Isometry

An isometry is a distance-preserving map between metric spaces. In a metric space (X, d), a map f: X → X is an isometry if for all x and y in X, d(f(x), f(y)) = d(x, y). Equivalently, an isometry preserves all distances; such maps are necessarily injective, and if f is onto, it is a bijection with an inverse that is also an isometry.

In Euclidean space R^n with the standard Euclidean distance, every isometry has the form f(x) = Qx + b, where Q is an orthogonal n×n matrix (Q^T Q = I) and b ∈ R^n is a translation vector. The determinant of Q is ±1; det = 1 yields orientation-preserving isometries (translations and rotations, possibly combined as screw motions), while det = −1 yields orientation-reversing isometries (reflections, glide reflections). In 2D, typical isometries are translations, rotations, reflections, and glide reflections; in 3D they include screw motions as well.

Isometries preserve not only distances but also the congruence of figures: two figures are congruent if one can be mapped to the other by an isometry. The set of all isometries of a space forms its isometry group, with composition as the operation. In Euclidean space this group is the semidirect product of the orthogonal group O(n) with the translation group R^n.

Common examples include translating every point by a fixed vector, rotating around an axis or a point, reflecting across a line or plane, and the combinations of these. Transformations such as scaling or shear do not preserve distances and are therefore not isometries. In the study of geometry, isometries are used to formalize the notion of congruence and rigid motion.