Curvatures

Curvature is a measure of how quickly a geometric object deviates from being straight or flat. It is defined differently for curves and surfaces, and it extends to higher-dimensional spaces in differential geometry.

For a smooth plane curve, curvature at a point can be defined as the rate at which the unit tangent vector changes with arc length: kappa(s) = |dT/ds|. A circle of radius r has constant curvature kappa = 1/r, while a straight line has curvature zero. For space curves, curvature is part of the Frenet-Serret apparatus, together with torsion, describing how the curve bends and twists in three dimensions.

On a smooth surface in three-dimensional space, curvature is captured by two principal curvatures, k1 and k2, corresponding to the maximal and minimal normal curvatures at a point. The Gaussian curvature K = k1 k2 is an intrinsic measure of how the surface bends, independent of its embedding. The mean curvature H = (k1 + k2)/2 is an extrinsic quantity describing the average bending in all directions. The plane has k1 = k2 = 0, so K = 0 and H = 0; a sphere of radius r has k1 = k2 = 1/r, so K = 1/r^2 and H = 1/r; a cylinder has K = 0 and H = 1/(2r).

Related concepts include intrinsic curvature, which depends only on the metric, and extrinsic curvature, which depends on embedding. Gauss’s Theorema Egregium states that Gaussian curvature is intrinsic. In broader contexts, curvature is described by curvature tensors in Riemannian geometry and plays a central role in physics, computer graphics, and geometric analysis. The total curvature of a simple closed plane curve is 2π, reflecting how the tangent direction winds around.