Zentrumstransformator

Zentrumstransformator is a term used in German-language mathematical literature to denote an affine transformation that re-centers a coordinate space around a chosen center vector c. Formally, for a space V over the real numbers and a center c in V, the Zentrumstransformator Z_c maps x in V to Z_c(x) = x − c. This operation translates the entire space so that the center becomes the origin of the new coordinate system.

As an affine transformation, Z_c is invertible with inverse Z_c^{-1}(y) = y + c. It is not a linear operator because it does not preserve the origin in the original coordinates, but it is isometric in Euclidean space, preserving distances and angles. When applied successively with centers c1 and c2, the resulting transformation is Z_{c2} ∘ Z_{c1} = Z_{c1+c2}.

Variants may include additional scaling or rotation. A normalized centered transform Z_{c,s}(x) = (x − c)/s, where s > 0, combines centering with scaling. In practice, Zentrumstransformator is often discussed as part of the broader class of affine transformations used to change reference frames or to zero-mean data.

Applications include data preprocessing for statistical analyses to improve numerical stability, computer graphics and 3D modeling to establish a local coordinate system for objects, and robotics to define local frames of reference relative to a chosen center.

Etymology: Zentrum means center in German, and Transformator denotes a transformer or operator. The term is used mainly in German-language texts and is related to the general concept of translations and affine transformations.