Curvature and the Shape Operator

line, curve, tangent and normal

What is Curvature?

Now we need to take a moment to answer a question that will become an integral part of what follows:

How do we measure curvature?

But before we can measure it, we need to first define it.

Curvature can be defined as the measure of the magnitued by which a 1D geometric object differs from a straight line. Curvature, therefore, is measured in units of inverse distance (for example, "the curvature of this curve is 3 units per meter").

The key to understanding how we measure the curvature, however, is to realize that we cannot measure the curvature of the entire curve at once. Curvature is only defined locally.

Rather, we separate the curve into an infinite number of points (so that this local region is as small as possible) and measure the curvature at each point.

We can approximate the curvature by measuring the distance between a given point (in this case the closest point on the x-axis) and the curve. As we increase the amount of lines between our reference point and the curve, we increase the accuracy of the final curvature calculation.