Mathematicians measure how sharply a curve bends by watching its tangent turn
Imagine driving along a winding road at a steady speed and noting how fast your steering direction swings. That rate of turning is exactly what mathematicians call curvature. A straight road scores zero, and a tight roundabout scores high, because smaller circles bend more sharply than larger ones.
Curvature captures how far a curve strays from a straight line, or a surface from a flat plane. At any point, a curve's direction is given by its tangent line. Curvature is the angle through which that tangent swings per unit of distance travelled along the curve, so it is measured in radians per unit length. For a circle the value is identical everywhere and equals one divided by the radius, which is why small circles curve more. At any point on a smooth curve, the best-fitting circle, called the osculating circle, shares the curve's curvature there.
The measurement belongs to the curve itself, not to how it is described. Tracing it forwards or backwards, or labelling points with different parameters, leaves the value unchanged. If you picture a particle gliding along the path at constant unit speed, curvature tells you how quickly its heading rotates. The curvature vector adds direction, pointing where the curve is turning while its length records how sharply.
Surfaces are trickier because bending depends on which direction you face, giving rise to maximum, minimum and mean curvature. Curvature can also be defined intrinsically, for spaces of two or more dimensions, without reference to any surrounding space at all.
The idea has a long pedigree. Ancient Greeks separated straight lines from circular ones, and Aristotle and Apollonius refined the notions. In the 14th century Nicole Oresme described curvature as departure from straightness, making it inversely proportional to a circle's radius, and tried to extend it to other curves. Calculus, developed by Newton and Leibniz in the 17th century, allowed systematic calculation. Cauchy showed that the centre of curvature is where two infinitely close normal lines meet. Euler carried the study to surfaces, Gauss realised curvature could be intrinsic to a surface, and Riemann generalised it to higher dimensions.
Source: Curvature
