EMAT 6680 :: Clay Kitchings
:: Cycloids and Parametric Equations :: Assignment 10, Problem 12

A **cycloid** is the locus of a point on a circle that
rolls along a line. Write parametric equations for the cycloid and graph it.
Consider also a GSP construction of the cycloid.

Our task is to
come up with a parametric equation (or equations) to plot a cycloid.

My first
inclination was to consider a situation such as |sin t|. So, I tried the following parametric
and got the corresponding graph:

Obviously I wasnÕt
Òfishing in the right hole.Ó I knew at once that my t needed to be in the x-row
and the other function (presumably a trig function of some type) needs to be in
the y-row.

I tried the
following:

Now, IÕve managed
to doÉ not very much! I need to
somehow find the absolute value here to try to make the function become more of
the cycloid path. However,
Graphing Calculator does not have an absolute value function built in the
program.

Therefore, I
decided to ÒsettleÓ for taking the square root as follows:

The ÒholesÓ in the
graph are not surprising since the sine function yields negative values on [,
2],
and the square root function is not defined for negative values. Therefore, the
graph isnÕt defined on those particular intervals.

My next idea was
to create another parametric equation and translate the graph by a factor of .

The following
screen capture displays the results:

Now we are getting
quite close to the cycloid graph.
However, I would like to make this graph using one equation instead of
two. I decided to revert back to my absolute value idea. Squaring the sine function and then
taking an additional square root produces something that resembles an absolute
value situation:

This parametric
equation represents a cycloid such as is shown in the GSP file at the beginning
of this assignment.