Curvature is of critical importance in numerous contexts. An example is shown in the figure above, a map of the Silverstone Formula 1 racetrack. The sharp bends (high curvature) force drivers to reduct speed drastically.

**The Concept of Curvature**

Curvature is a fundamental concept in differential geometry. The curvature of a plane curve is a measure of how much it deviates from a straight line. We can compute the curvature as the limit of the angle through which the tangent at a point turns as the point moves through a small distance along the curve.

The simplest example of a curve is a circle. As a point moves around a circle, the tangent curve turns through a full rotation. Thus, the curvature is

Thus, a circle has curvature equal to the reciprocal of the radius. A straight line is the limiting case of a circle as the radius becomes infinite, and the curvature tends to zero.

In general, the curvature of a differentiable curve at a point is the curvature of the circle that most closely approximates the curve near the point, the so-called *osculating circle.* We can choose three points on the curve and determine the circle that passes through them. As the three points coallesce to a single point, this circle becomes the osculating circle, and its radius determines the curvature at the point.

**Direct Calculation of Curvature**

We evaluate a function at point and at two neighbouring and where and (see Fig. 1). We define the mean gradients

The tangent angles over the intervals and are

Then using a standard result for the difference of two arc-tangents,

Using (1) and approximating and by ,

The arc-length between the midpoints of and is

Combining the above two equations, the curvature is

**The Osculating Circle**

We can derive an expression for the *osculating circle* at by taking three points close to . First, let us consider three general points, , , and . Suppose the circle passing through these three points has centre at , and radius . Then

These are three quadratic equations for three unknowns, . We can get two linear equations for by subtracting (3) from (4) and (4) from (5) and rearranging:

We can easily solve these simultaneous linear equations for , the centre of the circle. Then any one of the equations (3)–(5) can be used to find the radius (see Fig. 2, left panel).

**Coalescing Points **

The curvature of the graph of a function at a point with coordinates depends on the second derivative . But what is the precise relationship?

Let us suppose that the three points are close together, and expand to second order:

Therefore, to second order, we get

To simplify matters, let us assume the origin is moved to the point and the -values are equally spaced: and (see Fig. 2, right panel). Expanding to order we have

or, defining and ,

Using these in (6) and (7) and subtracing one from the other, gives

Adding the two gives

Now we can compute :

Finally, recalling that , we get the expression for curvature,

which agrees with (2).

**Curvature in Higher Dimensions**

The curvature of a two-dimensional surface is a measure of how much it deviates from a plane. More generally, for a Riemannian manifold of dimension we can define the curvature intrinsically, that is without considering the embedding in an external space. This is done by means of the Riemann curvature tensor. For a two-dimensional surface, we can define the maximal curvature, minimal curvature, and mean curvature.

Much more can be written about the topic of curvature. The Wikipedia article *Curvature* is an excellent place to begin deeper investigations.