A point in three-dimensional Euclidean space can be located by three coordinates.
Euclidean space is the fundamental space ofgeometry, intended to representphysical space. Originally, inEuclid'sElements, it was thethree-dimensional space ofEuclidean geometry, but in modernmathematics there areEuclidean spaces of any positive integerdimensionn, which are calledEuclideann-spaces when one wants to specify their dimension.[1] Forn equal to one or two, they are commonly called respectivelyEuclidean lines andEuclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from otherspaces that were later considered inphysics and modern mathematics.
AncientGreek geometers introduced Euclidean space for modeling the physical space. Their work was collected by theancient Greek mathematicianEuclid in hisElements,[2] with the great innovation ofproving all properties of the space astheorems, by starting from a few fundamental properties, calledpostulates, which either were considered as evident (for example, there is exactly onestraight line passing through two points), or seemed impossible to prove (parallel postulate).
After the introduction at the end of the 19th century ofnon-Euclidean geometries, the old postulates were re-formalized to define Euclidean spaces throughaxiomatic theory. Another definition of Euclidean spaces by means ofvector spaces andlinear algebra 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.[3] 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 areisomorphic. Therefore, it is usually possible to work with a specific Euclidean space, denoted or, which can be represented usingCartesian coordinates as therealn-space equipped with the standarddot product.
Euclidean space was introduced byancient Greeks as an abstraction of our physical space. Their great innovation, appearing inEuclid'sElements, was to build andprove 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 calledpostulates, oraxioms in modern language. This way of defining Euclidean space is still in use under the name ofsynthetic geometry.
In 1637,René Descartes introducedCartesian coordinates, and showed that these allow reducing geometric problems to algebraic computations with numbers. This reduction of geometry toalgebra was a major change in point of view, as, until then, thereal numbers 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 generalized Euclidean geometry to spaces of dimensionn, using both synthetic and algebraic methods, and discovered all of the regularpolytopes (higher-dimensional analogues of thePlatonic solids) that exist in Euclidean spaces of any dimension.[4]
Despite the wide use of Descartes' approach, which was calledanalytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstractvector spaces 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.
One way to think of the Euclidean plane is as aset of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to asmotions) on the plane. One istranslation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other isrotation 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 twofigures (usually considered assubsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations andreflections (seebelow).
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 inphysical theories, Euclidean space is anabstraction detached from actual physical locations, specificreference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions ofunits of length and otherphysical dimensions: the distance in a "mathematical" space is anumber, 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 areal vector spaceacts – thespace of translations which is equipped with aninner product.[1] The action of translations makes the space anaffine space, and this allows defining lines, planes, subspaces, dimension, andparallelism. The inner product allows defining distance and angles.
The set ofn-tuples of real numbers equipped with thedot product is a Euclidean space of dimensionn. Conversely, the choice of a point called theorigin and anorthonormal basis of the space of translations is equivalent with defining anisomorphism between a Euclidean space of dimensionn and viewed as a Euclidean space.
It follows that everything that can be said about a Euclidean space can also be said about Therefore, many authors, especially at elementary level, call thestandard Euclidean space of dimensionn,[5] or simplythe Euclidean space of dimensionn.
Origin-free illustration of the Euclidean plane
A reason for introducing such an abstract definition of Euclidean spaces, and for working with instead of is that it is often preferable to work in acoordinate-free andorigin-free manner (that is, without choosing a preferred basis and a preferred origin). Another reason is that there is no standard origin nor any standard basis in the physical world.
AEuclidean space is anaffine space over thereals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes calledEuclidean affine spaces to distinguish them from Euclidean vector spaces.[6]
IfE is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted Thedimension of a Euclidean space is thedimension of its associated vector space.
The elements ofE are calledpoints, and are commonly denoted by capital letters. The elements of are calledEuclidean vectors orfree vectors. They are also calledtranslations, although, properly speaking, atranslation is thegeometric transformation resulting from theaction of a Euclidean vector on the Euclidean space.
The action of a translationv on a pointP provides a point that is denotedP +v. This action satisfies
Note: The second+ in the left-hand side is a vector addition; each other+ denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of+, it suffices to look at the nature of its left argument.
The fact that the action is free and transitive means that, for every pair of points(P,Q), there is exactly onedisplacement vectorv such thatP +v =Q. This vectorv is denotedQ −P or
As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in§ Affine structure and its subsections. The properties resulting from the inner product are explained in§ Metric structure and its subsections.
