What is Curvature, centre of curvature and radius of curvature?

The idea with curvature is to quantify how curved a curve is at a particular point.

First of all, I think you'll be satisfied that a circle has constant curvature. A circle is unique in this respect (at least in two-dimensional Euclidean -ie flat or non-curved- space). For any other curve the curvature varies from one point to another.

The second thing is to compare this concept with the change (with respect to x) of the gradient of a curve in the x-y plane. A parabola (the curve of a quadratic equation) has a constant rate of change of gradient. We can find this by differentiating its gradient function with respect to x. On the other hand, its curvature clearly varies, decreasing away from the vertex or turning point (located at the origin, (0, 0), for y=x^2). In other words, the line becomes straighter away from the turning point. This is because as the line becomes steeper a small change in curvature will produce a large change in gradient. So although it is related to the ides of curvature, the second derivative is not quite what we want.

Returning to the circle, we can understand that the edge of a small circle is, in absolute terms, more curved than the edge of a larger one. The Earth, for example, seems to us almost flat, whereas the surface of a football does not.

So a logical way to quantify curvature is with reference to the radii of circles of different size. If we take the vertex (or turning point) of a parabola again, the curvature at this point is indicated by the size of the circle that will 'fit best' along this part of the curve.

Curvature is normally defined as the reciprocal of the radius of this circle (ie, 1/r). The centre of curvature is of course the centre of the circle. And the radius of curvature is radius of this circle of 'best fit' to the point particular point on the curve under consideration.

With regard to the concept of a 'circle of best fit', note that the fit only has to apply in the locality of the point on the curve in question. Note also that both the radius of curvature and the centre of curvature wil change along most curves.

We can use the equation of a circle to find out how the radius of curvature (simply equal to the circle's radius, r) is related to the gradient and rate of change of gradient.

For a circle in the x-y plane with centre (a, b), (x-a)^2+(y-b)^2=r^2.

Differentiating this with respect to x we get, 2(x-a)+2(y-b)dy/dx=0, which simplifies to x-a+(y-b)dy/dx=0.

Differentiating with respect to x again we get, 1+(y-b)d^2y/dx^2+(dy/dx)^2=0

We can combine thes three equations to eliminate x-a and y-b, yielding an equation for r in terms of of d^2y/dx^2 (abbreviated to y'' below) and dy/dx (abbreviated to y' below).

The key is rearranging the second of these equations, x-a+(y-b)y'=0, to (x-a)^2=(y-b)^2*y'^2. Then, using the first equation is replace (x-a)^2 gives us and then r^2-(y-b)^2=(y-b)^2*y'^2. This simplifies to r^2=(y-b)^2*(1+y'^2).

Rearranging the third equation to make (y-b)^2 the subject we get, (y-b)^2=(1+y'^2)^2/y''^2. Substituting this into the equation derived above gives us, r^2=(1+y'^2)^2/y''^2*(1+y'^2), which simplifies to r^2=(1+y'^2)^3/y''^2 and r=|(1+y'^2)^(3/2)/y''|.

So we know that a circle of radius |(1+y'^2)^(3/2)/y''| will 'fit' a point on a curve with these first and second derivatives, y' and y''. For example, for the parabola y=x^2, y'=2x, y''=2, so the radius of curvature, r, which varies with x is |(1+4x^2)^(3/2)/2|. The curvature, 1/r, then is |2/(1+4x^2)^(3/2)|, which decreases as the magnitude of x increases, as expected.

Define Radius Of Curvature