distancepreserving

Distancepreserving describes a property of a function between metric spaces that preserves distances exactly. Formally, a map f: (X, d_X) → (Y, d_Y) is distance-preserving if for all x1, x2 in X, d_Y(f(x1), f(x2)) = d_X(x1, x2). In many texts this is called an isometry.

If f is distance-preserving and onto its codomain, it is a (surjective) isometry between X and Y. If f is distance-preserving but not necessarily onto, it is typically called an isometric embedding of X into Y; such maps are injective and identify X with a subset of Y that preserves all pairwise distances.

Examples in Euclidean space include translations, rotations, reflections, and the identity map. Any rigid motion in R^n is distance-preserving, as are compositions of these motions. In these cases, the shape and size of geometric figures are preserved under the map.

Key properties include that distance-preserving maps are injective, preserve convergence and limits, and map balls to balls of the same radius. They are 1-Lipschitz maps, with equality of the Lipschitz bound for all point pairs.

Distance-preserving mappings play a central role in geometry and applied fields such as computer graphics and data analysis. While many spaces admit distance-preserving embeddings into larger spaces, not every metric space can be embedded isometrically into a given target space without distortion. See also isometry, isometric embedding, and metric space.