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Euclidean space is the fundamental space of
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, intended to represent
physical space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually conside ...
. Originally, that is, in Euclid's ''Elements'', it was the
three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (m ...
of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, but in modern
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
there are Euclidean spaces of any positive integer
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
, including the three-dimensional space and the ''
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the
ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...
mathematician
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
in his ''Elements'', with the great innovation of '' proving'' all properties of the space as
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s, by starting from a few fundamental properties, called ''
postulate An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s'', which either were considered as evident (for example, there is exactly one
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
passing through two points), or seemed impossible to prove (
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
). After the introduction at the end of 19th century of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, the old postulates were re-formalized to define Euclidean spaces through
axiomatic theory In mathematics and logic, an axiomatic system is any Set (mathematics), set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A Theory (mathematical logic), theory is a consistent, relatively-self-co ...
. Another definition of Euclidean spaces by means of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s and
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space. There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real -space \R^n, equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. An isomorphism from a Euclidean space to \R^n associates with each point an -tuple of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s which locate that point in the Euclidean space and are called the ''
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
'' of that point.


Definition


History of the definition

Euclidean space was introduced by
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cultu ...
as an abstraction of our physical space. Their great innovation, appearing in Euclid's ''Elements'' was to build and '' prove'' all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called
postulate An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s, or
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s in modern language. This way of defining Euclidean space is still in use under the name of
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compa ...
. In 1637,
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
introduced
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
was a major change in point of view, as, until then, the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s were defined in terms of lengths and distances. Euclidean geometry was not applied in spaces of dimension more than three until the 19th century.
Ludwig Schläfli Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
generalized Euclidean geometry to spaces of dimension , using both synthetic and algebraic methods, and discovered all of the regular
polytope In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
s (higher-dimensional analogues of the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s) that exist in Euclidean spaces of any dimension. Despite the wide use of Descartes' approach, which was called
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineerin ...
, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s allowed their use in defining Euclidean spaces with a purely algebraic definition. This new definition has been shown to be equivalent to the classical definition in terms of geometric axioms. It is this algebraic definition that is now most often used for introducing Euclidean spaces.


Motivation of the modern definition

One way to think of the Euclidean plane is as a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...
s satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as
motions In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and me ...
) on the plane. One is
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
around a fixed point in the plane, in which all points in the plane turn around that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (usually considered as
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s) of the plane should be considered equivalent (
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
) if one can be transformed into the other by some sequence of translations, rotations and
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
s (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) Bottom may refer to: Anatomy and sex * Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
). In order to make all of this mathematically precise, the theory must clearly define what is a Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in
physical Physical may refer to: *Physical examination In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...
theories, Euclidean space is an
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstr ...
detached from actual physical locations, specific
reference frames In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of
units of length A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric units, used in every country globally. In the United States the U.S. customary units ...
and other physical dimensions: the distance in a "mathematical" space is a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
, not something expressed in inches or metres. The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a
real vector space Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
, the ''space of translations'' which is equipped with an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. The action of translations makes the space an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles. The set \R^n of -tuples of real numbers equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
is a Euclidean space of dimension . Conversely, the choice of a point called the ''origin'' and an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of the space of translations is equivalent with defining an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
between a Euclidean space of dimension and \R^n viewed as a Euclidean space. It follows that everything that can be said about a Euclidean space can also be said about \R^n. Therefore, many authors, especially at elementary level, call \R^n the ''standard Euclidean space'' of dimension , or simply ''the'' Euclidean space of dimension . A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of \R^n is that it is often preferable to work in a ''coordinate-free'' and ''origin-free'' manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no origin nor any basis in the physical world.


