Parametric Lab
Skill goal: Trace curves given by x = f(t), y = g(t): circles, Lissajous figures, and projectile motion.
1 · Observe
Press Play and watch the point trace the curve as t grows.
2 · Manipulate
3 · Predict
Set Lissajous with a = 1, b = 2. Before pressing Play: will the curve cross itself? How many “lobes” will it have?
4 · See
5 · Explain
Parametric equations describe motion: at each time t the point is at (x(t), y(t)). A Lissajous figure with frequency ratio a:b closes into a pattern with a horizontal lobes and b vertical lobes. Projectile motion splits into horizontal motion at constant speed and vertical motion slowed by gravity — that is why the path is a parabola.
6 · Challenge
Q: x = sin(t), y = sin(2t) traces a figure-eight. Which ratio a:b makes it?