G.N. Cycloid. M., Science, 1980, 112 pp.
Gindikin S.G. Stories about physicists and mathematicians . - third edition, expanded. - M .: MCLMO, 2001 .-- S. 126-165. - ISBN 5-900916-83-9 .
Mathematical Encyclopedia (in 5 volumes) . - M .: Soviet Encyclopedia , 1982.- T. 5.
Markushevich A. I. Remarkable Curves , Popular Lectures in Mathematics , Issue 4, Science 1978 , p. 32.

Links

Cycloidal curves ( Animation ).

[1] Arkhipov G.I. , Sadovnichy V.A. , Chubarikov V.N. Lectures on Mathematical Analysis / Ed. V. A. Sadovnichogo. - 2nd ed. - M .: Higher school , 2000 .-- S. 261. - 695 p. - 8000 copies. - ISBN 5-06-003955-2 .

[2] Aleksandrova N.V. History of mathematical terms, concepts, designations: Dictionary Dictionary, ed. 3rd - St. Petersburg: LCI, 2008 .-- S. 213. - 248 p. - ISBN 978-5-382-00839-4 .

[3] Klyaus E.M., Pogrebysky I.B. , Frankfurt W.I. Pascal. - M .: Nauka , 1971. - S. 191. - ( Scientific and biographical literature ). - 10,000 copies.

[4] Pascal, Blaise. Histoire de la roulette, appellée autrement la trochoïde, ou la cycloïde, où l'on rapporte par quels degrez on est arrivé à la connoissance de la nature de cette ligne . 10 octobre 1658. P.1.