 So curvature. Curvature is so important. It's actually an easy concept to understand. Let's look at this graph that we have on this side. Gentle slope, gentle slope, and then a very sharp slope. So imagine you are a motor vehicle. You're a motor vehicle and this is the little road that you travel. If you go around this bend, you can go at quite a bit of speed. You're not going to slip off the road unless your tires are worn or the road surface is wet. If you carry some speed though through this corner and you try to take that sharp u-turn up there, you might have to slow down. There's something inherently different between this curve and that curve and we need to describe this mathematically. We can all agree though that there's less curvature to this section than there is there. So how can we describe this? Because I can drive fast here and I can drive very slow there. So I can't really express curvature in terms of the velocity or the speed in which I'm traveling. I have to do something else. And it's quite clever what we do. Imagine we take the same arc length, the same distance of road. Say I'm traveling from there to there and that's a 50 meter bend. Let's suggest I can measure it out. It's exactly 50 meter. Right at this position here I'm going to have a unit position, a unit velocity vector. So I'm going to have a velocity vector in that direction and if I divide it by its magnitude I'm left with a unit, the unit velocity vector there. And if I were to do that, take the first derivative there, the position vector. Remember there's the one position vector. There's the other position vector. Imagine that. So that is my unit tangent vector, my unit velocity vector. So from that point to that point, traveling 50 meters, there's a bit of change. This has a length one. That one has a length one. So I'm expressing these as unit vectors. So their magnitude doesn't change and there's just a little bit of change in that 50 meter in their direction. If I take another 50 meters here from there to there, imagine I could also measure that out and that's another 50 meters and I look at this unit velocity vector there and I look at the unit velocity vector there. So that's a bit out of scale and a bit skewed. But there's a drastic change. That one's pointing there and this one's pointing there. There's a much bigger change per arc length, per distance. So I'm not expressing it as time. As I said, you can go fast, slow, slow, fast. I can't express it in terms of time. I can't express curvature with respect to time. I've got to express it per arc length and you can clearly see that for the same distance traveled from the same arc length there's a much bigger change there. So now we've got this concept of curvature down. I can really do something and look at what I used here. I used this tangent normal vector. I used a tangent normal vector. Let's have a look. And I am going to use the Greek letter kappa. Everyone uses it. I use it. Everyone uses it. The Greek kappa and that means curvature. That's our symbol for curvature. And we're going to say that it is this norm of the rate of change of this unit tangent vector per arc length with respect to arc length. How do you do a derivative with respect to arc length? Because we know what the equation for arc length looks like. It's a long, depending on how many dimensions we have, this is obviously a long integral. So per unit length it's a very difficult thing to do. Let's just remember one thing there. Let's just remember or let's just look at the unit tangent vector. Now that's actually a composite function if you think about it because it has to do with time. It has to do with distance. So if I were just to take the first derivative of this unit tangent vector with respect to time, I'm just taking its time derivative, t prime. Just its time derivative, it being a composite function must actually use the chain rule. So what we're going to have here is a dt ds times ds dot product there dt. I have to use the chain rule if I want to do this derivative. I want to use the chain rule if I do this derivative. Now that's exactly what we wanted up there. So let's do that. Let's just get that on its own. ds and that's going to equal, let's just say that that's t prime of t, that one, divided by ds dt. ds to t for all ds dt is not equaling zero. Okay, I can't divide by zero. So I've got to actually, you know, it's a limit as this goes to zero actually, but I'm not interested in this. I'm actually interested in this. So that is actually just this and that, if you think about it. Nothing magical happening there. But I also know what this is. Remember this? This is speed. And what a speed is nothing other than the magnitude of the first derivative of the position vector, velocity in other words. The magnitude of velocity is speed and lo and behold I have my first equation for my first equation for curvature. It is the first derivative of this unit tangent vector. It's a norm divided by the magnitude of velocity, which is just from there. So there is my first equation for curvature. Very easy to do. Very easy to do. And you can well imagine why I'm going to get a much bigger value there because there's this massive change than I am going to get there. Now there's a second concept that goes with this and this is this radius of curvature, a row, a Greek letter row that looks like the smooth P. That is just one over the curvature. That's this one over the curvature. And what you can imagine that that's the radius of curvature that you have a circle at every point you can draw a circle. So if the curvature here is a small value, which you'll get from there, there's not a lot of change, obviously one over a small value is going to give you a big value. So there's this big circle here with a radius of curvature. And as you get there, this circle, its radius is going to get smaller and smaller and smaller. And at every point there is this concept of this radius of curvature. And it's very easy to do in as much as just one over where you can just swap these two around one over curvature. So there's our first equation for curvature. That's very easy. The derivation is quite easy and it's quite an easy concept to understand. But now we can express something that we felt intuitively. We can now express mathematically.