How Long Is A Bad Ellipse Question?


Something came to mind while thinking about that failed grading scheme for multivariable calculus. I’d taught it two summers, and the first time around — when I didn’t try the alternate grading scheme — I made what everyone assured me was a common mistake.

One of the techniques taught in multivariable calculus is how to compute the length of a curve. There are a couple of ways of doing this, but you can think of them as variations on the same idea: imagine the curve as a track, and imagine that there’s a dot which moves along that track over some stretch of time. Then, if you know how quickly the dot is moving at each moment in time, you can figure out how long the track is, in much the same way you’d know that your parents’ place is 35 miles away if it takes you 35 minutes of travelling at 60 miles per hour to get there. There are details to be filled in here, which is why this is fit in an advanced calculus course.

Anyway, the introduction of this, and the homeworks, start out with pretty simple curves — straight lines, for example, or circles — because they’re easy to understand, and the student can tell offhand if the answer she got was right, and the calculus involved is easy. You can focus energy on learning the concept instead of integrating bizarre or unpleasant functions. But this also makes it harder to come up with a fresh problem for the exams: the student knowing how to find the length of a parabola segment or the circumference of a circle might reflect mastering the idea, or just that they remembered it from class.

So for the exam I assigned a simple variant, something we hadn’t done in class but was surely close enough that I didn’t need to work the problem out before printing up and handing out the exams. I’m sure it will shock you that an instructor might give out on an exam a problem he hasn’t actually solved already, but, I promise you, sometimes even teachers who aren’t grad students taking summer courses will do this. Usually it’s all right. Here’s where it wasn’t.

The problem I gave was to work out the circumference of an ellipse, which I figured would involve calculations that looked a lot like those of the circumference of the circle, done in class, without being too identical. And they were successfully not identical. The circumference of an ellipse is actually a really, fundamentally hard problem, something that can’t be done using any formula anyone taking a multivariable calculus course would know. The search for an expression describing the circumference of an ellipse created the field known as Elliptic Integrals, looked on even by mathematicians with an almost unique terror as a frightfully complicated field. It’s a fruitful one, but even a glance at the introduction to the topic, from the indefatigable Handbook of Mathematical Functions by Abromowitz and Stegun, shows what a cruel thing I inflicted on my students. They made awfully good efforts before finally giving up.

I apologized to them the next class day, and owned up to what I’d done, and I think they all accepted this as one of those dumb things teachers sometimes do.

While I’m at it, I should point out that while there isn’t any elementary exact formula for the circumference of an ellipse, there are outstandingly good approximations. John D Cook’s blog just happened to bring up one from Srinivasa Ramanujan which is a marvel in reasonable clarity and ease of use, although it’s not going to beat out “2 π r” anytime soon.

For this formula, you let a and b be the longest and the shortest radius of the ellipse. To make the formula more tolerable we then define another parameter, \lambda = \frac{a - b}{a + b}. The circumference C of the ellipse is then, approximately, equal to

C \approx \pi \left(a + b\right)\left(1 + \frac{3\lambda^2}{10 + \sqrt{4 - 3\lambda^2}}\right)

Cook notes that the error here is on the order of \lambda^{10}, but we can make this more physically obvious. Halley’s comet gets as far from the Sun as Pluto gets (at its closest), and it gets closer to the sun than Venus is. The difference between the actual circumference of the ellipse going out that far, and Ramanujan’s estimate, is less than the length of United States Route 66, the path from Chicago to Los Angeles. Most of us can accept this sort of error.

My sort of error, less so, but I’ve tried to be more careful about assigning problems that actually can’t possibly be done since then, and I don’t remember doing worse than giving problems that are more tediously long than I would have wanted since. And I’ve been careful to say that if one can’t finish a question, then at least describe in detail how one wants to finish it, which would have made the ellipse problem something at least conceivably doable; occasionally students even take me up on that.

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