Curvature formula, part 5

So let's sum up where we are so far. We're looking at this formula and trying to understand why it corresponds to curvature, why it tells you how much a curve actually curves.

The first thing we did is we noticed that this numerator corresponds to a certain cross product—the cross product between the first derivative and the second derivative of the function parameterizing the curve. The way we started to understand that is we say, well, the function parameterizing the curve, S of T, produces vectors whose tips trace out that curve itself.

Now, if you think about how one tip moves to the next, the direction that it needs to go, um, for that tip to move to the next one, that's what the first derivative tells you. When you treat that in an infinitesimal way, this is why you get tangent vectors along the curve. There's an entire series of videos on that for the derivative of a position vector-valued function, and they explain why the first derivative of a parametric function gives you tangent vectors to that curve.

If we draw all of those tangent vectors just in their own space, their own little S prime of t space, you get all of these vectors, and we're rooting them at the same point to be able to relate them more easily. The way that you move from the tip of one of those to the next one is given by the second derivative. It kind of plays the same role for the first derivative as the first derivative plays for the original parametric function.

Specifically, if you have a circumstance where the tangent vectors are just turning, the only thing they're doing is purely turning around, which is when your curve is actually curving. This corresponds to a case when the second derivative function is pretty much perpendicular as a vector. The vector it produces is perpendicular to the first derivative vector.

So this is loosely why the cross product is kind of a good measure of curving, because it tells you how perpendicular these guys are. But there's a bit of a catch. The original formula for curvature—the whole reason we're doing it with respect to arc length and not with respect to the parameter T—is that curvature doesn't really care about how you parameterize the function.

If you imagine zipping along it really quickly, so your tangent, your first derivative vectors are all super long, it shouldn't matter compared to crawling along it like a turtle. The curvature should just always be the same. But this is a problem if you think back to the cross product that we're now looking at, where you're taking the cross product between the first derivative and the second derivative.

Because if you were traveling along this curve twice as quickly, what that would mean is your first derivative vector—so I'll kind of draw it again over here—would be twice as long to indicate that you're going twice as fast. Similarly, your second derivative vector, to kind of keep up with that changing rate, would also be twice as long. As a result, the parallelogram that they trace out—I should actually kind of going off screen here—the parallelogram that they trace out would be actually four times as big, right? Because both of the vectors get scaled up.

The way that we really want to be thinking about this is not the tangent vector due to the derivative, but normalizing this. And this should kind of make sense because we're thinking in terms of unit tangent vectors for the curvature as a whole. So if you imagine instead kind of cutting off the vector to make sure that it's got a unit length, a length of one, what that means is you're taking the derivative vector and dividing it by its own magnitude, by the magnitude of that derivative vector.

Then similarly, we want to scale everything else down, so you're taking this and kind of scaling it down by what you need. The resulting parallelogram they trace out is a more pure measurement of how perpendicular they are without caring about how long they are. What this guy would be, by the way, then, this is the second derivative vector—not normalized with respect to itself—but we're still dividing. You know, the thing we're dividing by, as we scale everything down, is still just the size of that first derivative vector.

So this cross product, if we take the cross product between S prime normalized, S double prime—oh no, no, no, sorry—S prime itself, and I should be saying vectors for all of these. These are all vectors. If we take the cross product between that and S double prime scaled down by that same value, that's still S prime.

So it's not normalized; this is just scaled down by S prime. This here is a more pure measurement of how perpendicular the second derivative vector is to the first, and the reason we don't really care about the second derivative being normalized is if it was the case that, you know, the second derivative was really, really strong and wasn't necessarily a unit vector, that's fine.

That's just telling us that the tangent vector turns much more quickly, and the curvature should be higher. In fact, it turns out that this whole expression is the derivative of the unit tangent vector T, that unit tangent vector that I've talked about a lot with respect to the parameter T.

So whatever the parameter of your original function is. Now, if you think back to the—I'm not sure if it was the last video or the one before that, but I talked about how when you take—when we're looking for this derivative of the tangent vector with respect to arc length, the way that you compute this is to first take its derivative with respect to the parameter, which is something we can actually do because everything is expressed in terms of that parameter, and then dividing it by the basically the change in arc length with respect to that parameter, which is the size of that first derivative function.

So if this whole thing is the derivative of the tangent vector with respect to T, what that means is when we take this and we divide that whole thing by the derivative by S prime, that should give us curvature. In fact, that's just worth writing on its own here—that's curvature.

Curvature is equal to—and what I'm going to do is I'm going to take, since we see this three different times, we see, you know, S, the magnitude of S prime here, magnitude of S prime here, magnitude of S prime here. Since we see that three times, I'm going to go over here and I'm going to put that on the denominator but cubing it—so S prime, that derivative vector, the magnitude of that derivative vector cubed.

On the top, we still have S prime that vector, S prime cross product with S prime that vector. This—I mean, you could think of this as yet another formula for curvature. I think I've given you like four at this point, or you could think of it as just kind of the same thing.

If we look back up to our original one that I was trying to justify, this is just the spelled-out version of it. What is this bottom component here? If we take X prime S + Y prime S, and if we think of the square root of that—so kind of taking it to the 1/2 power—that would be the magnitude of the derivative, and I kind of showed that in some of the previous videos.

What we're doing is we're cubing that—so that whole formula, that's the very explicit, you know, in terms of X and Y's. What's going on is really just expressing this idea. You take the cross product between the first derivative and the second derivative, and then because you're normalizing it, you know, normalizing with respect to the first derivative, you want to scale down the second derivative by that same amount, just so that the parallelogram we think of kind of shrinks and everything stays in proportion.

Once again, we're dividing by that S prime, basically because curvature is supposed to be with respect to S and not with respect to T. So that's a way of kind of getting a correction factor for how wrong you're going to be if you just think in terms of the parameter T instead of steps in terms of the arc length.

Hopefully, this makes the original formula a little bit less random-seeming, you know, in this two-dimensional case, and it also gives another strong conceptual tool for understanding yet another way that you can think about how much a curve itself actually curves.

I think I've probably done enough videos here going through all the different formulas for what curvature should be, and then in the next one or two, I'll go through some specific examples just to see what it looks like to compute that.