Isometry

An isometry preserves every pairwise distance exactly. If (X,d) and (Y,\rho) are metric spaces, a map f:X\to Y is an isometry when \rho(f(x),f(y))=d(x,y) for all x,y\in X. The map is injective, because points at distance zero must coincide. When it is also onto, the spaces are called isometric: they may look different or use different coordinates, but their metric structures are identical.

Isometries preserve more than individual lengths. They preserve open and closed balls, Cauchy sequences, completeness, geodesic distances, and any property defined purely through the distance function. In Euclidean and inner-product spaces, they consequently preserve angles, perpendicularity, circles, and higher-dimensional spheres. A distance-preserving map need not be onto; an inclusion of a line into a plane, with its usual metric, is an isometric embedding but not an isometry from the whole plane.

Every Euclidean isometry is a rigid motion consisting of an orthogonal linear transformation followed by a translation. In n-dimensional Euclidean space it has the form f(x)=Ax+b, where A satisfies A^TA=I and b is a fixed vector.¹ Orthogonal matrices preserve dot products and therefore Euclidean distances.

In the plane, the principal examples are translations, rotations, reflections, and glide reflections; combinations of these produce all plane rigid motions.² The determinant of A distinguishes orientation: determinant +1 means orientation-preserving, while determinant −1 reverses orientation. In three dimensions, rotations, reflections, screw motions, and their translated forms appear. Unlike a general affine transformation, an isometry cannot stretch one direction more than another, shear a figure, or alter its angles.

The same idea extends beyond ordinary Euclidean geometry by taking the relevant distance as the starting structure. An isometry of a normed vector space preserves the norm of every difference, so it is automatically affine under standard finite-dimensional hypotheses; the linear part preserves the norm. For a discrete metric space, every bijection is an isometry because distinct points all have the same distance.

On a Riemannian manifold, an isometry preserves the metric tensor, and hence lengths of curves, angles, geodesics, curvature, and volume. The resulting transformations form a group under composition, called the isometry group; for a Riemannian manifold this group has the structure of a Lie group under standard connectedness and smoothness assumptions.³ The Myers–Steenrod theorem explains why distance-preserving bijections between connected Riemannian manifolds are automatically smooth, even if smoothness is not assumed initially.⁴

Isometry is stronger than similarity because it fixes the scale rather than merely preserving shape. A similarity may multiply all distances by a common factor, whereas an isometry requires that factor to be exactly one. This distinction matters in rigidity questions: a triangle or polyhedron can be moved without deformation, but a flexible framework may change its shape while preserving selected edge lengths.

Distance data can also recover structure. In Euclidean geometry, the Cayley–Menger determinant tests whether a collection of distances can arise from points in a specified dimension, while distance matrices support reconstruction up to Euclidean isometry. In theoretical computer science, metric embeddings seek maps that preserve distances exactly or approximately; an exact embedding is an isometric embedding, while approximation introduces distortion. In differential geometry, local isometries need not be global ones: a covering map can preserve local metric geometry while identifying distant points.

Isometric embedding

A distance-preserving map that need not be surjective.

Rigid motion

An isometry of Euclidean space, commonly a translation, rotation, reflection, or composition of these.

Orthogonal matrix

A real matrix A satisfying AᵀA = I; it preserves the Euclidean inner product and distance.

Isometry group

The group of all bijective isometries of a metric or geometric space under composition.

Similarity

A transformation that multiplies all distances by the same positive factor, not necessarily one.