curved

Curved describes something that is not straight but bends or arches. In geometry, a curve is a path traced by a moving point in a plane or space; a line is the simplest curve, interpreted as a straight path with no bend.

In mathematics, curvature is a quantitative measure of bending. For a plane curve, curvature at a point describes how rapidly the curve deviates from a straight line there. The radius of curvature R is the reciprocal of curvature kappa (kappa = 1/R). A circle has constant curvature equal to 1/R, while a straight line has zero curvature. For space curves, curvature is defined similarly using the derivative of the unit tangent vector along the curve, and it can be combined with torsion to describe twisting.

On surfaces, curvature is improved to Gaussian curvature K, which is intrinsic to the surface and depends on how the surface bends in two directions. A flat plane has K = 0, a sphere has positive K, and a saddle-shaped surface has negative K. These concepts underpin differential geometry and general relativity, where curvature describes the shape of space or spacetime.

Curved shapes appear in daily life and engineering: roads and tracks bend to follow terrain, architectural arches and shells use curvature for strength and aesthetics, and curved lenses and mirrors affect focusing and imaging. In design and computer graphics, curves are used to model smooth outlines and motion paths. The term derives from Latin curvare, meaning to bend.