curving

Curving refers to the act of shaping something into a curve or bend, or to the geometric property of deviating from a straight line. In mathematics, curvature is a measure of how quickly a curve changes direction. For a plane curve given by x(t), y(t), the curvature κ is defined as κ = |x'y'' − y'x''| / (x'^2 + y'^2)^(3/2). For a space curve r(t), curvature and torsion describe bending and twisting, with κ = |r' × r''| / |r'|^3 and torsion τ defined via the Frenet frame. A circle has constant curvature 1/R, where R is its radius; a straight line has zero curvature.

Curvature on surfaces is described by mean curvature and Gaussian curvature, derived from principal curvatures—the maximum and minimum curvatures in a given direction. These quantities determine local shape, indicating whether a patch is curved outward, inward, or flat, and influence properties such as minimal surfaces and saddle points.

Applications span many fields. In transportation engineering, road and rail curves are designed with radii to balance safety and comfort, using curvature to control turning. In computer graphics and robotics, curves model paths, trajectories, and surfaces. In physics, an object moving along a curved path experiences centripetal acceleration toward the curve’s center, related to curvature and speed.

In education, “curving” or grading on a curve refers to adjusting scores to fit a preferred distribution, often to standardize results rather than reflect absolute performance. In design and typography, curving text or shapes is a common stylistic device.