Technical definition

A is a finite-dimensional
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. A Euclidean space is an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called ''Euclidean affine spaces'' for distinguishing them from Euclidean vector spaces. If is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted \overrightarrow E. The ''dimension'' of a Euclidean space is the
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of its associated vector space. The elements of are called ''points'' and are commonly denoted by capital letters. The elements of \overrightarrow E are called ''
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
s'' or ''
free vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
s''. They are also called ''translations'', although, properly speaking, a
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
is the
geometric transformation In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often ...
resulting of the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of a Euclidean vector on the Euclidean space. The action of a translation on a point provides a point that is denoted . This action satisfies P+(v+w)= (P+v)+w. Note: The second in the left-hand side is a vector addition; all other denote an action of a vector on a point. This notation is not ambiguous, as, for distinguishing between the two meanings of , it suffices to look on the nature of its left argument. The fact that the action is free and transitive means that for every pair of points there is exactly one
displacement vector In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
such that . This vector is denoted or \overrightarrow . As previously explained, some of the basic properties of Euclidean spaces result of the structure of affine space. They are described in and its subsections. The properties resulting from the inner product are explained in and its subsections.


Prototypical examples

For any vector space, the addition acts freely and transitively on the vector space itself. Thus a Euclidean vector space can be viewed as a Euclidean space that has itself as the associated vector space. A typical case of Euclidean vector space is \R^n viewed as a vector space equipped with the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
as an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to it. More precisely, given a Euclidean space of dimension , the choice of a point, called an ''origin'' and an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
of \overrightarrow E defines an isomorphism of Euclidean spaces from to \R^n. As every Euclidean space of dimension is isomorphic to it, the Euclidean space \R^n is sometimes called the ''standard Euclidean space'' of dimension .


Affine structure

Some basic properties of Euclidean spaces depend only of the fact that a Euclidean space is an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
. They are called affine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.


Subspaces

Let be a Euclidean space and \overrightarrow E its associated vector space. A ''flat'', ''Euclidean subspace'' or ''affine subspace'' of is a subset of such that \overrightarrow F = \left\ as the associated vector space of is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
(vector subspace) of \overrightarrow E. A Euclidean subspace is a Euclidean space with \overrightarrow F as the associated vector space. This linear subspace \overrightarrow F is also called the ''direction'' of . If is a point of then F = \left\. Conversely, if is a point of and \overrightarrow V is a
linear subspace In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
of \overrightarrow E, then P + V = \left\ is a Euclidean subspace of direction \overrightarrow V. (The associated vector space of this subspace is \overrightarrow V.) A Euclidean vector space \overrightarrow E (that is, a Euclidean space that is equal to \overrightarrow E) has two sorts of subspaces: its Euclidean subspaces and its linear subspaces. Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.


Lines and segments

In a Euclidean space, a ''line'' is a Euclidean subspace of dimension one. Since a vector space of dimension one is spanned by any nonzero vector, a line is a set of the form \left\, where and are two distinct points of the Euclidean space as a part of the line. It follows that ''there is exactly one line that passes through (contains) two distinct points.'' This implies that two distinct lines intersect in at most one point. A more symmetric representation of the line passing through and is \left\, where is an arbitrary point (not necessary on the line). In a Euclidean vector space, the zero vector is usually chosen for ; this allows simplifying the preceding formula into \left\. A standard convention allows using this formula in every Euclidean space, see . The ''
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
'', or simply ''segment'', joining the points and is the subset of points such that in the preceding formulas. It is denoted or ; that is PQ = QP = \left\.


Parallelism

Two subspaces and of the same dimension in a Euclidean space are ''parallel'' if they have the same direction (i.e., the same associated vector space). Equivalently, they are parallel, if there is a translation vector that maps one to the other: T= S+v. Given a point and a subspace , there exists exactly one subspace that contains and is parallel to , which is P + \overrightarrow S. In the case where is a line (subspace of dimension one), this property is
Playfair's axiom In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): ''In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the p ...
. It follows that in a Euclidean plane, two lines either meet in one point or are parallel. The concept of parallel subspaces has been extended to subspaces of different dimensions: two subspaces are parallel if the direction of one of them is contained in the direction to the other.


