Love and other parametric curves

The cupid curve from Wolfram Alpha.

This morning, Worlfram Alpha greeted me on facebook with what they call the “cupid curve”, pictured on the left. You can look it up on the actual site to get the parametric equations that give you this curve. Needless to say, they are quite involved. However, it is kind of cool that you can find parametric equations for lots of different themes. I then searched for “fun curves” which yielded 417 results. There are the bunny curve, the jellyfish curve, the My Little Pony curve, and the Borat curve, to name a few. They all come with their defining parametric equations (and sometimes with the equation for the area enclosed by the curve, like with the bulldog curve). Besides entertaining me for a large chunk of my morning, I’m not entirely sure how this could be useful for anyone. I guess in a Calculus class, when teaching about parametric curves, these could make for some fun examples. But doing calculus with these curves would be beyond anything you would want to teach (depends on the curve I guess).

A Lissajous curve with parametric equations x= sin(3t+\pi/2) and y=sint.

I think the most complicated thing I have made my students work with are Lissajous curves (pictured left), and they are not even that bad (or not nearly as bad as the cupid curve).  Googling “famous parametric curves” led me to this webpage by Gutsavo Gordillo, which also has all the parametric equations and Mathematica commands needed to graph them. This might be a better place to start with students. But I wonder if it would be an interesting project to give students these more complicated pictures, or to have them draw and define their own? Is that even reasonable? I actually have no idea, but it seems to me like it would be too difficult. I may still, next time I teach Calculus, have the students look at some of these examples and their parametric equations for fun (like I was just doing). Any ideas on how to use this database in Wolfram Alpha for teaching are welcome!

As always, please share your thoughts in the comments below, and hope your Valentine’s day is turning out to be as fun as mine!