Kurve

Kurve is the German noun for curve, indicating a bending line or path. In everyday usage it can describe a smoothly bending line, a road curve, or a trajectory. In mathematics and related disciplines, a Kurve refers to a one-dimensional object that resides in a plane or in space and is described by points that vary along a parameter.

In geometry, a plane curve lies entirely in a two-dimensional plane, while a space curve exists in three-dimensional space. A common way to specify a curve is by a parameterization r(t) = (x(t), y(t)) for a plane curve or r(t) = (x(t), y(t), z(t)) for a space curve. Differentiable curves have tangent vectors, and many curves are analyzed through quantities such as arc length, curvature, and, for space curves, torsion.

Famous planar curves include the circle, ellipse, parabola, and hyperbola; a familiar space curve is the helix. Special families of curves used in design and graphics include Bezier curves and B-splines, which are defined by control points and yield smooth paths.

Applications of curves are widespread: in engineering and road design to define alignments, in computer graphics to render shapes and motion, in physics and astronomy to describe trajectories, and in data analysis to model trends. The concept of Kurve thus encompasses both simple geometric shapes and complex, parameterized paths used across science and engineering.