Metric structure

The vector space \overrightarrow E associated to a Euclidean space is an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often den ...
. This implies a
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear ...
\begin \overrightarrow E \times \overrightarrow E &\to \R\\ (x,y)&\mapsto \langle x,y \rangle \end that is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
(that is \langle x,x \rangle is always positive for ). The inner product of a Euclidean space is often called ''dot product'' and denoted . This is specially the case when a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
has been chosen, as, in this case, the inner product of two vectors is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of their
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensiona ...
s. For this reason, and for historical reasons, the dot notation is more commonly used than the bracket notation for the inner product of Euclidean spaces. This article will follow this usage; that is \langle x,y \rangle will be denoted in the remainder of this article. The Euclidean norm of a vector is \, x\, = \sqrt . The inner product and the norm allows expressing and proving
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
and
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
properties of
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
. The next subsection describe the most fundamental ones. ''In these subsections,'' ''denotes an arbitrary Euclidean space, and \overrightarrow E denotes its vector space of translations.''


Distance and length

The ''distance'' (more precisely the ''Euclidean distance'') between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is d(P,Q) = \Bigl\, \overrightarrow \vphantom\Bigr\, . The ''length'' of a segment is the distance between its endpoints ''P'' and ''Q''. It is often denoted , PQ, . The distance is a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
, as it is positive definite, symmetric, and satisfies the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
d(P,Q)\le d(P,R) + d(R, Q). Moreover, the equality is true if and only if a point belongs to the segment . This inequality means that the length of any edge of a
triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...
is smaller than the sum of the lengths of the other edges. This is the origin of the term ''triangle inequality''. With the Euclidean distance, every Euclidean space is a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
.


Orthogonality

Two nonzero vectors and of \overrightarrow E (the associated vector space of a Euclidean space ) are ''perpendicular'' or ''orthogonal'' if their inner product is zero: u \cdot v =0 Two linear subspaces of \overrightarrow E are orthogonal if every nonzero vector of the first one is perpendicular to every nonzero vector of the second one. This implies that the intersection of the linear subspaces is reduced to the zero vector. Two lines, and more generally two Euclidean subspaces (A line can be considered as one Euclidean subspace.) are orthogonal if their directions (the associated vector spaces of the Euclidean subspaces) are orthogonal. Two orthogonal lines that intersect are said ''perpendicular''. Two segments and that share a common endpoint are ''perpendicular'' or ''form a
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
'' if the vectors \overrightarrow and \overrightarrow are orthogonal. If and form a right angle, one has , BC, ^2 = , AB, ^2 + , AC, ^2. This is the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
. Its proof is easy in this context, as, expressing this in terms of the inner product, one has, using bilinearity and symmetry of the inner product: \begin , BC, ^2 &= \overrightarrow \cdot \overrightarrow \\ &=\left(\overrightarrow +\overrightarrow \right ) \cdot \left(\overrightarrow +\overrightarrow \right)\\ &=\overrightarrow \cdot \overrightarrow + \overrightarrow \cdot \overrightarrow -2 \overrightarrow \cdot \overrightarrow \\ &=\overrightarrow \cdot \overrightarrow + \overrightarrow \cdot\overrightarrow \\ &=, AB, ^2 + , AC, ^2. \end Here, \overrightarrow \cdot \overrightarrow = 0 is used since these two vectors are orthogonal.


