Geometriefaktor

GeometrieFaktor (GF) is a dimensionless quantity used to describe how a geometric object is altered by a transformation. In differential geometry and analytic geometry, for a differentiable mapping f: R^n -> R^n with Jacobian Df(x) at point x, the local GeometrieFaktor is defined as GF(x) = sqrt(|det(Df(x))|) in two dimensions, and more generally GF(x) = (abs(det(Df(x))))^(1/n) in n dimensions. This GF equals the geometric mean of the principal stretches (the singular values of Df) and reduces to the familiar scale factor for uniform dilations: if f(x) = s x, then GF = s. For anisotropic distortions with principal stretch factors s1, s2, GF = sqrt(s1 s2). The determinant det(Df) controls area scaling, while GF provides a single scalar summary of average distortion.

Examples: a uniform scale by 2 gives GF = 2; a pure shear has det(Df) = 1, hence GF = 1; nonlinear maps yield GF varying with position.

Applications: in computer graphics and geometric modeling to measure how mappings distort textures; in finite element mesh generation and quality control to assess element distortion; in optical design and architectural geometry to track geometric deviations.

History: The term GeometrieFaktor is encountered in German-language texts as a compact descriptor for local distortion measures, aligned with the broader concept of scaling and area-change factors in mapping theory.

See also: Jacobian, determinant, singular value decomposition, distortion, conformal map.