How sharply does a road bend?

The circle that fits a curve best at a point — the limit of circles through three nearby points — is the osculating circle ("kissing circle"). If its radius is $R$, the curvature of the curve there is $\kappa = 1/R$: a gentle bend has a large circle and small curvature, a hairpin has a small circle and large curvature, and a straight road has curvature $0$.

For the graph of $y = f(x)$,

$$\kappa = \frac{|f''(x)|}{\bigl(1 + f'(x)^2\bigr)^{3/2}} .$$

For the parabola $y = x^2$ at its lowest point, $f'(0) = 0$ and $f''(0) = 2$, so $\kappa = 2$ and the osculating circle has radius $\tfrac12$.

Curvature has a striking global property. Walk once around a simple closed curve, adding up the curvature as you go (counting bends to the left as positive and to the right as negative): the total is always $2\pi$. Heinz Hopf proved this turning-tangent theorem in 1935. Road and railway engineers use curves whose curvature grows steadily — clothoids, or Euler spirals — so that the steering wheel turns smoothly instead of suddenly.

Gauss extended curvature to surfaces, and found that the curvature of a surface can be measured by creatures living inside it, without leaving it. Riemann extended it to spaces of any dimension, and Einstein used that curvature to describe gravity.