Angle

The (non-oriented) ''angle'' between two nonzero vectors and in \overrightarrow E is \theta = \arccos\left(\frac\right) where is the
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
of the
arccosine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
function. By
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was published by . The corresponding inequality fo ...
, the argument of the arccosine is in the interval . Therefore is real, and (or if angles are measured in degrees). Angles are not useful in a Euclidean line, as they can be only 0 or . In an
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
Euclidean plane, one can define the ''oriented angle'' of two vectors. The oriented angle of two vectors and is then the opposite of the oriented angle of and . In this case, the angle of two vectors can have any value
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is t ...
an integer multiple of . In particular, a
reflex angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
equals the negative angle . The angle of two vectors does not change if they are multiplied by positive numbers. More precisely, if and are two vectors, and and are real numbers, then \operatorname(\lambda x, \mu y)= \begin \operatorname(x, y) \qquad\qquad \text \lambda \text \mu \text\\ \pi - \operatorname(x, y)\qquad \text. \end If , , and are three points in a Euclidean space, the angle of the segments and is the angle of the vectors \overrightarrow and \overrightarrow . As the multiplication of vectors by positive numbers do not change the angle, the angle of two
half-line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
s with initial point can be defined: it is the angle of the segments and , where and are arbitrary points, one on each half-line. Although this is less used, one can define similarly the angle of segments or half-lines that do not share an initial point. The angle of two lines is defined as follows. If is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either or . One of these angles is in the interval , and the other being in . The ''non-oriented angle'' of the two lines is the one in the interval . In an oriented Euclidean plane, the ''oriented angle'' of two lines belongs to the interval .


Cartesian coordinates

Every Euclidean vector space has an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
(in fact, infinitely many in dimension higher than one, and two in dimension one), that is a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
(e_1, \dots, e_n) of
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
s (\, e_i\, = 1) that are pairwise orthogonal (e_i\cdot e_j = 0 for ). More precisely, given any
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
(b_1, \dots, b_n), the
Gram–Schmidt process In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner prod ...
computes an orthonormal basis such that, for every , the
linear span In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
s of (e_1, \dots, e_i) and (b_1, \dots, b_i) are equal. Given a Euclidean space , a ''Cartesian frame'' is a set of data consisting of an orthonormal basis of \overrightarrow E, and a point of , called the ''origin'' and often denoted . A Cartesian frame (O, e_1, \dots, e_n) allows defining Cartesian coordinates for both and \overrightarrow E in the following way. The Cartesian coordinates of a vector of \overrightarrow E are the coefficients of on the orthonormal basis e_1, \dots, e_n. For example, the Cartesian coordinates of a vector v on an orthonormal basis (e_1,e_2,e_3) (that may be named as (x,y,z) as a convention) in a 3-dimensional Euclidean space is (\alpha_1,\alpha_2,\alpha_3) if v = \alpha_1 e_1 + \alpha_2 e_2 + \alpha_3 e_3. As the basis is orthonormal, the -th coefficient \alpha_i is equal to the dot product v\cdot e_i. The Cartesian coordinates of a point of are the Cartesian coordinates of the vector \overrightarrow .


Other coordinates

As a Euclidean space is an
affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, one can consider an
affine frame In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define
affine coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
, sometimes called ''skew coordinates'' for emphasizing that the basis vectors are not pairwise orthogonal. An
affine basis In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
of a Euclidean space of dimension is a set of points that are not contained in a hyperplane. An affine basis define
barycentric coordinates In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
for every point. Many other coordinates systems can be defined on a Euclidean space of dimension , in the following way. Let be a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
(or, more often, a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
) from a
dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of to an open subset of \R^n. The ''coordinates'' of a point of are the components of . The
polar coordinate system In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
(dimension 2) and the
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
and
cylindrical A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infini ...
coordinate systems (dimension 3) are defined this way. For points that are outside the domain of , coordinates may sometimes be defined as the limit of coordinates of neighbour points, but these coordinates may be not uniquely defined, and may be not continuous in the neighborhood of the point. For example, for the spherical coordinate system, the longitude is not defined at the pole, and on the
antimeridian The 180th meridian or antimeridian is the meridian 180° both east and west of the prime meridian in a geographical coordinate system. The longitude at this line can be given as either east or west. On Earth, these two meridians form a ...
, the longitude passes discontinuously from –180° to +180°. This way of defining coordinates extends easily to other mathematical structures, and in particular to
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s.