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 viewed as a vector space equipped with thedot product as aninner product. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space isisomorphic to it. More precisely, given a Euclidean spaceE of dimensionn, the choice of a point, called anorigin and anorthonormal basis of defines an isomorphism of Euclidean spaces fromE to
As every Euclidean space of dimensionn is isomorphic to it, the Euclidean space is sometimes called thestandard Euclidean space of dimensionn.[5]
Some basic properties of Euclidean spaces depend only on the fact that a Euclidean space is anaffine space. They are calledaffine properties and include the concepts of lines, subspaces, and parallelism, which are detailed in next subsections.
LetE be a Euclidean space and its associated vector space.
Aflat,Euclidean subspace oraffine subspace ofE is a subsetF ofE such that
as the associated vector space ofF is alinear subspace (vector subspace) of A Euclidean subspaceF is a Euclidean space with as the associated vector space. This linear subspace is also called thedirection ofF.
IfP is a point ofF then
Conversely, ifP is a point ofE and is alinear subspace of then
is a Euclidean subspace of direction. (The associated vector space of this subspace is.)
A Euclidean vector space (that is, a Euclidean space that is equal to) 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.
In a Euclidean space, aline 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
whereP andQ are two distinct points of the Euclidean space as a part of the line.
It follows thatthere 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 throughP andQ is
whereO is an arbitrary point (not necessary on the line).
In a Euclidean vector space, the zero vector is usually chosen forO; this allows simplifying the preceding formula into
Theline segment, or simplysegment, joining the pointsP andQ is the subset of points such that0 ≤𝜆 ≤ 1 in the preceding formulas. It is denotedPQ orQP; that is
Two subspacesS andT of the same dimension in a Euclidean space areparallel if they have the same direction (i.e., the same associated vector space).[a] Equivalently, they are parallel, if there is a translation vectorv that maps one to the other:
Given a pointP and a subspaceS, there exists exactly one subspace that containsP and is parallel toS, which is In the case whereS is a line (subspace of dimension one), this property isPlayfair's axiom.
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.
The inner product of a Euclidean space is often calleddot product and denotedx ⋅y. This is specially the case when aCartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is thedot product of theircoordinate vectors. 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 will be denotedx ⋅y in the remainder of this article.
TheEuclidean norm of a vectorx is
The inner product and the norm allows expressing and provingmetric andtopological properties ofEuclidean geometry. The next subsection describe the most fundamental ones.In these subsections,Edenotes an arbitrary Euclidean space, and denotes its vector space of translations.
Thedistance (more precisely theEuclidean distance) between two points of a Euclidean space is the norm of the translation vector that maps one point to the other; that is
Thelength of asegmentPQ is the distanced(P,Q) between its endpointsP andQ. It is often denoted.
The distance is ametric, as it is positive definite, symmetric, and satisfies thetriangle inequality
Moreover, the equality is true if and only if a pointR belongs to the segmentPQ. This inequality means that the length of any edge of atriangle is smaller than the sum of the lengths of the other edges. This is the origin of the termtriangle inequality.
Two nonzero vectorsu andv of (the associated vector space of a Euclidean spaceE) areperpendicular ororthogonal if their inner product is zero:
Two linear subspaces of 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 saidperpendicular.
Two segmentsAB andAC that share a common endpointA areperpendicular orform aright angle if the vectors and are orthogonal.
IfAB andAC form a right angle, one has
This is thePythagorean theorem. 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:
Here, is used since these two vectors are orthogonal.
Positive and negative angles on the oriented plane
The (non-oriented)angleθ between two nonzero vectorsx andy in is
wherearccos is theprincipal value of thearccosine function. ByCauchy–Schwarz inequality, the argument of the arccosine is in the interval[−1, 1]. Thereforeθ is real, and0 ≤θ ≤π (or0 ≤θ ≤ 180 if angles are measured in degrees).
Angles are not useful in a Euclidean line, as they can be only 0 orπ.
In anoriented Euclidean plane, one can define theoriented angle of two vectors. The oriented angle of two vectorsx andy is then the opposite of the oriented angle ofy andx. In this case, the angle of two vectors can have any valuemodulo an integer multiple of2π. In particular, areflex angleπ <θ < 2π equals the negative angle−π <θ − 2π < 0.
