Calculus III - Lab 1b:
Parametric Equations

Parametric graphs

Explanation

A very well-known parametric curve is the cycloid.  This is the curve you get if you look at the path traced out by a point on the edge of a wheel as it rolls along a surface (double-click on the animation below to see this; you may want to slow it down a bit).

There are different ways to find the parametric equations for this, but a particularly cool way is to use vectors.  Basically, any point on the curve can be located by finding the vector from the origin to the center of the circle and the vector from the center of the circle to the point on its rim and adding those two vectors together (see the diagram below).  The resultant vector is a vector from the origin to the point on the rim; it will trace out the curve.

What is the point of doing this?  Well, it is very easy to find a formula for the coordinates of the center of the circle (it is just  moving in a straight line) and it is very easy to find the coordinates of a point on the circle with respect to the center of the circle.  

So, we can define a function wheel[t] that gives the coordinates of the point with respect to the edge of the circle using standard parametric equations for a circle (at time t).  We will assume that the point starts at the origin (i.e., at the bottom of the circle) and that the circle has a radius of 1 (the basic shape remains the same for any radius).  Execute the following command to make the definition; notice that there are two components in the { }'s - the first one is the x coordinate and the second one is the y coordinate:

wheel[t_] := {-  Sin[t], - Cos[t]}

(Be sure you understand why these need to be negative and why I chose x to be Sin and y to be Cos.)

Now, to find the parametric equations for the center of the circle at time t, you need to figure out how far it has traveled after t seconds (minutes/hours/years/whatever).  After time t, the circle will have rotated through an arc length of rt units (remember that arc length is s = rθ), which is how far the circle will have actually traveled (assuming the wheel doesn't "slip") in the x direction.  It's y coordinate will just be r (since the circle is sitting on the x axis).  Therefore, if we take r = 1 as above, we can define a function (execute it now):

centerLinePath[t_] := {  t, 1}

Now, to find parametric equations for the cycloid, we simply add these two vectors (i.e., parametric equations) together:

wheel[t] + centerLinePath[t]

This is the formula for a cycloid generated by a circle of radius 1.  You can graph this by using the following command:

ParametricPlot[
      Evaluate[wheel[t] + centerLinePath[t]], {t, 0, 4  π}, 
      PlotRange -> {0, 6}]

(The Evaluate part isn't really essential, but it does stop Mathematica from whining about not being able to compile the functions.  The PlotRange part tells Mathematica which y values to plot; otherwise it uses its own best guess which distors the graph quite a bit.)

Questions:

Find equations for a cycloid generated by a circle rolling along underneath the x axis (i.e., the graph will look the same, but will be upside down).  Graph it.

Find equations for a cycloid generated by a circle rolling along on top of the x axis, but of radius r.  Graph it for some specific value of r.

Find the arc length of one arch of a cycloid (either one) generated by a circle of radius r.

Find the area under one arch of a cycloid generated by a circle of radius r

Extra Credit:  Find parametric equations for the curve you would get if you let a circle of radius 1 roll around the inside of another circle (of radius r), centered at the origin.  (This is called a hypocycloid.)  Graph some of these for various values of r.

Hint:  Use the same technique we have used for the cycloid.  In fact, you can use the same wheel function we defined earlier (if you start it from the right place) for the inner circle.  So, all you really need to do here is to come up with equations  from the origin to the center of the rolling circle, call it centerCircleInsidePath, and then add them together. 

The one tricky part to this is relating the two angles involved here.  Remember that the distance traveled by the center of the rolling circle is the same as the amount of arc length the rolling circle has "unrolled" (i.e., no slippage).  So, if you let the angle for the rolling circle be t (that way you don't need to change your equations) and the angle for the outer circle be θ, then the center of the rolling circle has moved a distance of (r - 1) θ while the rolling circle has "unrolled" an arc length of t (since its radius is 1).  Setting these equal, you can solve for θ in terms of t.

Extra Credit:  How would this change if the circle (of radius 1) were rolling around the outside of the circle (of radius r) centered on the origin?  Graph some of these for various values of r.

Hint:  This is almost identical to the last problem except for a few sign changes.