Parametrisches

Parametrisches refers to descriptions or models that are defined by parameters. In mathematics, engineering, computer graphics and related fields, a parametric description expresses a geometric object as the image of a parameterized map. For a curve in the plane, a parameter t yields a point (x(t), y(t)); for a surface, two parameters (u, v) yield (x(u,v), y(u,v), z(u,v)). The parameter domain is typically an interval or a rectangle in R^n. This approach enables direct control over the object’s geometry through the parameters.

Parametrization facilitates calculations such as locating points, computing tangents and normals, or performing integration, and it contrasts with implicit representations like F(x, y, z) = 0 or with purely coordinate-based descriptions. It is widely used to express design intent and to manage dependencies and constraints in computational workflows.

Applications are diverse. In computer-aided design (CAD) and parametric design, objects are defined by parameters so that changing one parameter automatically updates the dependent geometry. In computer graphics, parametric curves and surfaces such as Bezier curves, B-splines and NURBS are standard tools for modeling and rendering. In statistics, parametric models describe data using a fixed set of parameters, enabling estimation, hypothesis testing and inference under assumed distributions.

Examples include the circle parameterized by x = r cos t and y = r sin t, or a torus parameterized by (u, v) -> ((R + r cos v) cos u, (R + r cos v) sin u, r sin v). The choice of parameterization can influence numerics, stability and ease of manipulation.

Etymology: the term derives from parameter, from Greek metron "measure," forming a quantity that governs another quantity.