???????????????? ( cycloid ) ?????? ??? ???????? ?????? ??? ?????? ???????? ???????????, ????????????? ????????? ??? ?????????? ??????? ??????? ?????? ?????. ??? ???????? ?????????? .

?????? [ ???? ]

???????????????? ??????? ????????????? ???????? ????????????? (Nicholas of Cusa) ??????? ?????? ??????????? (Marin Mersenne) ????? ?????????????. 1599 ??? ??????????? ??????????????. 1634 ??? ??. ??. ??. ???????? (G.P. de Roberval) ???????????????????? ?????????? ??????, ??? ??????????? ?????????? ???????? ??? ?????? ?????? ???? ??????????. 1658 ??? ???????????? ?????? (Christopher Wren) ???? ?????????? ???????? ?????, ??????????? ?????????? ??????????? ???? ?????? ?????? ?? ??????????. 17 ??? ?????????? ?????????????????? ???????? ?????????? ???? ??????????????????? ???????????????? "?? ?????? ???? ?????????????" (The Helen of Geometers) ?? ??????????????. ^[1]

??????????? [ ???? ]

??????????? ????? ????????? r ???? ??????? ???????????? ???????????????????, ( x , y ) ??????????? ????? ???????????????????? ?????????? ???????????:

${\displaystyle x=r(t-\sin t)\,}$

${\displaystyle y=r(1-\cos t)\,}$

?????? ?????????? ????????? ?????? ( ????????? ) ?????????? ???????? t

??????????? t ??? ???????????, ??????? ?????????? ?????:

${\displaystyle x=rt}$

${\displaystyle y=r}$

???????????????????? ?????????? ??????????????????? t -? ??????? ?????????? ???? ????????????? ????????:

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

???????????????????? ???????????? ???? ????????? ??????? ??????:

${\displaystyle 0\leq t\leq 2\pi .\,}$

?????? [ ???? ]

r ???? ??????? ???????????? ??????????????? ???????????????????? ???????????? ????????? ?????????? ???????????:

${\displaystyle x=r(t-\sin t),\,}$

${\displaystyle y=r(1-\cos t),\,}$ ( ${\displaystyle 0\leq t\leq 2\pi .\,}$)

???????????,

${\displaystyle {\frac {dx}{dt}}=r(1-\cos t),}$

???????????? ???????? ???:

${\displaystyle {\begin{aligned}A&=\int _{t=0}^{t=2\pi }y\,dx=\int _{t=0}^{t=2\pi }r^{2}(1-\cos t)^{2}\,dt\\&=\left.r^{2}\left({\frac {3}{2}}t-2\sin t+{\frac {1}{2}}\cos t\sin t\right)\right|_{t=0}^{t=2\pi }\\&=3\pi r^{2}.\end{aligned}}}$

???????? ????? [ ???? ]

???????????????????? ?????????? ???????? ????? S ?????:

${\displaystyle {\begin{aligned}S&=\int _{t=0}^{t=2\pi }\left(\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dx}{dt}}\right)^{2}\right)^{1/2}\,dt\\&=\int _{t=0}^{t=2\pi }r{\sqrt {2-2\cos(t)}}\,dt\\&=\int _{t=0}^{t=2\pi }2r\sin \left({\frac {t}{2}}\right)\,dt\\&=8r.\end{aligned}}}$

?????????? ????????? [ ???? ]

???????????????????? ?????????? ?? ????????? ???? ?????????????.

• ???????? ???????????????? (Curtate cycloid):

??? ??????, ??? ???????? ?????? ???????? ???????????, ??????? ?????????????? ??????? ??? ?????? ?????? ????????? ?????????? ??????.

• ?????? ???????????????? (Prolate cycloid):

??? ?????? ??? ???????? ?????? ???????? ???????????, ??????? ???????????? ?????? ??????? ??? ?????? ?????? ????????? ?????????? ??????.

• ???????? (Trochoid):

????????????????, ???????? ????????????????, ?????? ???????????????? ???? ?????????? ??????????.

• ??????????????????? (Hypocycloid):

??? ?????? ??????? ??????? ?????????? ???????????? ?????? ?????????? ???????? ???????????, ??????? ?????????? ????????? ??? ?????? ?????? ????????? ?????????? ??????.

• ???????????????????? (Epicycloid):

??? ?????? ??????? ??????? ?????????? ??????????????? ?????? ?????????? ???????? ???????????, ??????? ?????????? ????????? ??? ?????? ?????? ????????? ?????????? ??????.

• ??????????? (Hypotrochoid):

????????? ??????????????????????? ???????. ????? ????? ???? ????????????? ??????, ??????? ?????????????? ?????? ?????????????.

• ?????????????? :

????????? ???????????????????????? ???????. ????? ????? ???? ????????????? ??????, ??????? ?????????????? ?????? ?????????????.

???? ???????????? ????????? ????????? ????????????????? .

????????????????, ???????????????????, ?????????????????????? ???????? ???????? ????????? ????????????? ?????????.

??????????? [ ???? ]

?????? ?????? [ ???? ]

• An application from physics : Ghatak, A. & Mahadevan, L. Crack street: the cycloidal wake of a cylinder tearing through a sheet. Physical Review Letters, 91, (2003). link.aps.org
• Edward Kasner & James Newman (1940) Mathematics and the Imagination , pp 196–200, Simon & Schuster .
• Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry . New York: Penguin Books. pp. 445?47. ??????????? ?????????? ???   0-14-011813-6 .

???? ?????????? [ ???? ]

• O'Connor, John J.; Robertson, Edmund F., "Cycloid" , MacTutor History of Mathematics archive , ????? ????????? ????????????? .
• Weisstein, Eric W., "Cycloid" , MathWorld . Retrieved April 27, 2007.
• Cycloids at cut-the-knot
• A Treatise on The Cycloid and all forms of Cycloidal Curves , monograph by Richard A. Proctor, B.A. posted by Cornell University Library .
• Cycloid Curves by Sean Madsen with contributions by David von Seggern, Wolfram Demonstrations Project .
• Cycloid on PlanetPTC (Mathcad ????????????? 2012-06-04 at the ??????? ?????????