## Visualizing Parametric Equations for the Cycloid

### Play with the applet to get a feel for it.

• It is color coordinated so that things with the same color are related.
• Try to figure out the relationships.

### 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.
• Are there values of these parameters for which the yellow dot sometimes moves to the left? What are they?
• Are there values for which the yellow dot is sometimes stationary? What are they?
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:
• Express the directed segment joining the origin to the left end of this purple line as a vector.
• Express the directed segment along the purple line from the left end to the right end as a vector. (It will depend upon theta).
• Express the blue segment directed from the center of the circle to the end as a vector. (This will also depend upon theta).
• Now express the position of the yellow dot as a vector-valued function of theta.

### 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.