A distance preserving transformation.

A transformation that maps every segment to a congruent segment. Also called a congruence mapping.

If (''M''1, ''d''1) and (''M''2, ''d''2) are metric spaces, an isometry from ''M''1 to ''M''2 is a function ''f'' : ''M''1 â†’ ''M''2 such that ''d''2(''f''(''x''), ''f''(''y'')) = ''d''1(''x'', ''y'') for all ''x'', ''y'' in ''M''1. Every isometry is injective, although not every isometry is surjective.

a one-to-one mapping of one metric space into another metric space that preserves the distances between each pair of points; "the isometries of the cube"

a distance-preserving bijection between metric spaces

a function f with the stated property, which is then necessarily injective but may fail to be surjective

a linear transformation which preserves length

an automorphism of the lattice which preserves the inner product

a plane transformation which preserves distances, and therefore shapes of figures

a special type of dilation that preserves shape and size

a transformation of a geometric space that leaves distances between points unchanged

a transformation that preserves distance

A transformation that is distancepreserving. The distance between two points is the same as the distance between the corresponding images of these points.

In the study of Riemannian geometry in mathematics, a local isometry from one (pseudo-)Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first. When such a map is also a diffeomorphism, such a map is called an isometry (or isometric isomorphism), and provides a notion of isomorphism ("sameness") in the category Rm of Riemannian manifolds.