Randkurve

Randkurve is a German-language term used to describe a boundary curve that delineates a region of interest in a plane or space. It denotes the outer limit where a property changes from admissible to inadmissible, or where a region ends. The concept is generic and appears across disciplines, from pure geometry to applied visualization.

In mathematical contexts, a Randkurve can be represented in several forms. It may be given parametrically as r(t), implicitly by an equation F(x, y) = 0, or in polar or other coordinates. Often the Randkurve coincides with the set of points where a constraint becomes active in an optimization problem, i.e., where boundary constraints are tight. The curve is studied for properties such as smoothness, curvature, convexity, and how it responds to perturbations of the defining data.

Applications are diverse. In geometry and computer graphics, Randkurven outline shapes, polygons, or smooth contours and serve as the boundary of objects. In image analysis, the term can describe an object's outline after edge detection. In optimization and economic modeling, Randkurven help visualize feasible regions or payoff boundaries where competing constraints intersect the objective.

Construction and analysis often rely on numerical methods. Marching squares, contouring algorithms, and spline fitting are common techniques to approximate Randkurven from discrete data. The exact interpretation of a Randkurve is therefore context-dependent, with the common thread being its role as the boundary that separates distinct regions or states.