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Distance-preserving mathematical transformation
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed
Isometry
Function between two metric spaces that only respects their large-scale geometry
In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale
Quasi-isometry
Isometry group of Euclidean space
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
Euclidean_group
Fundamental space of geometry
{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}
Euclidean_space
Term in stochastic calculus
In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable
Itô_isometry
Automorphism group of a metric space or pseudo-Euclidean space
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
Isometry_group
Topics referred to by the same term
Isometry group Quasi-isometry Dade isometry Euclidean isometry Euclidean plane isometry Itō isometry Isometric (disambiguation) Isometries in physics This
Isometry_(disambiguation)
In 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
Partial_isometry
Groups of point isometries in 3 dimensions
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
Point groups in three dimensions
Point_groups_in_three_dimensions
In mathematics, a piecewise isometry is a dynamical system that consists of finitely many Euclidean isometries acting in different places, including rotations
Piecewise_isometry
Isometry of the Eluclidean plane
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
Euclidean_plane_isometry
Smooth manifold with an inner product on each tangent space
surface is called a local isometry. A property of a surface is called an intrinsic property if it is preserved by local isometries and it is called an extrinsic
Riemannian_manifold
Matrix property in linear algebra
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors
Restricted_isometry_property
Rotation composed with a reflection
rotation-reflection, rotoreflection, rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and
Improper_rotation
Polynomial with all terms of degree two
T : V → V′ (isometry) such that Q ( v ) = Q ′ ( T v ) for all v ∈ V . {\displaystyle Q(v)=Q'(Tv){\text{ for all }}v\in V.} The isometry classes of n-dimensional
Quadratic_form
Theorem in manifold theory
at p. The lemma allows the exponential map to be understood as a radial isometry, and is of fundamental importance in the study of geodesic convexity and
Gauss's lemma (Riemannian geometry)
Gauss's_lemma_(Riemannian_geometry)
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
preserves the metric. Flows generated by Killing vector fields are continuous isometries of the manifold. This means that the flow generates a symmetry, in the
Killing_vector_field
Group of transformations under which the object is invariant
For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups
Symmetry_group
Feature of a system that is preserved under some transformation
spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed
Symmetry_(physics)
3D symmetry group
and octahedron. It is the hyperoctahedral group for n = 3. See also the isometries of the cube. With the 4-fold axes as coordinate axes, a fundamental domain
Octahedral_symmetry
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
Dade_isometry
Type of topological group
discrete isometry group is an isometry group such that for every point of the metric space the set of images of the point under the isometries is a discrete
Discrete_group
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
Conjugation of isometries in Euclidean space
Conjugation_of_isometries_in_Euclidean_space
Every rigid motion is a screw displacement
direct Euclidean isometry in three dimensions involves a translation and a rotation. The screw displacement representation of the isometry decomposes the
Chasles'_theorem_(kinematics)
Subgroup of a root system's isometry group
group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated
Weyl_group
isometric linear operators on a given Hilbert space. It states that every isometry is a direct sum of copies of the unilateral shift and a unitary operator
Wold's_decomposition
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
Fixed points of isometry groups in Euclidean space
Fixed_points_of_isometry_groups_in_Euclidean_space
Mathematical transformation that preserves distances
rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the
Rigid_transformation
Group of flat spacetime symmetries
Poincaré (1905), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. It is a ten-dimensional non-abelian Lie group
Poincaré_group
Result of differential geometry proved by Gauss
invariant under isometries. Finally, an equation linking Gaussian curvature to Christoffel symbols shows that it is also invariant under isometries. Let S ,
Theorema_Egregium
Topics referred to by the same term
