|
A cycloid is a curve that a point on the perimeter of a wheel traces when rolling along the $x$ -axis without slipping. If the radius of the rolling wheel is $a$ , then the cycloid may be presented in the parametric form
where $\varphi$ expresses the angle rotated by the wheel around its center.
In what follows, a blue curve indicates a cycloid (or a portion thereof) and red line segments indicate radii of the wheel.
Below is a picture of the wheel on the $x$ -axis with $\varphi = 0$ .
As the wheel rolls, $\varphi$ increases. To obtain the cycloid, we keep track of the path along which the fixed point of the wheel has travelled.
After the wheel has completed a full turn, the cycloid takes a sharp turn due to the fact that the point hits the $x$ -axis, then begins travelling upwards again.
Thus, below is the graph of a cycloid for $a=1$ .
The graph of a cycloid for any $a$ can be obtained by replacing $1$ with $a$ and $2$ with $2a$ on the $y$ -axis of the graph above.
The length of one arc of the cycloid formed by one revolution of the circle (e.g. $0 \le \varphi \le 2\pi$ ) is
Therefore, the length of one arc of the cycloid is $8a$ , i.e. four times the diameter of the circle.
| 
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
(0,1) ... ...,-0.3){$x$} \psdots(0,0)(0,1) \rput[l](-1,0){.} \rput[a](0,2){.} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/7100/js/img5.png "\begin{pspicture}(-1,0)(1,2) \pscircle(0,1){1} \psline[linecolor=red](0,0)(0,1) ... ...,-0.3){$x$} \psdots(0,0)(0,1) \rput[l](-1,0){.} \rput[a](0,2){.} \end{pspicture}")
(0.00126,0.019215)(0... ....6) \rput[b](3.2,-0.3){$x$} \rput[l](-1,0){.} \rput[a](2.2,2){.} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/7100/js/img6.png "\begin{pspicture}(-1,0)(3.2,2) \pscurve[linecolor=blue](0,0)(0.00126,0.019215)(0... ....6) \rput[b](3.2,-0.3){$x$} \rput[l](-1,0){.} \rput[a](2.2,2){.} \end{pspicture}")
![\begin{pspicture}(-7,-1)(7,3) \pscurve[linecolor=blue]{-}(-6.5374,0.61732)(-6.43... ...a](6.6,-0.25){$x$} \rput[r](-0.22,2.35){$y$} \rput[l](-6.5,0){.} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/7100/js/img7.png "\begin{pspicture}(-7,-1)(7,3) \pscurve[linecolor=blue]{-}(-6.5374,0.61732)(-6.43... ...a](6.6,-0.25){$x$} \rput[r](-0.22,2.35){$y$} \rput[l](-6.5,0){.} \end{pspicture}")
