This article is about distance-preserving functions. For other mathematical uses, see isometry (disambiguation). For non-mathematical uses, see Isometric.
Not to be confused with Isometric projection.
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.[a] The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion.
^Coxeter 1969, p. 46
3.51Any direct isometry is either a translation or a rotation. Any opposite isometry is either a reflection or a glide reflection.
^Coxeter 1969, p. 29
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In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed...
In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric...
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space E n {\displaystyle \mathbb {E} ^{n}} ; that is, the transformations...
mathematical functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel...
{1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).} An isometry of Euclidean vector spaces is a linear isomorphism. An isometry f : E → F {\displaystyle f\colon E\to F}...
In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical...
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors...
A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set...
If h is a translation, then its conjugation by an isometry can be described as applying the isometry to the translation: the conjugation of a translation...
in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a...
rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and...
rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the...
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In mathematical finite group theory, the Dade isometry is an isometry from class function on a subgroup H with support on a subset K of H to class functions...
bijective distance-preserving function is called an isometry. One perhaps non-obvious example of an isometry between spaces described in this article is the...
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follows: The Gaussian curvature of a surface is invariant under local isometry. A sphere of radius R has constant Gaussian curvature which is equal to...
are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow...