Euclidean space

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A point in three-dimensional Euclidean space can be located by three coordinates. Coord system CA 0.svg
A point in three-dimensional Euclidean space can be located by three coordinates.

Euclidean space is the fundamental space of geometry. Originally, this was the three-dimensional space of Euclidean geometry, but, in modern mathematics, there are Euclidean spaces of any nonnegative integer dimension, [1] including the three-dimensional space and the Euclidean plane (dimension two). It has been introduced by the Ancient Greek mathematician Euclid of Alexandria, [2] and the qualifier Euclidean has been added for distinguishing it from other spaces that are considered in physics and modern mathematics.

Geometry Branch of mathematics that studies the shape, size and position of objects

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

Three-dimensional space geometric three-parameter model of the physical universe

Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension.

Euclidean geometry Mathematical system attributed to Euclid

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

Contents

Ancient Greek geometers introduced Euclidean space for modeling the physical universe. Their great innovation was to prove all properties of the space (theorems) by starting from a few fundamental properties, called postulates , which either were considered as evidences (for example, there is exactly one straight line passing through two points), or seemed impossible to prove (parallel postulate).

In religion and esotericism, the term "physical universe" or "material universe" is used to distinguish the physical matter of the universe from a proposed spiritual or supernatural essence.

Mathematical proof rigorous demonstration that a mathematical statement follows from its premises and assumed axioms

A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with accepted rules of inference. Proofs are examples of exhaustive deductive reasoning or exhaustive inductive reasoning which establish logical certainty, and are distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Enumerating many confirmatory cases is not enough for a proof, which must demonstrate that the statement is always true. An unproven proposition that is believed to be true is known as a conjecture.

Theorem In mathematics, a statement that has been proved

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

After the introduction at the end of 19th century of non-Euclidean geometries, the old postulates have been formalized for defining Euclidean spaces through an axiomatic theory. Another definition of Euclidean spaces by mean of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definitions. This is this definition that is more commonly used in modern mathematics, and detailed in this article. [3]

Vector space Algebraic structure which is fundamental for linear algebra

A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below, in § Definition. For specifying that the scalars are real or complex numbers, the terms real vector space and complex vector space are often used.

Linear algebra branch of mathematics

Linear algebra is the branch of mathematics concerning linear equations such as

For 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. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real n-space equipped with the dot product. An isomorphism from a Euclidean space to associates to each point a n-tuple of real numbers, which locate them in the Euclidean space and are called Cartesian coordinates .

Isomorphism In mathematics, invertible homomorphism

In mathematics, an isomorphism is a homomorphism or morphism that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called "the" inner product of Euclidean space even though it is not the only inner product that can be defined on Euclidean space; see also inner product space.

In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence of n elements, where n is a non-negative integer. There is only one 0-tuple, an empty sequence, or empty tuple, as it is referred to. An n-tuple is defined inductively using the construction of an ordered pair.

Definition

History of the definition

Euclidean space was introduced by ancient Greeks 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 postulates, or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry.

Greek mathematics Mathematics texts written by and ideas stemming from Ancient Greek mathematicians

Greek mathematics refers to mathematics texts written by and ideas stemming from the Classical and Hellinistic (i.e. post-Alexander the Great Greek culture, extant from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture and language. The word "mathematics" itself derives from the Ancient Greek: μάθημα, romanized: máthēmaAttic Greek: [má.tʰɛː.ma]Koine Greek: [ˈma.θi.ma], meaning "subject of instruction". The study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations.

Euclids <i>Elements</i> Mathematical treatise by Euclid

The Elements is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions, and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century.

An axiom or postulate 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 Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'

In 1637, René Descartes introduced Cartesian coordinates and showed that this allows reducing geometric problems to algebraic computations with numbers. This reduction of geometry to algebra was a major change of point of view, as, until then, the real numbers—that is, rational numbers and non-rational numbers together–were defined in terms of geometry, as lengths and distance.

