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### 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/
```
Associated Topics:
High School Trigonometry

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