growth or over evolutionary time do not lead to changes in proportion. Isometry and isometric embeddings in mathematics, a distance-preserving representation
Isometric
Topics referred to by the same term
to: Dade (surname) Dade City, Florida Miami-Dade County, Florida Dade isometry Dade's conjecture Dade (1135–1139), era name used by Emperor Chongzong
Dade
Geometric transformation combining reflection and translation
between points are not changed under glide reflection, it is a motion or isometry. When the context is the two-dimensional Euclidean plane, the hyperplane
Glide_reflection
Geometry concept
two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a
Point groups in two dimensions
Point_groups_in_two_dimensions
Surjective isometries are affine mappings
and the mapping f : V → W {\displaystyle f\colon V\to W} is a surjective isometry, then f {\displaystyle f} is affine. It was proved by Stanisław Mazur and
Mazur–Ulam_theorem
Mapping from a Euclidean space to itself
spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis
Reflection_(mathematics)
Mathematical concept
two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another. A rotation in the plane can be formed
Rotations and reflections in two dimensions
Rotations_and_reflections_in_two_dimensions
automorphisms of the underlying real surface; if one allows orientation-reversing isometries, this yields a group twice as large, of order 168(g − 1), which is sometimes
Hurwitz_surface
If geometry is regarded as the study of isometry groups, then a centre is a fixed point of all the isometries that move the object onto itself. The centre
Centre_(geometry)
Metric of a homogenous universe
d\phi ^{2})} The isometry group of this spacetime is R × S O ( 3 ) {\displaystyle \mathbb {R} \times SO(3)} . Remarkably, the isometry group does not act
Kantowski–Sachs_metric
Mathematical concept
g_{n}\rangle _{n}} One can prove that the Frobenius characteristic map is an isometry by explicit computation. To show this, it suffices to assume that f , g
Frobenius_characteristic_map
(pseudo-)Riemannian manifold whose geodesics are reversible
manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied
Symmetric_space
Relationship between two figures of the same shape and size, or mirroring each other
congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation
Congruence_(geometry)
Basic result in the algebraic theory of quadratic forms, on extending isometries
quadratic forms: any isometry between two subspaces of a nonsingular quadratic space over a field k may be extended to an isometry of the whole space.
Witt's_theorem
Linear algebra concept
}A=I{\text{ or }}AA^{\operatorname {T} }=I.\,} A semi-orthogonal matrix is an isometry. This means that it preserves the norm either in row space, or column space
Semi-orthogonal_matrix
Mathematical space with a notion of distance
bijective distance-preserving function is called an isometry. One perhaps non-obvious example of an isometry between spaces described in this article is the
Metric_space
Planar movement within a Euclidean space without rotation
of the coordinate system. In a Euclidean space, any translation is an isometry. A slide is a translation along a screw axis, around which a rotation may
Translation_(geometry)
Type of orthographic projection
drawings free of optical distortion. This would lead him to formulate isometry. Isometry means "equal measures" because the same scale is used for height,
Axonometric_projection
Property of an object that is not congruent to its mirror image
orientation concept: an isometry is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality
Chirality_(mathematics)
PO(2k+1) is isometries of RP2k = P(R2k+1), while PO(2k) is isometries of RP2k−1 = P(R2k) – the odd-dimensional (vector) group is isometries of even-dimensional
Projective_orthogonal_group
Calculus of stochastic differential equations
Itô isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itô isometry. The
Itô_calculus
The isometry group of a Riemannian manifold is a Lie group
distance-preserving surjective map (that is, an isometry of metric spaces) between two connected Riemannian manifolds is a smooth isometry of Riemannian manifolds. A simpler
Myers–Steenrod_theorem
One-dimensional complex manifold
The isometry group of a uniformized Riemann surface (equivalently, the conformal automorphism group) reflects its geometry: genus 0 – the isometry group
Riemann_surface
52-dimensional exceptional simple Lie group
fundamental representation is 26-dimensional. The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective
F4_(mathematics)
G {\displaystyle g\in G} to k g {\displaystyle kg} . This action is an isometry of the word metric. The proof is simple: the distance between k g {\displaystyle
Word_metric
Mathematical shape
Animation showing the local isometry of a helicoid segment and a catenoid segment.