René Descartes 17th-century French philosopher, mathematician, and scientist

René Descartes was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. One of the most notable intellectual figures of the Dutch Golden Age, Descartes is also widely regarded as one of the founders of modern philosophy.

Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

Real number Number representing a continuous quantity

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more.

Euclidean geometry was not applied in spaces of more than three dimensions until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of n dimensions using both synthetic and algebraic methods, and discovered all of the regular polytopes (higher-dimensional analogues of the Platonic solids) that exist in Euclidean spaces of any number of dimensions. [4]

Despite the wide use of Descartes' approach, which was called analytic geometry, the definition of Euclidean space remained unchanged until the end of 19th century. The introduction of abstract vector 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.

Motivation of the modern definition

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles. For example, there are two fundamental operations (referred to as motions) on the plane. One is translation, 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 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 subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations, rotations and reflections (see below).

In order to make all of this mathematically precise, the theory must clearly define what is an Euclidean space, and the related notions of distance, angle, translation, and rotation. Even when used in physical theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments, and so on. A purely mathematical definition of Euclidean space also ignores questions of units of length and other physical dimensions: the distance in a "mathematical" space is a number, not something expressed in inches or metres.

The standard way to mathematically define an Euclidean space, as carried out in the remainder of this article, is to define an Euclidean space as a set of points on which acts a real vector space, the space of translations which is equipped with an inner product. [1] The action of translations makes the space an affine space, and this allow defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles.

The set of n-tuples of real numbers equipped with the dot product is a Euclidean space of dimension n. Conversely, the choice of a point called the origin and an orthonormal basis of the space of translations is equivalent with defining an isomorphism between a Euclidean space of dimension n 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, specially at elementary level, call the standard Euclidean space of dimension n, [5] or simply the Euclidean space of dimension n.

A reason for introducing such an abstract definition of Euclidean spaces, and for working with it instead of 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 Euclidean vector space is a finite-dimensional inner product space over the real numbers.

A Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space. Euclidean spaces are sometimes called Euclidean spaces for distinguishing them from Euclidean vector spaces. [6]

The dimension of a Euclidean space is the dimension of its associated vector space.

If E is a Euclidean space, its associated vector space is also called its space of translations, and often denoted

The elements of E are called points and are commonly denoted by capital letters. The elements of are called translations, Euclidean vectors or free vectors .

The action of a translation v on a point P provides a point that is denoted P + v. This action satisfies

(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 (P, Q there is exactly one vector v such that P + v = Q. This vector v is denoted QP or

As previously explained, some of the basic properties of Euclidean spaces result of 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.

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 associated vector space.

A typical case of Euclidean vector space is viewed as a vector space equipped with the dot product as an inner product. The importance of this particular example of Euclidean space lies is the fact that every Euclidean space is isomorphic to it. More precisely, given a Euclidean space E of dimension n, the choice of a point, called a origin and a orthonormal basis of defines an isomorphism of Euclidean spaces from E to

As every Euclidean space of dimension n is isomorphic to it, the Euclidean space is sometimes called the standard Euclidean space of dimension n. [5]

Affine structure

Some basic properties of Euclidean spaces depend only of the fact that an Euclidean space is an affine space. they are called affine properties and include the concepts of lines, subspaces, and parallelism. which are detailed in next subsections.

Subspaces

Let E a Euclidean space, and its associated vector space.

A flat, Euclidean subspace or affine subspace of E is a subset F of E such that

is a linear subspace of An Euclidean subspace F is a Euclidean space, with as associated vector space. This linear subspace is called the direction of F.

If P is a point of F then

Conversely, if P is a point of E, and V is a linear subspace of then

is a Euclidean subspace of direction V.

A Euclidean vector space (that is a Euclidean space such that ) 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

where P and Q are two distinct points.

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 P and Q is

where O is an arbitrary point (not necessary on the line).

