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A Project on Cycloids

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Date: 01/16/99 at 19:36:59
From: Jenny
Subject: Cycloids and their formulas

Hi,

My math class was assigned to research and write a math fair project. I
chose to do cycloids, something I knew nothing about and thought I
would try. However, they are confusing me.

I know that the parametric formulas are x = a(theta - sin*theta) and
y = a(1 - cos*theta), where (a) is the radius of the circle that is
rolling on the straight line. I read that theta is the variable angle
through which the circle rolls, but how can I find theta? Can I
substitute all the values from 0 to 2pi as degree measures in? Now, how
is this formula used and what does it figure out? Would it find all of
the points on a cycloid? If I used any value for (a), would the values
of x and y give me the coordinated to graph the cycloid? Also, does
this equation have anything to do with time? Also, for one cusp of the
cycloid to be drawn, theta would take the angle measures of 0-360
degrees.

Thanks.
```

```
Date: 01/17/99 at 06:25:48
From: Doctor Jaime
Subject: Re: Cycloids and their formulas

Hello Jenny,

For each value of theta that you choose, you get a point (x,y) of the
cycloid. But you must be aware that if you are taking degrees, you get
a complete cycloid when x goes from 0 to 360*a units, and if you choose
radians you get a complete cycloid when x goes from 0 to 2pi*a units.

Yes, if you take for theta the values from 0 to 2pi when computing cos
and sin in radians, you get a complete cycloid, and if you give values
for theta higher than 2pi you get more cycloids (and the same for
negative values of theta). You can also use degrees, but then the
cycloid will occupy the space when x goes from 0 to 360*a and other
intervals with equal length. Note that for each value of a, you get a
cycloid with different size.

The cycloid is related to time in interesting ways. First, it is a
Brachistochrone, that is a curve of descent in minimum time. It also
has the tautochrone property; that is, the property that a particle P
sliding on a cycloid will exhibit simple harmonic motion and the period
will be independent of the starting point.

Note that to draw the cusp of a cycloid, you must label the graph in
this case with 0-360 intervals.

At these sites you can find information on the cycloid, including its
history:

the MacTutor History of Mathematics Archives:
http://www-history.mcs.st-and.ac.uk/history/Curves/Cycloid.html

from Xah Lee:
http://www.best.com/~xah/SpecialPlaneCurves_dir/Cycloid_dir/cycloid.html

from the University of Melbourne's School of Physics
http://www.ph.unimelb.edu.au/lecdem/mg3.htm

from Alexander Bogomolny:
http://www.cut-the-knot.org/pythagoras/cycloids.shtml

Hope this helps.

- Doctor Jaime, The Math Forum
http://mathforum.org/dr.math/
```
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High School Functions
High School Projects

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