Isometries

An
isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
between two
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s is a bijection preserving the distance, that is d(f(x), f(y))= d(x,y). In the case of a Euclidean vector space, an isometry that maps the origin to the origin preserves the norm \, f(x)\, = \, x\, , since the norm of a vector is its distance from the zero vector. It preserves also the inner product f(x)\cdot f(y)=x\cdot y, since x \cdot y=\frac 1 2 \left(\, x+y\, ^2-\, x\, ^2-\, y\, ^2\right). An isometry of Euclidean vector spaces is a
linear isomorphism In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
. An isometry f\colon E\to F of Euclidean spaces defines an isometry \overrightarrow f \colon \overrightarrow E \to \overrightarrow F of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, if and are Euclidean spaces, , , and \overrightarrow f\colon \overrightarrow E\to \overrightarrow F is an isometry, then the map f\colon E\to F defined by f(P)=O' + \overrightarrow f\left(\overrightarrow\right) is an isometry of Euclidean spaces. It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.


Isometry with prototypical examples

If is a Euclidean space, its associated vector space \overrightarrow E can be considered as a Euclidean space. Every point defines an isometry of Euclidean spaces P\mapsto \overrightarrow , which maps to the zero vector and has the identity as associated linear map. The inverse isometry is the map v\mapsto O+v. A Euclidean frame allows defining the map \begin E&\to \R^n\\ P&\mapsto \left(e_1\cdot \overrightarrow , \dots, e_n\cdot\overrightarrow \right), \end which is an isometry of Euclidean spaces. The inverse isometry is \begin \R^n&\to E \\ (x_1\dots, x_n)&\mapsto \left(O+x_1e_1+ \dots + x_ne_n\right). \end ''This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.'' This justifies that many authors talk of \R^n as ''the'' Euclidean space of dimension .


Euclidean group

An isometry from a Euclidean space onto itself is called ''Euclidean isometry'', ''Euclidean transformation'' or ''rigid transformation''. The rigid transformations of a Euclidean space form a group (under
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
), called the ''Euclidean group'' and often denoted of . The simplest Euclidean transformations are
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
P \to P+v. They are in bijective correspondence with vectors. This is a reason for calling ''space of translations'' the vector space associated to a Euclidean space. The translations form a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of the Euclidean group. A Euclidean isometry of a Euclidean space defines a linear isometry \overrightarrow f of the associated vector space (by ''linear isometry'', it is meant an isometry that is also a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
) in the following way: denoting by the vector \overrightarrow , if is an arbitrary point of , one has \overrightarrow f(\overrightarrow )= f(P)-f(O). It is straightforward to prove that this is a linear map that does not depend from the choice of The map f \to \overrightarrow f is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
from the Euclidean group onto the group of linear isometries, called the
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
. The kernel of this homomorphism is the translation group, showing that it is a normal subgroup of the Euclidean group. The isometries that fix a given point form the
stabilizer subgroup In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of the Euclidean group with respect to . The restriction to this stabilizer of above group homomorphism is an isomorphism. So the isometries that fix a given point form a group isomorphic to the orthogonal group. Let be a point, an isometry, and the translation that maps to . The isometry g=t^\circ f fixes . So f= t\circ g, and ''the Euclidean group is the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
of the translation group and the orthogonal group.'' The
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
is the normal subgroup of the orthogonal group that preserves
handedness In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to it being stronger, faster or more Fine motor skill, dextrous. The other hand, comparatively often the weaker, less dextrous or sim ...
. It is a subgroup of
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
two of the orthogonal group. Its inverse image by the group homomorphism f \to \overrightarrow f is a normal subgroup of index two of the Euclidean group, which is called the ''special Euclidean group'' or the ''displacement group''. Its elements are called ''rigid motions'' or ''displacements''. Rigid motions include the
identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...
, translations,
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s (the rigid motions that fix at least a point), and also screw motions. Typical examples of rigid transformations that are not rigid motions are
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
s, which are rigid transformations that fix a hyperplane and are not the identity. They are also the transformations consisting in changing the sign of one coordinate over some Euclidean frame. As the special Euclidean group is a subgroup of index two of the Euclidean group, given a reflection , every rigid transformation that is not a rigid motion is the product of and a rigid motion. A
glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection ...
is an example of a rigid transformation that is not a rigid motion or a reflection. All groups that have been considered in this section are
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s and
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
s.