In a Euclidean vector space, the zero vector is usually chosen for O; this allows simplifying the preceding formula into

A standard convention allows using this formula in every Euclidean space, see Affine space § Affine combinations and barycenter.

The line segment , or simply segment, joining the points P and Q is the subset of the points such that 0 ≤ λ ≤ 1 in the preceding formulas. It is denoted PQ or QP; that is

Parallelism

Two subspaces S and T of the same dimension in a Euclidean space are parallel if they have the same direction. [lower-alpha 1] Equivalently, they are parallel, if there is a translation v vector that maps one to the other:

Given a point P and a subspace S, there exists exactly one subspace that contains P and is parallel to S, which is In the case where S is a line (subspace of dimension one), this property is Playfair'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.

Metric structure

The vector space associated to a Euclidean space E is an inner product space. This implies a symmetric bilinear form

that is positive definite (that is is always positive for x ≠ 0).

The inner product of a Euclidean space is often called dot product and denoted xy. This is specially the case when a Cartesian coordinate system has been chosen, as, in this case, the inner product of two vectors is the dot product of their coordinate 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 denoted xy in the remainder of this article.

The Euclidean norm of a vector v is

The inner product and the norm allows expressing and proving all metric and topological properties of Euclidean geometry. The next subsection describe the most fundamental ones. In these subsections,Edenotes an arbitrary Euclidean space, and 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

The length of a segment PQ is the distance d(P, Q) between its endpoints. It is often denoted |PQ|.

The distance is a metric, as it satisfies the triangular inequality

Moreover, the equality is true if and only if R belongs to the segment PQ. This inequality means that the length of any edge of a triangle is smaller than the sum of the lengths of the other edges. This is the origin of the term triangular inequality.

With the Euclidean distance, every Euclidean space is a complete metric space.

Orthogonality

Two nonzero vectors u and v of are perpendicular or orthogonal 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 subspace is reduced to the zero vector.

Two lines, and more generally two Euclidean subspaces are orthogonal if their direction are orthogonal. Two orthogonal lines that intersect are said perpendicular.

Two segments AB and AC that share a common endpoint are perpendicular or form a right angle if the vectors and are orthogonal.

If AB and AC form a right angle, one has

This is the Pythagorean 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:

Angle

Positive and negative angles on the oriented plane 45, -315, and 405 co-terminal angles.svg
Positive and negative angles on the oriented plane

The (non-oriented) angleθ between two nonzero vectors x and y in is

where arccos is the principal value of the arccosine function. By Cauchy–Schwarz inequality, the argument of the arcsine is in the interval [–1, 1]. Therefore θ is real, and 0 ≤ θπ (or 0 ≤ θ ≤ 180} if angles are measured in degrees).

Angles are not useful in a Euclidean line, as they can be only 0 or π.

In an oriented Euclidean plane, one can define the oriented angle of two vectors. The oriented angle of two vectors x and y is then the opposite of the oriented angle of y and x. In this case, the angle of two vectors can have any value modulo an integer multiple of 2π. In particular, a reflex angle π < θ < 2π equals the negative angle π < θ – 2π < 0.

The angle of two vectors does not change if they are multiplied by positive numbers. More precisely, if x and y are two vectors, and λ and μ are real numbers, then

If A, B and C are three points in a Euclidean space, the angle of the segments AB and AC is the angle of the vectors and As the multiplication of vectors by positive numbers do not change the angle, the angle of two half-lines with initial point A can be defined: it is the angle of the segments AB and AC, where B and C 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 points.

The angle of two lines is defined as follows. If θ is the angle of two segments, one on each line, the angle of any two other segments, one on each line, is either θ or πθ. One of these angles is in the interval [0, π/2], and the other being in [π/2, π]. The non-oriented angle of the two lines is the one in the interval [0, π/2]. In an oriented Euclidean plane, the oriented angle of two lines belongs to the interval [–π/2, π/2].

