Cycloids

Date: 06/16/98 at 16:07:53
From: David Marek
Subject: Cycloids

What are cycloids, and how do they differ from sine curves?  Do they 
have a general equation?

Date: 06/17/98 at 12:53:11
From: Doctor Peterson
Subject: Re: Cycloids

Hi, David. Your question is phrased in an interesting way, because the 
difference between these two curves is worth thinking about.

Both can be thought of in terms of a point on the circumference of a 
rolling circle. To make a cycloid, you just follow the path of the 
point itself:


    |
  2r|          ***                     ***                +++
    |   *               *       *               *       *     +
    |*                     * *                     * * +   +   +
    *                       *                       *   +     +
   0*-----------------------*-----------------------*-----+++-----
    0                    2 pi r                  4 pi r


The equation is best expressed in parametric form, with both x and y 
depending on a "time" variable t:

    x = r(t - sin t)
    y = r(1 - cos t)

To make a sine curve, you ignore the left-right motion of the point 
around the circle, and just plot its height, with the x-coordinate 
being the location of the center of the circle; you can think of it as 
projecting the point onto the vertical diameter of the circle and 
drawing a point there:


    |
  2r|          ***                     ***                +++
    |       *       *               *       *           +--*  +
    |     *           *           *           *        +   +   +
    |   *               *       *               *       +     +
   0**---------------------***---------------------***----+++-----
    0                     2 pi r                 4 pi r


The parametric equations here would be

    x = rt
    y = r(1 - cos t)

Yes, this is really y = r(1 - cos x/r) - not the sine, but it's the 
same shape. Showing it this way makes the relation especially clear.

For more about cycloids, see these sites on the Web:

Cycloid - MacTutor Math History Archives, St. Andrews
 http://www-history.mcs.st-and.ac.uk/history/Curves/Cycloid.html   
   (For Xah Lee's Special Plane Curves: Cycloid, don't miss the link 
    at the bottom of this page under Other Web sites to California, 
    USA.)

A Java applet from International Education Software (IES)
 http://www.ies.co.jp/math/java/cycloid/cycloid.html   

Mechanics - School of Physics, Univ. of Melbourne:
 http://www.ph.unimelb.edu.au/lecdem/mg3.htm   

-Doctors Peterson and Sarah,  The Math Forum
 http://mathforum.org/dr.math/