Consider a wheel of radius **r**. While rolling, the wheel
rolls on an ant and the ant is stuck on its rim. Assume the ant
is not moving,say the wheel squashed it. What does the path of
the ant look like as the wheel rolls in a forward direction? First
consider the following:

The wheel rolls a distance **rt**. This is the arc length
of the arc between the Ant's new position, **Ant',** and its
its old position,**T**. The new position of the ant is at **Ant'**.Notice
it has moved a horizontal distance given by **rt-rsint.** It
has moved a vertical distance given by **r-rcost**. Now we
have x,y coordinate pairs that describes the path of the ant as
the wheel rolls. The coordinates are in terms of the angle the
wheel rolls,** t**.The equations: **x=rt-rsint** and **y=r-rcost
**are called the **parametric equations**
for the curve that describes the ant's path, **t
is called a parameter**.

We can use the graphing calculator to graph parametric equations: The equation I put in was:

. This is assuming the radius of my wheel is 1 unit. The graph that results when the parameter t ranges from 0 to 20 is:

The curve that results is called a cycloid. This parametric equation can also be plotted using a spreadsheet program.

A calculator such as the TI-83 also can plot this parametric equation. The commands you need to put in to plot a parametric equation are:

Note: when a key is to be pressed it will be displayed as green.

1. Press **Mode v v v (**three down arrows) **> **(right arrow)
**Enter **to
choose Par - this put you in Parametric Mode.

2. Press **Y=** and clear anything there.Now press the following
keys:

x,T,,n
**- sin **x,T,,n **) Enter**

This enters the formula we obtained for x assuming a radius of 1. Now the cursor is in place to enter the formula we obtained for y. To do this Press:

**1 - cos **x,T,,n **)**

Now we need to turn STAT Plot off for PLOT1. Press:

**2nd Y= **(Stat Plot)

**If you see Plot1...On then Press**

**Enter **To change to Off

**> Enter **to select Off

**Now when you see Plot1..Off Press**

**Window **set
Tmin to 0 and Tmax to 12.6, xmin = 0 and xmax = 13, and ymin =0,
and ymax = 2.

Now Press:
**Graph**.

If everything went ok you should see the cycloid.

To see spreadsheet solution,or to return to the spreadsheet solution

**For
historical information on the cycloid Click here.**