Cartesian coordinates

Every Euclidean vector space has an orthonormal basis (in fact, infinitely many in dimension higher than one, and two in dimension one), that is a basis of unit vectors () that are pairwise orthogonal ( for ij). More precisely, given any basis the Gram–Schmidt process computes an orthonormal basis such that, for every i, the linear spans of and are equal.

Given a Euclidean space E, a Cartesian frame is a set of data consisting of an orthonormal basis of and a point of E, called the origin and often denoted O. A Cartesian frame allows defining Cartesian coordinates for both E and in the following way.

The Cartesian coordinates of a vector v are the coefficients of v on the basis As the basis is orthonormal, the ith coefficient is the dot product

The Cartesian coordinates of a point P of E are the Cartesian coordinates of the vector

Other coordinates

3-dimensional skew coordinates Repere espace.png
3-dimensional skew coordinates

As a Euclidean space is an affine space, one can consider an affine frame on it, which is the same as a Euclidean frame, except that the basis is not required to be orthonormal. This define affine coordinates, sometimes called skew coordinates for emphasizing that the basis vectors are not pairwise orthogonal.

An affine basis of a Euclidean space of dimension n is a set of n + 1 points that are not contained in a hyperplan. An affine basis define barycentric coordinates for every point.

Many other coordinates systems can be defined on a Euclidean space E of dimension n, in the following way. Let f be a homeomorphism (or, more often, a diffeomorphism) from a dense open subset of E to an open subset of The coordinates of a point x of E are the components of f(x). The polar coordinate system (dimension 2) and the spherical and cylindrical coordinate systems (dimension 3) are defined this way.

For points that are outside the domain of f, 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 longitude passes discontinuousely from –180° to +180°.

This way of defining coordinates extends easily to other mathematical structures, and in particular to manifolds.

Isometries

An isometry between two metric spaces is a bijection preserving the distance, [lower-alpha 2] that is

In the case of a Euclidean vector space, an isometry preserves the norm

since the norm of a vector is its distance from the zero vector. It preserves also the inner product

since

An isometry Euclidean vector spaces is a linear isomorphism. [lower-alpha 3] [7]

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, if E and F are Euclidean spaces, OE, OF, 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.

Isometry with prototypical examples

If E is a Euclidean space, it associated vector space can be considered as a Euclidean space. Every point OE defines an isometry of Euclidean spaces

which maps O to the zero vector and has the identity as associated linear map. The inverse isometry is the map

A Euclidean frame allows defining the map

which is an isometry of Euclidean spaces. The inverse isometry is

This means that, up to an isomorphism, there is exactly one Euclidean space of a given dimension.

This justifies that many authors talk of as the Euclidean space of dimension n.

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), called the Euclidean group and often denoted E(n) of ISO(n).

The simplest Euclidean transformations are translations

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 of the Euclidean group.

A Euclidean isometry f of a Euclidean space E defines a linear isometry of the associated vector space (by linear isometry, it is meant an isometry that is also a linear map) in the following way: denoting by QP the vector , if O is an arbitrary point of E, one has

It is straightforward to prove that this is a linear map that does not depend from the choice of O.

The map is a group homomorphism from the Euclidean group onto the group of linear isometries, called the orthogonal 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 point P form the stabilizer subgroup of the Euclidean group with respect to P. 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 P be a point, f an isometry, and t the translation that maps P to f(P). The isometry fixes P. So and the Euclidean group is the semidirect product of the translation group and the orthogonal group.

The special orthogonal group is the normal subgroup of the orthogonal group that preserves handeness. It is a subgroup of index two ion the orthogonal group. Its inverse image by the group homomorphism 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, translations, rotations (the rigid motions that fix at least a point), and also screw motions.

Typical examples of rigid transformations that are not rigid motions are reflections, 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 r, every rigid transformation that is not a rigid motion is the product of r and a rigid motion. A glide reflection is an example of a rigid transformation that is not a rigid motion not a reflection.

All groups that have been considered in this section are Lie groups and algebraic groups.

