...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Tuesday, June 04, 2002

Ah, curvature. It was something of a bane for me in Vector Calculus class. I understood it enough to pass, but I didn't really grok it. I think I do, now.

So. Say you've got a squiggly (but not pathologically so) curve that you've drawn on a piece of paper. Say further that you want to be able to talk about how curved it is in various areas. Now, obviously, to do that, you've got to figure out what you mean by curvature. That's actually easier than it looks, if you squint just right. We can use our intuitive idea of what 'how curved something is' means as a guide.

Now. Here's the key insight. Look at a small part of your squiggle. Generally speaking, it's going to look like a small arc - like a piece of a circle. And that's all we need to get going. We're basically going to characterize the curvature of a piece of our squiggle by fitting a circle to it. We're going to characterize how curved a squiggle is at a point by saying how big a circle we can fit to it at that point. And we'll call that circle the 'circle of curvature', for future reference.

Let's think about circles for a bit. First, to completely describe a circle, all we need is a radius (well, it's center is nice too, but let's ignore that). Remember that. Now, our intuitive idea of 'curvature' tells us that a Really Big circle is less curved than a small one (make sure you see this!). After all, a chunk (aka arc) of a Really Big circle looks a lot like a straight line, and our intuition tells us that straight lines are, well, straight - they ain't curved.

So apply that back to squiggles. If a really big circle fits to a point on our squiggle, it's not curved much there. If a really small one fits to the squiggle at a point, then the squiggle is seriusly curved in that spot. In other words, the bigger the circle, the less the curvature. So, we can define curvature to be 1/(radius of the circle of curvature), and we now have a definition that fits with what we wanted: a careful way of thinking about how curvy something (or someone, if that's the application*) is.

Now, to actually find the curvature given an equation for a curve requires some calculus, which is much less interesting (and more mechanical) than having the concept above in the first place, IMHO. So I won't talk about that.

I wrote all that out to make sure the concept is properly sorted out in my head. Hopefully that'll do it.

* - What's math without some sexual innuendo?


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