Topology

The Euclidean distance makes a Euclidean space a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, and thus a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
. This topology is called the
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition The Euclidean norm on \R^n is the non-negative f ...
. In the case of \mathbb R^n, this topology is also the
product topology In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-seemin ...
. The
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
s are the subsets that contains an
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
around each of their points. In other words, open balls form a base of the topology. The
topological dimension In mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topological invariant, topologically invariant way. Informal discussion F ...
of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
. Moreover, the theorem of
invariance of domain Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n. It states: :If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorph ...
asserts that a subset of a Euclidean space is open (for the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension. Euclidean spaces are
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
and
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
. That is, a closed subset of a Euclidean space is compact if it is bounded (that is, contained in a ball). In particular, closed balls are compact.


Axiomatic definitions

The definition of Euclidean spaces that has been described in this article differs fundamentally of
Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's one. In reality, Euclid did not define formally the space, because it was thought as a description of the physical world that exists independently of human mind. The need of a formal definition appeared only at the end of 19th century, with the introduction of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
. Two different approaches have been used.
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
suggested to define geometries through their
symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
. The presentation of Euclidean spaces given in this article, is essentially issued from his
Erlangen program In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is nam ...
, with the emphasis given on the groups of translations and isometries. On the other hand,
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
proposed a set of
axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
, inspired by
Euclid's postulates Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axiom ...
. They belong to
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compa ...
, as they do not involve any definition of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. Later G. D. Birkhoff and
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician a ...
proposed simpler sets of axioms, which use
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s (see
Birkhoff's axioms In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protrac ...
and
Tarski's axioms Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory (i.e., that part of Euclidean geometry that is formulabl ...
). In ''
Geometric Algebra In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the ge ...
'',
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
has proved that all these definitions of a Euclidean space are equivalent. It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following. In Hilbert's axioms, congruence is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...
on segments. One can thus define the ''length'' of a segment as its equivalence class. One must thus prove that this length satisfies properties that characterize nonnegative real numbers. Artin proved this with axioms equivalent to those of Hilbert.


Usage

Since
ancient Greeks Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of cultu ...
, Euclidean space is used for modeling
shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ...
s in the physical world. It is thus used in many
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
s such as
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, and
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
. It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as
architecture Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
,
geodesy Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
,
topography Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps. Topography is a field of geoscience and planetary sci ...
,
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
,
industrial design Industrial design is a process of design applied to physical Product (business), products that are to be manufactured by mass production. It is the creative act of determining and defining a product's form and features, which takes place in advan ...
, or
technical drawing Technical drawing, drafting or drawing, is the act and Academic discipline, discipline of composing Plan (drawing), drawings that Visual communication, visually communicate how something functions or is constructed. Technical drawing is essent ...
. Space of dimensions higher than three occurs in several modern theories of physics; see
Higher dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
. They occur also in configuration spaces of
physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
s. Beside
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
, Euclidean spaces are also widely used in other areas of mathematics.
Tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s of
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s are Euclidean vector spaces. More generally, a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
is a space that is locally approximated by Euclidean spaces. Most
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
can be modeled by a manifold, and embedded in a Euclidean space of higher dimension. For example, an
elliptic space Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
can be modeled by an
ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface;  that is, a surface that may be defined as the ...
. It is common to represent in a Euclidean space mathematical objects that are ''a priori'' not of a geometrical nature. An example among many is the usual representation of
graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
.


Other geometric spaces

Since the introduction, at the end of 19th century, of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
, many sorts of spaces have been considered, about which one can do geometric reasoning in the same way as with Euclidean spaces. In general, they share some properties with Euclidean spaces, but may also have properties that could appear as rather strange. Some of these spaces use Euclidean geometry for their definition, or can be modeled as subspaces of a Euclidean space of higher dimension. When such a space is defined by geometrical
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s,
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
the space in a Euclidean space is a standard way for proving
consistency In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
of its definition, or, more precisely for proving that its theory is consistent, if
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ...
is consistent (which cannot be proved).