Helicoid
Theorem that the diffeomorphism group of the 3-sphere has the homotopy-type of O(4)
the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in 1983 by Allen Hatcher
Smale_conjecture
Property that is not changed by mathematical transformations
For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to"
Invariant_(mathematics)
This result goes back, in different form, before the notion of quasi-isometry was formally introduced, to the work of Albert S. Schwarz (1955) and John
Švarc–Milnor_lemma
Unit-distance-preserving maps are isometries
homomorphism from the unit distance graph of the plane to itself must be an isometry of the plane. The theorem is named after Frank S. Beckman and Donald A
Beckman–Quarles_theorem
Mathematical study of linear operators
Hilbert spaces is a canonical factorization as the product of a partial isometry and a non-negative operator. The polar decomposition for matrices generalizes
Operator_theory
Geometric symmetry operation
fixed. In Euclidean or pseudo-Euclidean spaces, a point reflection is an isometry (preserves distance). In the Euclidean plane, a point reflection is the
Point_reflection
Icelandic mathematician (1927–2023)
symmetric spaces as well as some new results on the representations of their isometry groups. He also introduced a Fourier transform on these spaces and proved
Sigurður Helgason (mathematician)
Sigurður_Helgason_(mathematician)
Concept in mathematics
group of isometries of X {\displaystyle X} acts by homeomorphisms on ∂ X {\displaystyle \partial X} . This action can be used to classify isometries according
Hyperbolic_metric_space
Non-Euclidean geometry
other according to the previous paragraph, and in each case an explicit isometry can be explicitly given. Here is a list of the better-known models which
Hyperbolic_space
Topological space in group theory
automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is
Homogeneous_space
proved the local version. Ambrose proved a global version that allows for isometries between general Riemannian manifolds with varying curvature, in 1956.
Cartan–Ambrose–Hicks_theorem
Model of hyperbolic geometry
or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group PSU(1
Poincaré_disk_model
Formula relating the norm and the inner product in an inner product space
H;} that is, linear isometries preserve inner products. If A : H → Z {\displaystyle A:H\to Z} is instead an antilinear isometry then ⟨ A h , A k ⟩ Z
Polarization_identity
(named after Grigory Margulis) is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifold (e.g. the hyperbolic n-space)
Margulis_lemma
Symmetric subdivision in hyperbolic geometry
vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
Model of n-dimensional hyperbolic geometry
Minkowski bilinear form. In a different language, it is the group of linear isometries of the Minkowski space. In particular, this group preserves the hyperboloid
Hyperboloid_model
Method for visually representing three-dimensional objects
(1759–1837), the concept of isometry had existed in a rough empirical form for centuries. From the middle of the 19th century, isometry became an "invaluable
Isometric_projection
Classification of a two-dimensional repetitive pattern
topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations. Two such isometry groups are of the same
Wallpaper_group
Surjective bounded operator on a Hilbert space preserving the inner product
an isometry. The other weaker condition, UU* = I, defines a coisometry. Thus a unitary operator is a bounded linear operator that is both an isometry and
Unitary_operator
Australian and American mathematician (born 1975)
introduced the notion of a "restricted linear isometry," which is a matrix that is quantitatively close to an isometry when restricted to certain subspaces.[CT05]
Terence_Tao
itself which preserves the distance between each pair of points (i.e., an isometry). In general, every kind of structure in mathematics will have its own
Symmetry_in_mathematics
Rigidity theorem in differential geometry
A compact Bonnet pair: two non-congruent immersed real analytic tori that are related by a mean curvature preserving isometry.
Bonnet_theorem
Real square matrix whose columns and rows are orthogonal unit vectors
matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In
Orthogonal_matrix
Group of symmetries of a regular polygon
multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors
Dihedral_group
Non-periodic tiling of the plane
However, the tiling produced in this way is not unique, not even up to isometries of the Euclidean group, e.g. translations and rotations. When going to
Ammann_A1_tilings
Product of the principal curvatures of a surface
surface S in R3. A local isometry is a diffeomorphism f : U → V between open regions of R3 whose restriction to S ∩ U is an isometry onto its image. Theorema
Gaussian_curvature
Theorem about the dual of a Hilbert space
