Visualizing Parametric Equations for the Cycloid

Play with the applet to get a feel for it.

Set the applet as follows

  1. Set a = 1 and set b = 1.
    (When a and b are equal we get the cycloid.)
  2. Check the 'trace' box.
  3. Check the 'velocity vector' box.

The following questions will guide you through a visual exploration of the cycloid and related curves.

  1. Describe a real world system that this applet models.
  2. What is the yellow dot in your model?
  3. Notice that the radius of the wheel and the length of the blue strut can be adjusted.
  4. Select 'trace' and animate the system.
  5. For what values of the parameters does the yellow curve cross the x-axis?
  6. When it does cross the axis, what is the direction of the tangent vector? Please explain.
  7. When 'trace' is selected, a purple line is drawn. What does it represent, in the real world system you described above?
  8. Carry out the following vector manipulations:

Exploring the cusp of the cycloid

Set the parameters a and b back to 1 and turn trace on.
  1. Grab the circle with your mouse and drag it to the right.
  2. The 'cusps' are the sharp points on the cycloid. What happens to the velocity vector at these points?
  3. Now set b to 3.5 and drag the circle to the right.
    What happened to the cusp? What happens to the velocity vector at the bottom of the trajectory?
  4. Repeat this with b = .5.