The angle of two vectors does not change if they aremultiplied by positive numbers. More precisely, ifx andy are two vectors, andλ andμ are real numbers, then
IfA,B, andC are three points in a Euclidean space, the angle of the segmentsAB andAC is the angle of the vectors and As the multiplication of vectors by positive numbers do not change the angle, the angle of twohalf-lines with initial pointA can be defined: it is the angle of the segmentsAB andAC, whereB andC 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. Ifθ is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is eitherθ orπ −θ. One of these angles is in theinterval[0,π/2], and the other being in[π/2,π]. Thenon-oriented angle of the two lines is the one in the interval[0,π/2]. In an oriented Euclidean plane, theoriented angle of two lines belongs to the interval[−π/2,π/2].
Every Euclidean vector space has anorthonormal basis (in fact, infinitely many in dimension higher than one, and two in dimension one), that is abasis ofunit vectors () that are pairwise orthogonal ( fori ≠j). More precisely, given anybasis theGram–Schmidt process computes an orthonormal basis such that, for everyi, thelinear spans of and are equal.[7]
Given a Euclidean spaceE, aCartesian frame is a set of data consisting of an orthonormal basis of and a point ofE, called theorigin and often denotedO. A Cartesian frame allows defining Cartesian coordinates for bothE and in the following way.
The Cartesian coordinates of a vectorv of are the coefficients ofv on the orthonormal basis For example, the Cartesian coordinates of a vector on an orthonormal basis (that may be named as as a convention) in a 3-dimensional Euclidean space is if. As the basis is orthonormal, thei-th coefficient is equal to the dot product
The Cartesian coordinates of a pointP ofE are the Cartesian coordinates of the vector
As a Euclidean space is anaffine space, one can consider anaffine frame on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This defineaffine coordinates, sometimes calledskew coordinates for emphasizing that the basis vectors are not pairwise orthogonal.
Anaffine basis of a Euclidean space of dimensionn is a set ofn + 1 points that are not contained in a hyperplane. An affine basis definebarycentric coordinates for every point.
Many other coordinates systems can be defined on a Euclidean spaceE of dimensionn, in the following way. Letf be ahomeomorphism (or, more often, adiffeomorphism) from adenseopen subset ofE to an open subset of Thecoordinates of a pointx ofE are the components off(x). Thepolar coordinate system (dimension 2) and thespherical andcylindrical coordinate systems (dimension 3) are defined this way.
For points that are outside the domain off, 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 theantimeridian, the longitude passes discontinuously from –180° to +180°.
This way of defining coordinates extends easily to other mathematical structures, and in particular tomanifolds.
An isometry of Euclidean spaces defines an isometry of the associated Euclidean vector spaces. This implies that two isometric Euclidean spaces have the same dimension. Conversely, ifE andF are Euclidean spaces,O ∈E,O′ ∈F, and is an isometry, then the map defined by
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.
An isometry from a Euclidean space onto itself is calledEuclidean isometry,Euclidean transformation orrigid transformation. The rigid transformations of a Euclidean space form a group (undercomposition), called theEuclidean group and often denotedE(n).
The simplest Euclidean transformations aretranslations
They are in bijective correspondence with vectors. This is a reason for callingspace of translations the vector space associated to a Euclidean space. The translations form anormal subgroup of the Euclidean group.
A Euclidean isometryf of a Euclidean spaceE defines a linear isometry of the associated vector space (bylinear isometry, it is meant an isometry that is also alinear map) in the following way: denoting byQ −P the vector ifO is an arbitrary point ofE, one has
It is straightforward to prove that this is a linear map that does not depend from the choice ofO.
The map is agroup homomorphism from the Euclidean group onto the group of linear isometries, called theorthogonal group. 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 pointP form thestabilizer subgroup of the Euclidean group with respect toP. 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.
LetP be a point,f an isometry, andt the translation that mapsP tof(P). The isometry fixesP. So andthe Euclidean group is thesemidirect product of the translation group and the orthogonal group.
Thespecial orthogonal group is the normal subgroup of the orthogonal group that preserveshandedness. It is a subgroup ofindex two of the orthogonal group. Its inverse image by the group homomorphism is a normal subgroup of index two of the Euclidean group, which is called thespecial Euclidean group or thedisplacement group. Its elements are calledrigid motions ordisplacements.