space can be extended to be a dense subset of a unique (up to bijective isometry) complex Hilbert space, called its complexification, which is why Hilbert
Riesz_representation_theorem
the norm (length) of every vector is preserved (a property known as an isometry). It is named after Stefan Banach and Stanisław Mazur. Every real, separable
Banach–Mazur_theorem
American mathematician
Dade isometry and Dade's conjecture. While an undergraduate at Harvard University, he became a Putnam Fellow twice, in 1955 and 1957. The Dade isometry is
Everett_C._Dade
In mathematics, invertible homomorphism
depending on the type of structure under consideration. For example: An isometry is an isomorphism of metric spaces. A homeomorphism is an isomorphism of
Isomorphism
Matrix decomposition
axes. These directions happen to be mutually orthogonal. Apply first an isometry V ∗ {\displaystyle \mathbf {V} ^{*}} sending these directions to the
Singular_value_decomposition
Type of matrix representation
an isometry when its action is restricted onto the support of A {\displaystyle A} , that is, it means that U {\displaystyle U} is a partial isometry. As
Polar_decomposition
Differentiable manifold
a nilpotent group of isometries acting transitively on it. The requirement that the transitive nilpotent group acts by isometries leads to the following
Nilmanifold
This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. List of curves topics
List of differential geometry topics
List_of_differential_geometry_topics
Manifold that "locally looks like" Euclidean space
Riemannian manifold has a smooth unit-length vector field, and that an isometry from one of the above model examples is provided by considering an integral
Flat_manifold
group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group of a 3-sphere. 1889 Édouard
Point groups in four dimensions
Point_groups_in_four_dimensions
Isometry group of a compact Riemannian manifold with negative Ricci curvature is finite
manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite. The theorem is a corollary of Bochner's
Bochner's theorem (Riemannian geometry)
Bochner's_theorem_(Riemannian_geometry)
Linear or affine transformation which is its own inverse
an isometry. The two extreme cases for which this always applies are the identity function and inversion in a point. The other involutive isometries are
Affine_involution
On surjectivity of linear map to anti-dual
Hausdorff pre-Hilbert space to be a Hilbert space in terms of the canonical isometry of a pre-Hilbert space into its anti-dual. Suppose that H is a topological
Fundamental theorem of Hilbert spaces
Fundamental_theorem_of_Hilbert_spaces
Conjecture in ergodic theory
S^{2}} . Any matrix U ∈ S U ( 2 ) {\displaystyle U\in SU(2)} defines an isometry of the sphere S 2 {\displaystyle S^{2}} , which in turn defines an operator
Spectral_gap_conjecture
Geometrical object in four-dimensional space
and is often called the minimal Clifford torus; its images under the isometries of S3 are also minimal. The Clifford torus is named after William Kingdon
Clifford_torus
Four-dimensional analogue of the tetrahedron
In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol {3,3,3}. It is a 5-vertex four-dimensional object bounded by five tetrahedral cells
5-cell
Function that returns its argument unchanged
function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type C 1
Identity_function
Isomorphism of an object to itself
In metric geometry an automorphism is a self-isometry. The automorphism group is also called the isometry group. In the category of Riemann surfaces, an
Automorphism
ISOMETRY
ISOMETRY
ISOMETRY
ISOMETRY
Boy/Male
Indian
An authority on genealogy and the Quran
Surname or Lastname
Cornish
Cornish : habitational name from a minor place named Kellow, from Cornish kellow, plural of kelli ‘wood’, ‘grove’.English : habitational name from Kelloe in Durham, named from Old English celf ‘calf’ + hlÄw ‘hill’.Scottish : from the lands of Kelloe in Berwickshire, or in some cases possibly a variant of Kellogg.
Surname or Lastname
English (West Midlands)
English (West Midlands) : habitational name from a place in the West Midlands, recorded in Domesday Book as Tibintone, probably ‘settlement (Old English tūn) associated with Tibba’, an Old English personal name of obscure origin.
Surname or Lastname
English and French
English and French : nickname for a wise man, from Middle English, Old French sage ‘learned’, ‘sensible’, from Latin sagus ‘prophetic’, akin to sagax ‘sharp’, ‘perceptive’.Irish : variant of Savage, via the Gaelicized form Sabhaois.German : habitational name from a place near Oldenburg, so named from an old word, sege ‘sedge’, ‘reed’.
Girl/Female
Tamil
Bhavyasri | பாவà¯à®¯à®¸à¯à®°à¯€
Grand, Splendid
Girl/Female
Tamil
Marichika | மாரிசிகா
Mirage, Ray
Girl/Female
Biblical
Anger, heat, a wall.
Boy/Male
Muslim
Clouds at night, Name of a companion of the prophet
Girl/Female
Arabic, Muslim
Another Name for Hazrat Fatimah Zahra
Girl/Female
Muslim
Bright Sky
ISOMETRY
ISOMETRY
ISOMETRY
ISOMETRY
ISOMETRY