Topology

The Euclidean distance makes a Euclidean space a metric space, and thus a [[topological space]. This topology is called the Euclidean topology. In the case of this topology is also the product topology.

The open sets are the subsets that contains an open ball around each of their points. In other words, open balls form a base of the topology.

The topological dimension of a Euclidean space equals its dimension. This implies that Euclidean spaces of different dimensions are not homeomorphic. Moreover, the theorem of invariance of domain asserts that a subset of a Euclidean space is open (for the subspace topology) if and only if it is homeomorphic to an open subset of a Euclidean space of the same dimension.

Euclidean spaces are complete and locally compact. 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'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.

Two different approach have been used. Felix Klein suggested to define geometries through their symmetries. The presentation of Euclidean spaces given in this article, is essentially issued from his Erlangen program, with the emphasis given on the groups of translations and isometries.

On the other hand, David Hilbert proposed a set of axioms, inspired by Euclid's postulates. They belong to synthetic geometry, as they do not involve any definition of real numbers. Later G. D. Birkhoff and Alfred Tarski proposed simpler sets of axioms, which use real numbers (see Birkhoff's axioms and Tarski's axioms).

In Geometric Algebra , Emil Artin has proved that all these definitions of a Euclidean space are equivalent. [8] 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 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. It is what did Artin, with axioms that are not Hilbert's ones, but are equivalent.

Usage

Since ancien Greeks, Euclidean space is used for modeling shapes in the physical world. It is thus used in many sciences such as physics, mechanics, astronomy, ... It is also widely used in all technical areas that are concerned with shapes, figure, location and position, such as architecture, geodesy, topography, navigation, industrial design, technical drawing, robotics...

Space-time of general relativity is not a Euclidean space, but it is a manifold, and, as such, is locally approximated by Euclidean spaces.

Space of dimensions higher than three occurs in several modern theories of physics; see Higher dimension. They occur also in configuration spaces of physical systems.

Beside Euclidean geometry, Euclidean spaces are also widely used in mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. More generally, a manifold is a space that is locally a approximated by Euclidean spaces. Most non-Euclidean geometries can be modeled by a manifold, and embedded in a Euclidean space of higher dimension. For example, an elliptic space can be modeled by an ellipsoid. It is common to represent in a Euclidean space mathematics objects that are a priori not of a geometrical nature. An example among many is the usual representation of graphs.

Other geometric spaces

Since the introduction, at the end of 19th century, of Non-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 geometrical axioms, embedding the space in a Euclidean space is a standard way for proving consistency of its definition, or, more precisely for proving that its theory is consistent, if Euclidean geometry is consistent (which cannot be proved).

Affine space

A Euclidean space is an affine space equipped with a metric. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field, 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 numbers as an extension of Euclidean spaces. For example, a circle and a line have always two intersection points (possibly not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed fields. The shapes that are studied in algebraic geometry in these affine spaces are therefore called affine algebraic varieties.

Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) goemetry and number theory. For example, the Fermat's Last Theorem can be stated "a Fermat curve of degree higher than two has no point in the affine plane over the rationals."

Geometry in affine spaces over a finite fields has also been widely studied. For example, elliptic curves over finite fields are widely used in cryptography.

Projective space

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 "two coplanar lines meet in exactly in one point". Projective space share with Euclidean and affine spaces the property of being isotropic, 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 lines in a vector space of dimension one more.

As for affine spaces, projective spaces are defined over any field, and are fundamental spaces of algebraic geometry.

Non-Euclidean geometries

Non-Euclidean geometry refers usually to geometrical spaces where the parallel postulate is false. They include elliptic geometry, where the sum of the angles of a triangle is less than 180°, and hyperbolic geometry, where this sum is more than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistent (if Euclidean geometry is not contradictory) is one of the paradoxes that are at the origin of the foundational crisis in mathematics of the beginning of 20th century, and motivated the systematization of axiomatic theories in mathematics.

