A Project on Cycloids

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.html   

Hope this helps.

- Doctor Jaime, The Math Forum
  http://mathforum.org/dr.math/