Affine space

A Euclidean space is an affine space equipped with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
. Affine spaces have many other uses in mathematics. In particular, as they are defined over any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, they allow doing geometry in other contexts. As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s as an extension of Euclidean spaces. For example, a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
and a
line Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...
have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
is built in complex affine spaces and affine spaces over
algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
s. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic varieties. Affine spaces over the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s and more generally over
algebraic number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
s provide a link between (algebraic) geometry and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
. For example, the
Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
can be stated "a
Fermat curve In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation :X^n + Y^n = Z^n.\ Therefore, in terms of the affine plane its equation is :x^n ...
of degree higher than two has no point in the affine plane over the rationals." Geometry in affine spaces over a
finite fields In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtra ...
has also been widely studied. For example,
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s over finite fields are widely used in
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
.


Projective space

Originally, projective spaces have been introduced by adding "
points at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each Pencil (mathematics), pencil of parallel l ...
" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "two
coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. Howe ...
lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of being
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
, that is, there is no property of the space that allows distinguishing between two points or two lines. Therefore, a more isotropic definition is commonly used, which consists as defining a projective space as the set of the
vector line In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s in a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of dimension one more. As for affine spaces, projective spaces are defined over any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, and are fundamental spaces of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.


Non-Euclidean geometries

''Non-Euclidean geometry'' refers usually to geometrical spaces where the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segment ...
is false. They include
elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines a ...
, where the sum of the angles of a triangle is more than 180°, and
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is
consistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent i ...
(if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the
foundational crisis in mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...
of the beginning of 20th century, and motivated the systematization of axiomatic theories in mathematics.


Curved spaces

A
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
is a space that in the neighborhood of each point resembles a Euclidean space. In technical terms, a manifold is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
, such that each point has a
neighborhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
that is
homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to an
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of a Euclidean space. Manifolds can be classified by increasing degree of this "resemblance" into
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
s,
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s,
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s, and
analytic manifold In mathematics, an analytic manifold, also known as a C^\omega manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic ge ...
s. However, none of these types of "resemblance" respect distances and angles, even approximately. Distances and angles can be defined on a smooth manifold by providing a smoothly varying Euclidean metric on the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
. Generally,
straight line In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
s do not exist in a Riemannian manifold, but their role is played by
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
s, which are the "shortest paths" between two points. This allows defining distances, which are measured along geodesics, and angles between geodesics, which are the angle of their tangents in the tangent space at their intersection. So, Riemannian manifolds behave locally like a Euclidean space that has been bent. Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a
sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. In this case, geodesics are arcs of great circle, which are called orthodromes in the context of
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
. More generally, the spaces of
non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...
can be realized as Riemannian manifolds.


Pseudo-Euclidean space

An
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
of a real vector space is a
positive definite bilinear form Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
, and so characterized by a
positive definite quadratic form Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posit ...
. A
pseudo-Euclidean space In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real -space together with a non-degenerate quadratic form . Such a quadratic form can, given a suitable choice of basis , be applied to a vector , giving q(x) ...
is an affine space with an associated real vector space equipped with a
non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent defin ...
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
(that may be indefinite). A fundamental example of such a space is the
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...
, which is the
space-time In physics, spacetime is a mathematical model that combines the three-dimensional space, three dimensions of space and one dimension of time into a single four-dimensional manifold. Minkowski diagram, Spacetime diagrams can be used to visualize S ...
of
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
. It is a four-dimensional space, where the metric is defined by the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
x^2+y^2+z^2-t^2, where the last coordinate (''t'') is temporal, and the other three (''x'', ''y'', ''z'') are spatial. To take
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
into account,
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
uses a
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
that has Minkowski spaces as
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s. The
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
of this manifold at a point is a function of the value of the
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
at this point.


See also

*
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
, a generalization to infinite dimension, used in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...


Footnotes


References

* * * * * * {{Authority control Euclidean geometry Linear algebra Homogeneous spaces Norms (mathematics)