Curved spaces

A manifold is a space that in the neighborhood of each point resemble to a Euclidean space. In technical terms, a manifold is a topological space, such that each point has a neighborhood that is homeomorphic to an open subset of a Euclidean space. Manifold can be classify, by increasing degree of this "resemblance" into topological manifolds, differentiable manifolds, smooth manifolds, and analytic manifolds. 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 spaces at the points of the manifold (these tangent are thus Euclidean vector spaces). This results in a Riemannian manifold. Generally, straight lines do not exist in a Riemannian manifold, but their role is played by geodesics, 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 that has been bended.

Euclidean spaces are trivially Riemannian manifolds. A well illustrating example is the surface of a sphere. In this case, geodesics are arcs of great circle, which are called orthodromes in the context of navigation. More generally, the spaces of non-Euclidean geometries can be realized as Riemannian manifolds.

Pseudo-Euclidean space

The inner product that is defined to define Euclidean spaces is a positive definite bilinear form. If it is replaced by an indefinite quadratic form which is non-degenerate, one gets a pseudo-Euclidean space.

A fundamental example of such a space is the Minkowski space, which is the space-time of Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form

where the last coordinates is the time, and the three other are space coordinates.

To take the gravity into account, general relativity uses a pseudo-Riemannian manifold that has Minkowski spaces as tangent spaces. The curvature of this manifold at a point is a function of the value of the gravitational field at this point.

See also

Footnotes

  1. It may depend on the context or the author whether a subspace is parallel to itself
  2. 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.
  3. 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 As f is an isometry, this gives a linear combination of and which simplifies to zero.

Related Research Articles

Normed vector space vector space on which a norm is defined

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of distance in the real world. A norm is a real-valued function defined on the vector space that has the following properties:

  1. The zero vector, 0, has zero length; every other vector has a positive length.
  2. Multiplying a vector by a positive number changes its length without changing its direction. Moreover,
  3. The triangle inequality holds. That is, taking norms as distances, the distance from point A through B to C is never shorter than going directly from A to C, or the shortest distance between any two points is a straight line.
Affine transformation coordinates transformation

In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Hyperplane geometric object

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing.

In mathematics, a quadric or quadric surface, is a generalization of conic sections. It is a hypersurface in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric.

Normal (geometry) in geometry, an object that is perpendicular to a given object

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

Projective space Completion of the usual space with "points at infinity"

In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seems to meet at infinity. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines.

Isometry distance-preserving function between metric spaces

In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

Affine space geometric structure that generalizes the Euclidean 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 related to parallelism and ratio of lengths for parallel line segments.

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

Affine connection Construct allowing differentiation of tangent vector fields of manifolds

In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan and Hermann Weyl. The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.

Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.

Hyperboloid model

In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model, is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space and m-planes are represented by the intersections of the (m+1)-planes in Minkowski space with S+. The hyperbolic distance function admits a simple expression in this model. The hyperboloid model of the n-dimensional hyperbolic space is closely related to the Beltrami–Klein model and to the Poincaré disk model as they are projective models in the sense that the isometry group is a subgroup of the projective group.

Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps." Accordingly, a complex affine space, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin.

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, ..., en), be applied to a vector x = x1e1 + ... + xnen, giving

Point reflection Geometric symmetry operation

In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invariant under point reflection through its center, it is said to possess central symmetry or to be centrally symmetric.

Two-dimensional space Geometric model of the planar projection of the physical universe

Two-dimensional space is a geometric setting in which two values are required to determine the position of an element. The set 2 of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidian space. For a generalization of the concept, see dimension.

In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn, while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series.

In mathematics and physics, a vector is an element of a vector space.

References

  1. 1 2 Solomentsev 2001.
  2. Ball 1960, pp. 50–62.
  3. Berger 1987.
  4. Coxeter 1973.
  5. 1 2 Berger 1987, Section 9.1.
  6. Berger 1987, Chapter 9.
  7. Berger 1987, Proposition 9.1.3.
  8. Artin 1988.