Rigid motions include theidentity, translations,rotations (the rigid motions that fix at least a point), and alsoscrew motions.
Typical examples of rigid transformations that are not rigid motions arereflections, 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 reflectionr, every rigid transformation that is not a rigid motion is the product ofr and a rigid motion. Aglide reflection is an example of a rigid transformation that is not a rigid motion or a reflection.
Thetopological dimension of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are nothomeomorphic. Moreover, the theorem ofinvariance of domain asserts that a subset of a Euclidean space is open (for thesubspace topology) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension.
Euclidean spaces arecomplete andlocally compact. That is, a closed subset of a Euclidean space is compact if it isbounded (that is, contained in a ball). In particular, closed balls are compact.
The definition of Euclidean spaces that has been described in this article differs fundamentally ofEuclid'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 ofnon-Euclidean geometries.
Two different approaches have been used.Felix Klein suggested to define geometries through theirsymmetries. The presentation of Euclidean spaces given in this article, is essentially issued from hisErlangen program, with the emphasis given on the groups of translations and isometries.
InGeometric Algebra,Emil Artin has proved that all these definitions of a Euclidean space are equivalent.[9] 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 anequivalence relation on segments. One can thus define thelength 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.
BesideEuclidean geometry, Euclidean spaces are also widely used in other areas of mathematics.Tangent spaces ofdifferentiable manifolds are Euclidean vector spaces. More generally, amanifold is a space that is locally approximated by Euclidean spaces. Mostnon-Euclidean geometries can be modeled by a manifold, andembedded in a Euclidean space of higher dimension. For example, anelliptic space can be modeled by anellipsoid. It is common to represent in a Euclidean space mathematical objects that area priori not of a geometrical nature. An example among many is the usual representation ofgraphs.
Since the introduction, at the end of 19th century, ofnon-Euclidean geometries, 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 geometricalaxioms,embedding the space in a Euclidean space is a standard way for provingconsistency of its definition, or, more precisely for proving that its theory is consistent, ifEuclidean geometry is consistent (which cannot be proved).
A Euclidean space is an affine space equipped with ametric. Affine spaces have many other uses in mathematics. In particular, as they are defined over anyfield, they allow doing geometry in other contexts.
As soon as non-linear questions are considered, it is generally useful to consider affine spaces over thecomplex numbers as an extension of Euclidean spaces. For example, acircle and aline have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most ofalgebraic geometry is built in complex affine spaces and affine spaces overalgebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore calledaffine algebraic varieties.
Originally, projective spaces have been introduced by adding "points at infinity" to Euclidean spaces, and, more generally to affine spaces, in order to make true the assertion "twocoplanar lines meet in exactly one point". Projective space share with Euclidean and affine spaces the property of beingisotropic, 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 thevector lines in avector space of dimension one more.
As for affine spaces, projective spaces are defined over anyfield, and are fundamental spaces ofalgebraic geometry.
Non-Euclidean geometry refers usually to geometrical spaces where theparallel postulate is false. They includeelliptic geometry, where the sum of the angles of a triangle is more than 180°, andhyperbolic geometry, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory isconsistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of thefoundational crisis in mathematics of the beginning of 20th century, and motivated the systematization ofaxiomatic theories in mathematics.
Distances and angles can be defined on a smooth manifold by providing asmoothly varying Euclidean metric on thetangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in aRiemannian manifold. Generally,straight lines do not exist in a Riemannian manifold, but their role is played bygeodesics, 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 asphere. In this case, geodesics arearcs of great circle, which are calledorthodromes in the context ofnavigation. More generally, the spaces ofnon-Euclidean geometries can be realized as Riemannian manifolds.
^It may depend on the context or the author whether a subspace is parallel to itself
^If the condition of being a bijection is removed, a function preserving the distance is necessarily injective, and is an isometry from its domain to its image.
^Proof: one must prove that. For that, it suffices to prove that the square of the norm of the left-hand side is zero. Using the bilinearity of the inner product, this squared norm can be expanded into a linear combination of and Asf is an isometry, this gives a linear combination of and which simplifies to zero.
Coxeter, H.S.M. (1973) [1948].Regular Polytopes (3rd ed.). New York: Dover.Schläfli ... discovered them before 1853 -- a time when Cayley, Grassman and Möbius were the only other people who had ever conceived of the possibility of geometry in more than three dimensions.
