In my multivariable calculus we’re talking about parametric curves.
I’m using this applet for displaying parametric curves. You can use predefined curves or enter your own. Although the applet is on my web page, it was created by Marc Renault, a friend who teaches down the road at Shippensburg University. I only tweaked it slightly when I posted it on my page.
Marc also created this nice applet illustrating the creation of the cycloid.
Speaking of the cycloid, after the deriving the parametric equations for the cycloid I spent 10 minutes telling my class about the tautochrone and brachistochrone problems.
They loved the story about the history of the brachistochrone problem—the way that Johann Bernoulli taunted Newton, Newton’s 12-hour after-work solution, Newton’s annoyance (“I do not love to be dunned [pestered] and teased by foreigners about mathematical things…”), his anonymous solution, and Bernoulli’s reply (“I recognize the lion by his paw.”).
This afternoon, after teaching the class, I discovered that the tautochrone problem is mentioned in Moby Dick: “[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time.” Cool!