It's easy to find parametric equations for some curves. Graphs of functions, straight lines, and circles are the most familiar examples. This project focuses on circles.
First,
consider a point moving counterclockwise around a circle of radius 1 centered
at the origin.
The usual
parametric equations are x = cos(t), y = sin(t).
Exercises:
1. Suppose that t represents time, measured in seconds. How long does it take the point to move once around the circle?
2. If you replace t by 2t, what happens to the motion of the point? (You might want to graph both versions simultaneously. You can arrange that on the Graph Format screen.)
3. Modify the equations so that the point travels once around the circle every 5 seconds.
4. Modify the equations so that the rotation time is 5 seconds and the radius is 2.
5. Modify the equations so that the rotation time is 5 seconds and the radius is 2 and the center is (0,2).
6. Modify the equations so that the rotation time is k seconds, the radius is r, and the center is (a,b). Where is the moving point at time t=0? When does the moving point revisit this spot?
7. How can you make the point travel clockwise around the circle?