Curvature intuition

- [Voiceover] Hello everyone. So what I would like to do here, is talk about curvature. And, I've drawn on the X Y plane here, a certain curve. So this is our X-axis. This is our Y-axis. And this is a curve running through space. And I'd like you to imagine that this is a road of some kind, and you are driving on it. And you are at a certain point, so let's say this point right here. And if you imagine, what it feels like to drive along this road, and where you need to have your steering wheel, you're turning it a little bit to the right. Not a lot, because it's kind of a gentle curve at this point, you're not curving a lot. But the steering wheel isn't straight, you are still turning on the road. And now, imagine that your steering wheel's stuck. That it's not gonna move. And however you're turning it, you're stuck in that situation. What's gonna happen, hopefully you're on an open field or something, because your car is going to trace out some kind of circle. Right? You know your steering wheel can't do anything different, you're just turning it a certain rate, and that's gonna have you tracing out some giant circle. And this depends on where you are, right? If you had been at a different point on the curve, where the curve was rotating a lot, let's say you were back a little bit towards the start. At the start here, you have to turn the steering wheel to the right, but you're turning it much more sharply to stay on this part of the curve, than you were to be on this relatively straight part. And the circle that you draw as a result, is much smaller. And this turns out to be a pretty nice way to think about a measure, for just how much the curve actually curves. And one way you could do this, is you can think, "Okay, what is the radius of that circle, the circle you would trace out if your steering wheel locked at any given point.". And if you kind of follow the point, along different parts of the curve, and see, "Oh, what's the different circle that my car would trace out if it was stuck at that point.". You would get circles of varying different radii. Right? And this radius actual has a very special name. I will call this "R". This is called the "Radius of Curvature". And you can kind of see how this is a good way to describe, how much you are turning. You may have heard with a car, descriptions of the turning radius. You know, if you have a car with a very good turning radius, it's very small, 'cause what that means if you turned it all the way you could trace out just a very small circle. But a car with a bad turning radius, you know, you don't turn very much at all, so you'd have to trace out a much larger circle. And, curvature itself isn't this "R". It's not the radius of curvature. But what it is, is it's the reciprocal of that, it's one over "R". And there is a special symbol for it. It's kind of a "K", and I'm not sure in handwriting I am going to distinguish it from an actual "K", maybe give it a little curly there. It's the Greek letter Kappa. And this is curvature. And I want you to think for a second, why we would take one over "R". "R" is a perfectly fine description of how much the road curves. But why is it that you would think one divided by "R", instead of "R" itself? And the reason basically, is you want curvature to be a measure of, how much it curves in the sense that, more sharp turns should give you a higher number. So, if you're at a point, where you are turning the steering wheel a lot, you want that to result in a much higher number. But, radius of curvature will be really small, when you are turning a lot. But if you are at a point that's basically a straight road, you know, there's some slight curve to it, but it's basically a straight road, you want the curvature to be a very small number. But in this case, the radius of curvature is very large. So it's pretty helpful to just have one divided by "R", as the measure of how much the road is turning. And, in the next video, I am going to go ahead and start describing a little bit more mathematically, how we capture this value. 'Cause as a loose description, if you are just kind of drawing pictures, it's perfectly fine to say, "Oh yeah, you imagine a circle that's kind of closely hugging the curve, it's what your steering wheel would do if you were locked.". But in math, we will describe this curve parametrically. It'll be the output of a certain vector valued function. And I wanna know how you could capture this idea. This, one over "R" curvature idea, in a certain formula. And that's what the next few videos are going to cover.