Isometrien

Isometrien (singular: Isometrie; English: isometry) denote distance-preserving maps between metric spaces. If f: X → Y is an isometry, then for all x, y in X the distance satisfies d_Y(f(x), f(y)) = d_X(x, y). Isometrien preserve the intrinsic geometry of spaces, including lengths of curves and angles, and they map congruent figures to congruent figures. They are never expansive or contractive and, in particular, are 1-1.

In Euclidean space, isometrien are the rigid motions of geometry. In R^n with the standard Euclidean metric, every bijective isometry has a simple form: f(x) = Ax + b, where A is an orthogonal matrix (A^T A = I) and b is a constant vector. Translations correspond to b ≠ 0 with A = I, rotations to A orthogonal with b = 0, and reflections or glide reflections involve a determinant of −1 or a combination of reflection and translation. The set of all isometries of R^n forms the Euclidean group E(n), a semidirect product O(n) ⋉ R^n.

Isometrien are defined for general metric spaces, not only Euclidean space. In this broader context, an isometry is any distance-preserving map, which need not be onto. Isometrien are central to the study of congruence, rigidity, and symmetry, and they form the isometry group of a space, capturing its geometric symmetries.

Applications of isometrien span mathematics and applied fields, including computer graphics, robotics, crystallography, and the analysis of shapes, where preserving distances ensures faithful representations of objects and motions.