Curvature intuition

Hello everyone. So what I'd like to do here is talk about curvature. I've drawn on the xy plane here a certain curve. So this is our x-axis, this is our y-axis, this is a curve running through space, and I'd like you to imagine that this is a road of some kind and you're driving on it, and you're at a certain point. So let's say this point right here.

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.

Now imagine that your steering wheel is stuck, that it's not going to move, and however you're turning it, you're stuck in that situation. What's going to 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 at a certain rate, and that's going to have you tracing out some giant circle.

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. The circle that you draw as a result is much smaller.

This turns out to be a pretty nice way to think about a measure for just how much the curve actually curves. One way you could do this is you can think, "Okay, what is the radius of that circle?" The circle that you would trace out if your steering wheel locked at any given point. 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 get circles of varying different radii, right?

This radius actually has a very special name. I'll call this r; this is called the radius of curvature. You can kind of see how this is a good way to describe how much you're turning. Radius of curvature, you know, you may have heard with the car descriptions of the turning radius. If you have a car with a very good turning radius, it's very small because what that means is 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. Curvature itself isn't this r; it's not the radius of curvature. But what it is is it's the reciprocal of that, one over r. There's a special symbol for it; it's kind of a k. I'm not sure in handwriting how I'm 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.

I want you to think for a second why, you know, 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? 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're 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're turning it a lot. If you're at a point that's basically like 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 really helpful to just have one divided by r as the measure of how much the road is turning. In the next video, I'm going to go ahead and start describing a little bit more mathematically how we capture this value because, as a loose description, if you're 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 want to know how you can capture this idea, this one over r curvature idea in a certain formula. That's what the next few videos are going to cover.