How sharply does a road bend?
The question
Driving along a winding road, how sharply does it bend at each point? Is there a circle that fits the road best at each point — and what is its radius?
Try the experiment before reading on. There’s no wrong way to explore.
Experiment
The curvature of a curve
A point travels along the curve. The osculating circle is the circle that fits the curve best at that point; its radius is 1/κ.
- Signed curvature
- κ = 0.447
- Radius of curvature
- 1/|κ| = 2.239
κ is positive where the curve bends to the left as you travel along it, and negative where it bends to the right. N is the tangent T turned 90° to the left; the centre of the osculating circle lies 1/|κ| away, on the side the curve bends towards.
∫ κ ds = 6.283 = 1.000 × 2π
Turning number 1: going once around, the tangent turns through 1 × 360° in total.
Hopf's Umlaufsatz: for a smooth closed curve, ∫ κ ds = 2π × (turning number). A simple closed curve — one that never crosses itself — traversed counter-clockwise always has turning number 1. The limaçon with an inner loop has 2; the figure eight has 0.
Arc length and ∫ κ ds are computed numerically from exact derivatives at thousands of points, so the last digit may wobble.
Signed curvature κ against arc length s
The signed area under this graph is ∫ κ ds (teal above the axis counts as positive, terracotta below as negative). Very large values near a cusp are cut off.
Hints, if you want them
What’s going on
What’s going on
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 , the curvature of the curve there is : 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 .
For the graph of ,
For the parabola at its lowest point, and , so and the osculating circle has radius .
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 . 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.
Where this leads
- Advanced undergraduate · Lesson availableDifferential geometry of curves and surfacesCurvature and torsion of space curves, the first and second fundamental forms of surfaces, Gaussian and mean curvature, Gauss's Theorema Egregium, geodesics and the Gauss–Bonnet theorem.
- First-year graduate · Lesson availableRiemannian geometryCurved spaces studied from the inside: Riemannian metrics, the Levi-Civita connection, geodesics and the exponential map, curvature tensors, Jacobi fields, the global Gauss–Bonnet theorem, Hopf–Rinow, and comparison theorems such as Bonnet–Myers.
- First-year graduate · Lesson availableThe geometry of spacetime: special and general relativityPhysics as geometry: Minkowski spacetime and the Lorentz group, proper time and causal structure, Lorentzian manifolds, the equivalence principle, Einstein's field equations, the Schwarzschild solution, geodesics of light and matter, and cosmological models.