2DKurven

2D curves, sometimes written as 2D Kurven in German mathematical literature, denote continuous sets of points traced in the two-dimensional Euclidean plane. In mathematics and related fields a curve is often described by a parameterization, a rule that assigns a point (x(t), y(t)) for values of t in an interval. The image of this parameterization traces the curve as t varies.

Curves can be represented in several forms. In parametric form, a curve is given by x = x(t), y = y(t). In explicit form, a function y = f(x) defines the height at each horizontal position, while implicit form describes the curve as a level set F(x, y) = 0. A regular (or smooth) curve has a nonzero velocity vector (dx/dt, dy/dt) except possibly at endpoints.

Examples include straight lines, circles, ellipses, parabolas, and hyperbolas. More complex planar curves include spirals, cycloids, or Bézier curves used in computer graphics. A curve can be closed (forming a loop, such as a circle) or open; it can be simple (no self-intersections) or self-intersecting.

Two standard notions are arc length and curvature. The length of a parametric curve t in [a,b] is ∫ from a to b sqrt((dx/dt)^2 + (dy/dt)^2) dt. The curvature κ(t) measures how quickly the direction of the curve changes, given by |x' y'' − y' x''| / ((x')^2 + (y')^2)^(3/2) when derivatives exist.

2D curves are central in geometry, computer graphics, GIS, and robotics. They can be embedded in 3D space by fixing z = 0, and in practice are often discretized as polylines or sampled points for computation.