Isometry

An isometry is a map between metric spaces that preserves the distance between every pair of points. It preserves geometric shape and scale, though it need not map onto the entire target space.

Metric space: A set equipped with a distance function satisfying nonnegativity, identity, symmetry, and the triangle inequality. Isometries are defined by preserving the distance function on metric spaces.
Metric: A function assigning distances to pairs of points while satisfying the metric axioms. An isometry preserves the values assigned by the source and target metrics.
Similarity transformation: A transformation that scales all distances by the same positive factor. Unlike an isometry, a non-unit similarity changes the overall scale.
Congruence (geometry): Equality of geometric figures up to a distance-preserving transformation. Congruent Euclidean figures are related by an isometry of the plane or space.
Isometric embedding: A distance-preserving map from one metric space into another, not necessarily onto. It captures the general case when an isometry's image need not fill its target.
Injective function: A function that maps distinct inputs to distinct outputs. A distance-preserving map must be injective, since distinct points have positive distance.
Bi-Lipschitz map: A map that bounds distances above and below by fixed positive multiples. Bi-Lipschitz maps control distortion without requiring exact distance preservation.
Riemannian isometry: A diffeomorphism between Riemannian manifolds that preserves their metric tensors. This smooth-geometric form preserves the lengths of curves and the induced distances.
Surjective isometry: A distance-preserving map whose image is the entire target space. Surjectivity distinguishes spaces that are fully distance-preservingly equivalent.
Surjective function: A function whose image equals its entire codomain. Distance preservation alone does not guarantee that every target point is reached.

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