curves

A curve is a one-dimensional geometric object that, locally, resembles a line. In the plane, it is the set of points traced by a continuous function of a parameter, often described as the graph of a function or as the image of a parametric equation. More generally, a curve can be embedded in higher-dimensional space.

Curves are commonly classified by how they are described: plane curves can be given implicitly by F(x,y)=0 or explicitly as y=f(x); parametric plane curves by x=x(t), y=y(t); space curves by r(t)=(x(t),y(t),z(t)). Algebraic curves satisfy polynomial equations F(x,y)=0 in the plane, or polynomial vector equations in space.

Important geometric notions include arc length, curvature, and, for space curves, torsion. If r(t) is a smooth parametrization, the curvature κ measures turning and torsion τ measures twisting. Curvature can be computed from derivatives; arc length s is the integral of the speed |r'(t)|.

Common examples include lines, circles, ellipses, and parabolas (conic sections), as well as special curves such as the cycloid, catenary, logarithmic spiral, and helix. Many curves are studied for their algebraic properties or differential-geometric properties.

Curves play central roles in mathematics, physics, engineering, and computer graphics. They are used to model trajectories, design smooth paths, fit data, and study intrinsic properties of spaces in differential and algebraic geometry.