curve

A curve is a one-dimensional geometric object that can be traced with a continuous stroke. In mathematics, a curve is often defined as a continuous image of an interval, or as a smooth map from an interval into a plane or space. Curves may be described parametrically by a vector-valued function r(t) and, in the plane, as graphs y=f(x) or implicitly by equations F(x,y)=0. Straight lines are included as degenerate curves.

Planar curves lie in the plane; space curves lie in three-dimensional space. Examples include circles, ellipses, parabolas, and more complex curves such as helices or Bézier curves. A curve can be closed, returning to its starting point, and simple if it does not cross itself. Regular or smooth curves have continuous derivatives, while corners or cusps are singularities.

Key notions include arc length, curvature and torsion. Arc length measures the length of the curve; curvature measures how quickly the tangent direction changes. For space curves, torsion describes twisting out of the plane. Parametric forms such as r(t) = (x(t), y(t)) or r(t) = (x(t), y(t), z(t)) are common.

Algebraic curves are defined as the common zeros of polynomial equations in two variables, such as y^2 = x^3 − x. They are studied over various fields and can possess singularities and a genus that captures their topological complexity.

Applications span physics, engineering, computer graphics, CAD, road and track design, and data analysis, where curves model trajectories, shapes, and interpolating paths.