Curvature formula, part 3

So continuing on with where we were in the last video, um, we're looking for this unit tangent vector function given the parameterization.

So the specific example that I have is a function that parameterizes a circle with radius capital R, but I also kind of want to show in parallel what this looks like more abstractly.

So here I'll just kind of write down in the abstract half what we did here, what we did for the unit tangent vector.

Um, so we actually have the same thing over here where the unit tangent vector should be the derivative function, which we know gives a tangent, right? It's just it might not be unit, but then we normalize it.

We take the magnitude of that tangent vector function and in our specific case with the circle, once we did this and we kind of took the x component squared, y component squared and simplified it all out, we got the function R.

But in the general case, we might not be so lucky because the magnitude of this derivative is going to be the square root of x prime of t squared, right? It's this the x component of the derivative plus y prime of t squared, just taking the magnitude of a vector here.

So when we take the entire function and divide it by that, what we get doesn't simplify as it did in the case of a circle.

Instead, we have that x prime of t, which is the x component of our s prime of t, and we have to divide it by that entire magnitude, which was this whole expression, right? You have to divide it by that whole square root expression.

I'm just going to kind of write dot dot dot with the understanding that this square root expression is what goes up there.

And similarly, over here, we'd have y prime of t divided by that entire expression again, right?

So simplification doesn't always happen; that was just kind of a lucky happenstance of our circle example.

And then now what we want, once we have the unit tangent vector as a function of that same parameter, what we're hoping to find is the derivative of that unit tangent vector with respect to arc length, the arc length s.

And to find its magnitude, that's going to be what curvature is.

But the way to do this is to take the derivative with respect to the parameter t, so d big T d little t, and then divide it out by the derivative of our function s with respect to t, which we already found.

And the reason I'm doing this, you know, loosely, if you're just thinking of the notation, you might say, oh, you can kind of cancel out the dts from each one.

But another way to think about this is to say when we have our tangent vector function as a function of t, the parameter t, we're not sure of what its change is with respect to s, right? That's something we don't know directly.

But we do directly know its change with respect to a tiny change in that parameter.

So then, if we just kind of correct that by saying, hey, how much does the length of the curve change? How far do you move along the curve as you change that parameter?

And maybe if I go back up to the picture here, this ds dt is saying for a tiny nudge in time, right, what is the ratio of the size of the movement there with respect to that tiny time?

So the reason that this comes out to be a very large vector, right, it's not a tiny thing, is because you're taking the ratio; maybe this tiny change was just an itty bitty smidgen vector, but you're dividing it by like 1 millionth or whatever the size of dt that you're thinking.

And in this specific case for our circle, we saw that the magnitude of this guy, you know, if we took the magnitude of that guy, it's going to be equal to R, which is a little bit poetic, right?

That the magnitude of the derivative is the same as the distance from the center.

And what this means in our specific case, if we want to apply this to our circle example, we take dt d big T, the tangent vector function, and I'll go ahead and write here, we have the derivative of our tangent vector with respect to the parameter.

And we go up and we look here, we say, okay, the unit tangent vector has the formula negative sin of T and cosine of T.

So the derivative of negative sin of T is negative cosine, so over here, this guy should look like negative cosine of T.

And the other component, the y component, the derivative of cosine t as we're differentiating our tangent vector function is negative sine of T.

And what this implies is that the magnitude of that derivative of the tangent vector with respect to T, well, what's the magnitude of this vector?

You've got a cosine, you've got a sine; there's nothing else in there.

You're going to end up with cosine squared plus sine squared, so this magnitude just equals one.

And when we do what we're supposed to over here and divide it by the magnitude of the derivative, right, we take this and we divide it by the magnitude of the derivative ds dt, well, we've already computed the magnitude of the derivative; that was R.

That's what, that's how we got this R is we took the derivative here and took its magnitude and found it.

So we find that in the specific case of the circle, the curvature function that we want is just constantly equal to 1/R, which is good and hopeful, right?

Because I said in the original video on curvature that it's defined as 1 divided by the radius of the circle that hugs your curve most closely.

And if your curve is actually a circle, it's literally a circle, then the circle that hugs it most closely is itself, right?

So I should hope that its curvature ends up being 1 divided by R.

And in the more general case, if we take a look at what this ought to be, you can maybe imagine just how horrifying it's going to be to compute this, right?

We've got our tangent vector function, which myself, I, you know, it's almost too long for me to write down; I just put these dot dot dots where you're filling in x prime of t squared plus y prime of t squared, and you're going to have to take this, take its derivative with respect to T, right?

It's not going to get any simpler when you take its derivative, take the magnitude of that and divide all of that by the magnitude of the derivative of your original function.

And I think what I'll do, I'm not going to go through all of that here; it's a little bit much, and I'm not sure how helpful it is to walk through all those steps.

Um, but for the sake of having it for anyone who's curious, I think I'll put that into an article and you can kind of go through the steps at your own pace and see what the formula comes out to be.

And I'll just tell you right now, maybe kind of a spoiler alert, what that formula comes out to be is x prime, the derivative of that first component, multiplied by y prime, the second derivative of that second component, minus y prime, first derivative of that second component multiplied by x prime.

And all of that is divided by the the, uh, kind of magnitude component, the x prime squared plus y prime squared, that whole thing to the three halves.

And you can maybe see why you're going to get terms like this, right?

Because when you're taking the derivative of, when you're taking the derivative of the unit tangent vector function, you have the square root term in it, the square root that has x primes and y primes.

So that's where you're going to get your x prime y prime as the chain rule takes you down there.

And you can maybe see why this whole x prime squared, y prime squared term is going to maintain itself.

And it turns out it comes in here at a three halves power.

And what I'm going to do in the next video, um, I'm going to go ahead and describe kind of an intuition for why this formula isn't random, why if you break down what this is saying, it really does give a feeling for the curvature, the, you know, amount that curve curves that we want to try to measure.

So it's almost like this is a third way of thinking about it, right?

The first one I said you have whatever circle most closely hugs your curve, and you take one divided by its radius.

And then the second way you're thinking of this dT/ds, the change in the unit tangent vector with respect to arc length, and taking its magnitude.

And of course, all of these are the same, but they're just kind of three different ways to think about it or things that you might plug in when you come across a function.

And I'll go through an example; I'll go through something where we're really computing the curvature of something that's not just a circle.

Um, but with that, I'll see you next video.