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Difference between revisions of "Cycloid"

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The plane transcendental curve that is the trajectory of a point of a circle rolling along a straight line (Fig. a).
 
The plane transcendental curve that is the trajectory of a point of a circle rolling along a straight line (Fig. a).
  
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The parametric equations are:
 
The parametric equations are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275401.png" /></td> </tr></table>
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$$x=rt-r\sin t,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275402.png" /></td> </tr></table>
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$$y=r-r\cos t,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275403.png" /> is the radius of the circle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275404.png" /> the angle of rotation of the circle. In Cartesian coordinates the equation is:
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where $r$ is the radius of the circle and $t$ the angle of rotation of the circle. In Cartesian coordinates the equation is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275405.png" /></td> </tr></table>
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$$x=r\arccos\frac{r-y}{r}-\sqrt{2ry-y^2}.$$
  
A cycloid is a periodic curve: the period (basis) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275406.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275408.png" /> are cusps. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c0275409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c02754010.png" /> are the so-called vertices. The area is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c02754011.png" />, the radius of curvature is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c02754012.png" />.
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A cycloid is a periodic curve: the period (basis) is $OO_1=2\pi r$. The points $O,O_k=(2k\pi r,0)$, $k=\pm1,\pm2,\ldots,$ are cusps. The points $A=(\pi r,2r)$ and $A_k=((2k+1)\pi r,2r)$ are the so-called vertices. The area is $S_{OAO_1O}=3\pi r^2$, the radius of curvature is $r_k=4r\sin(t/2)$.
  
 
If the curve is described by a point lying outside (inside) a circle rolling along a line, then it is called an extended, (or elongated, Fig. b), a contracted, (or shortened, Fig. c) cycloid or sometimes a trochoid.
 
If the curve is described by a point lying outside (inside) a circle rolling along a line, then it is called an extended, (or elongated, Fig. b), a contracted, (or shortened, Fig. c) cycloid or sometimes a trochoid.
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The parametric equations are
 
The parametric equations are
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c02754013.png" /></td> </tr></table>
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$$x=rt-d\sin t,$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c02754014.png" /></td> </tr></table>
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$$y=r-d\cos t,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c02754015.png" /> is the distance of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027540/c02754016.png" /> from the centre of the rolling circle.
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where $d$ is the distance of the point $M$ from the centre of the rolling circle.
  
 
The cycloid is a tautochronic (or isochronic) curve, that is, a curve for which the time of descent of a material point along this curve from a certain height under the action of gravity does not depend on the original position of the point on the curve.
 
The cycloid is a tautochronic (or isochronic) curve, that is, a curve for which the time of descent of a material point along this curve from a certain height under the action of gravity does not depend on the original position of the point on the curve.

Revision as of 10:51, 7 August 2014

The plane transcendental curve that is the trajectory of a point of a circle rolling along a straight line (Fig. a).

Figure: c027540a

The parametric equations are:

$$x=rt-r\sin t,$$

$$y=r-r\cos t,$$

where $r$ is the radius of the circle and $t$ the angle of rotation of the circle. In Cartesian coordinates the equation is:

$$x=r\arccos\frac{r-y}{r}-\sqrt{2ry-y^2}.$$

A cycloid is a periodic curve: the period (basis) is $OO_1=2\pi r$. The points $O,O_k=(2k\pi r,0)$, $k=\pm1,\pm2,\ldots,$ are cusps. The points $A=(\pi r,2r)$ and $A_k=((2k+1)\pi r,2r)$ are the so-called vertices. The area is $S_{OAO_1O}=3\pi r^2$, the radius of curvature is $r_k=4r\sin(t/2)$.

If the curve is described by a point lying outside (inside) a circle rolling along a line, then it is called an extended, (or elongated, Fig. b), a contracted, (or shortened, Fig. c) cycloid or sometimes a trochoid.

Figure: c027540b

Figure: c027540c

The parametric equations are

$$x=rt-d\sin t,$$

$$y=r-d\cos t,$$

where $d$ is the distance of the point $M$ from the centre of the rolling circle.

The cycloid is a tautochronic (or isochronic) curve, that is, a curve for which the time of descent of a material point along this curve from a certain height under the action of gravity does not depend on the original position of the point on the curve.


Comments

References

[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
How to Cite This Entry:
Cycloid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cycloid&oldid=11749
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article