Overview

Isometry is a mathematical notion for a transformation that preserves distances. Informally, an isometry moves or repositions a figure without stretching, shrinking, or otherwise changing the distance between any pair of points. In Euclidean geometry these are often called rigid motions; in metric and differential geometry the concept generalizes to maps that preserve the metric or the length of curves.

Key properties and common types

An isometry f between metric spaces (X, d_X) and (Y, d_Y) satisfies d_Y(f(x), f(x')) = d_X(x, x') for every pair of points x, x' in X. Important features include:

  • Distance preservation: all pairwise distances remain unchanged.
  • Bijectivity in many settings: an isometry of a space onto itself is typically bijective and invertible, with the inverse also an isometry.
  • Composition closure: composing two isometries yields another isometry; they form a group under composition in the case of self-isometries.

In the Euclidean plane and space the familiar types are:

  • Translations — shifting every point by a fixed vector.
  • Rotations — turning the plane or space about a fixed center or axis.
  • Reflections — flipping across a line or plane.
  • Glide reflections — a combination of a reflection and a translation parallel to the reflecting line.

Historical and conceptual context

The idea of preserving lengths under transformation goes back to classical geometry, where congruence of figures was central. Over time the notion was abstracted: in metric spaces it is a distance-preserving map, and in Riemannian geometry an isometry is a diffeomorphism that preserves the metric tensor, hence lengths of curves and angles. Studying groups of isometries links geometry with algebra and symmetry theory.

Examples, applications and significance

Isometries appear in many areas: proving two polygons are congruent in elementary geometry; describing symmetries of molecules and crystals in chemistry; manipulating objects in computer graphics and CAD without distortion; and classifying homogeneous spaces in differential geometry. In applied settings exact isometries may be idealizations, but they serve as useful models for rigid-body motions in physics and engineering.

Distinctions and notable remarks

Not every distance-preserving map between subsets can be extended to the whole ambient space; such maps may be called isometric embeddings rather than global isometries. Also, orientation may be preserved or reversed: rotations and translations preserve orientation, whereas reflections reverse it. In many proofs and constructions the algebraic structure of the isometry group—orthogonal and Euclidean groups in the classical cases—plays a central role.

For additional formal definitions and further reading about related topics, see related resources.