isometries

An isometry is a mapping between metric spaces that preserves distances. Formally, a function f: X → Y between metric spaces with distance d_X and d_Y satisfies d_Y(f(x1), f(x2)) = d_X(x1, x2) for all x1, x2 in X. If f is bijective, it is a bijective isometry; such maps have distance-preserving inverses. In Euclidean space, every isometry is affine: f(x) = Ax + b, where A is an orthogonal matrix (A^T A = I) and b ∈ R^n. Because A preserves inner products, these maps preserve lengths and angles, and consequently straight lines and shapes; they are continuous and bicontinuous.

In the plane, isometries are translations, rotations, reflections, and glide reflections. They can be described by the pair (A, b) with A ∈ O(2). If det(A) = 1 the isometry preserves orientation (translations and rotations); if det(A) = −1 it reverses orientation (reflections and glide reflections). More generally, the set of all Euclidean isometries forms the Euclidean group E(n) = O(n) ⋉ R^n, the semidirect product of the orthogonal group with translations.

Differences from similarities: isometries preserve distances exactly, while similarities allow a uniform scale factor. In other metric spaces, an isometry is defined purely by distance preservation and need not have an affine form. Applications include geometry, crystallography, computer graphics, and navigation, where symmetry and rigid motion are essential.