How might you describe the difference in shape between an ellipse and a circle?
One way is to think about how the object is curving (changing direction) at each point. In the case of the circle,
the shape or the curvature (rate of change in direction) is exactly the same at every point whereas in the ellipse the shape
is not uniform so that at some points its curvature is more or less than at some other points. Think of
a basketball versus a football. The basketball has a uniformity of shape that the football does not.
At the tips of the football the change in direction per unit length is much greater than at the middle of the ball.
Given two circles with different radii, the curvature of the smaller circle will be greater than that of the larger circle.
For example, if you had two circles drawn on the ground with radii 2 feet and 20 feet, and you moved along the perimeter with a constant tangential velocity of v = 2 feet per second, you would be changing direction much faster on the smaller circle than on the larger circle. In fact, the respective angular velocities w=v/r (rate of change in direction) would be 1 radian per second versus 0.1 radian per second. That is, you would be changing direction 10 times faster around the smaller circle.
It is especially easy to define curvature in terms of a circle because the circle has constant curvature that is defined completely by the radius of the circle. In particular, we define the curvature at any point on a circle as the inverse of the radius. Thus a circle with small radius has a large curvature and a circle with large radius has a small curvature. Using k to represent curvature, we define k = 1/r as the curvature at any point on the circle of radius r.
Now what about the ellipse where the curvature varies at different points? How might we define the curvature at some chosen point P on the ellipse? We do it by considering the circle (C,R) with center C and radius R which is tangent to the ellipse at P and has the same curvature as the ellipse at P. We call (C,R) the circle of curvature at P where C and R are called the center of curvature and radius of curvature at P respectively. Thus as we move around the ellipse, (C,R) will vary in size with R being small around the narrow ends of the ellipse and larger around the flatter regions of the ellipse where the curvature is smaller (the radius of curvature being larger).
In fact, we can apply this definition to any plane curve, not just ellipses. In general, given a plane curve F defined by F={(x,y): y = f(x)},
we define the curvature of F at any point P=(p,f( p)) as the limit, as ∆p goes to zero, of the circle through P and Q=(p+∆p,f(p+∆p))that is tangent to the curve F at P. That is, the limit, as Q approaches P along F, of the circle through P and Q which is tangent to F at P.
In the paper Curvature Derivation, we have derived the formula for the center and radius of curvature of the circle of curvature at a point P of general curve F using the tools of elementary calculus. At the end of the paper there are some examples worked out and illustrated. Hopefully, the paper will provide the reader with a detailed analytical understanding of and appreciation for the concept of curvature in the plane.
