Cycloid

Date: 01/30/98 at 18:04:34
From: Chris
Subject: What is a cycloid and what does it do?

Hi, 

My school is having an Ed-Fair (education fair) in a few weeks. I 
decided to do my project on math. My math teacher suggested I might 
want to do a project on cycloids. I took her recommendation and now I 
am stuck.

This is what I know so far: A cycloid is the curve traced by a fixed 
point on the radius of a circle (or it's extension) while rolling  
across the x axis (given that it doesn't slip). Also, two equations 
can be made from cycloids. To understand these equations you should 
know x = the distance of the curve along the x axis, r = radius of 
circle, y = the distance of the curve along the y axis, theta = the 
angle between the line of the radius and the vertical line through 
the center of the circle. The equations are x = r(theta - sin * theta) 
and y = r(1 - cos * theta). 

Could you please tell me the purpose of cycloids and more about them?

Date: 01/31/98 at 08:03:38
From: Doctor Pete
Subject: Re: What is a cycloid and what does it do?

Hi,

So far, you're doing fine.  Some ideas that would help when presenting 
your project are:

  * draw a picture of a cycloid.  Include the coordinate axes and the
    circle.  How would you visually interpret the values x, y, r, and
    theta?

  * generalize cycloids.  That is, what would happen if you took a
    point along the radius of the circle, but not "on" the circle's
    circumference?  What is the curve traced by this point?  What if
    this point is inside the circle?  Outside?

  * What kinds of curves would you get if you rolled circles around
    other circles, as opposed to a straight line?  How do these curves
    change as the relative sizes of these circles change?  In
    particular, what curve do you get when you roll two circles of
    equal size around each other?

If you want more specific information about cycloids, here is some 
stuff I know.  

Take a marble and a curved ramp. If you roll the marble down the ramp, 
it takes a certain amount of time for it to reach the bottom. Now, how 
should the ramp be curved so that the marble will roll to the bottom 
in the shortest amount of time? The answer is that the ramp should be 
half a cycloid (turned upside-down). It makes sense, because initally 
you want the marble to go fast, so the ramp should be steep, but once 
it's going fast, you want it to move forward, so you need to curve it 
out since the steepness doesn't take it very far. 

This problem was considered by Bernoulli and was referred to as the 
brachistochrome problem.  Isaac Newton was the first to solve it 
correctly; it is a classic use of the calculus of variations.

-Doctor Pete,  The Math Forum
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