Görbékrl

Görbékrl is the study of curves within mathematics, addressing their shape, position, and motion in the plane and in space. Conceptually, a curve is described either parametrically as r(t) = (x(t), y(t), z(t)) in three-space or implicitly by equations F(x, y, z) = 0. Planar curves lie in two dimensions, while spatial curves extend into three dimensions.

Curves are categorized by smoothness (smooth, piecewise-smooth), by the nature of their equations (analytic, algebraic, transcendental), and by their ambient setting (plane, space). Well-known families include conic sections, algebraic curves such as cubic and quartic curves, and computational curves used in graphics, notably Bézier and B-spline curves.

Core geometric concepts include arc length, curvature, torsion, and, for space curves, the Frenet-Serret frame, which describes how the curve bends and twists along its path.

Applications span computer graphics, computer-aided design, robotics and motion planning, physics, and architectural design, wherever the precise description of a path or shape is required.

Historically, the study of curves emerged in classical geometry through conic sections and was extended by differential and algebraic geometry in the 18th to 20th centuries.

See also: Conic sections; Bézier curves; B-splines; Algebraic geometry; Differential geometry.