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 ...
, 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 of an element (i.e., point). This is the informa ...
of
Euclidean geometry
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 axioms ...
, but in modern
mathematics there are Euclidean spaces of any positive integer
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 coor ...
, including the three-dimensional space and the ''
Euclidean plane'' (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 p ...
mathematician
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
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 t ...
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 objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
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 segmen ...
).
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 ge ...
, the old postulates were re-formalized to define Euclidean spaces through
axiomatic theory. 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. Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the
real -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 alge ...
. An isomorphism from a Euclidean space to
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 ...
s which locate that point in the Euclidean space and are called the ''
Cartesian coordinates'' 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 cult ...
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
axioms 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, compass ...
.
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. Ma ...
introduced
Cartesian coordinates 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 ...
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 ...
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 ...
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 e ...
s) that exist in Euclidean spaces of any dimension.
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 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 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 ...
, 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
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).
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 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 mathe ...
, 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 unit ...
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 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 ...
. 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 related ...
, and this allows defining lines, planes, subspaces, dimension, and
parallelism. The inner product allows defining distance and angles.
The set
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 alge ...
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 examp ...
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 i ...
between a Euclidean space of dimension 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
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
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 ...
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 ...
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 related ...
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
The ''dimension'' of a Euclidean space is the
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 coor ...
of its associated vector space.
The elements of are called ''points'' and are commonly denoted by capital letters. The elements of
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 ...
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 a ...
s''. They are also called ''translations'', although, properly speaking, a
translation
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 ...
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 b ...
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
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
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
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 alge ...
as an
inner product
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 ...
. The importance of this particular example of Euclidean space lies in the fact that every Euclidean space is
isomorphic 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 examp ...
of
defines an isomorphism of Euclidean spaces from to
As every Euclidean space of dimension is isomorphic to it, the Euclidean space
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 related ...
. 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
its associated vector space.
A ''flat'', ''Euclidean subspace'' or ''affine subspace'' of is a subset of such that
as the associated vector space of is a
linear subspace (vector subspace) of
A Euclidean subspace is a Euclidean space with
as the associated vector space. This linear subspace
is also called the ''direction'' of .
If is a point of then
Conversely, if is a point of and
is a
linear 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.
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 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
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
A standard convention allows using this formula in every Euclidean space, see .
The ''
line segment'', or simply ''segment'', joining the points and is the subset of points such that in the preceding formulas. It is denoted or ; that is
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:
Given a point and a subspace , there exists exactly one subspace that contains and is parallel to , which is
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 ...
.
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 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 ...
. 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 bilinea ...
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
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 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 alge ...
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 in the remainder of this article.
The Euclidean norm of a vector is
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 mathem ...
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 ...
properties of
Euclidean geometry
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 axioms ...
. The next subsection describe the most fundamental ones. ''In these subsections,'' ''denotes 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 is the distance between its endpoints ''P'' and ''Q''. It is often denoted
.
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 mathem ...
, 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 ...
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 edges and three 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, any three points, when non- colline ...
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 bou ...
.
Orthogonality
Two nonzero vectors and of
(the associated vector space of a Euclidean space ) 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 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'' if the vectors
and
are orthogonal.
If and 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:
Here,
is used since these two vectors are orthogonal.
Angle
The (non-oriented) ''angle'' between two nonzero vectors and in
is
where is the
principal value 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 domains). Sp ...
function. By
Cauchy–Schwarz inequality, 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 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 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
If , , and are three points in a Euclidean space, the angle of the segments and 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-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 examp ...
(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 ...
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 v ...
s (
) that are pairwise orthogonal (
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 ...
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 produ ...
computes an orthonormal basis such that, for every , the
linear spans of
and
are equal.
Given a Euclidean space , a ''Cartesian frame'' is a set of data consisting of an orthonormal basis of
and a point of , called the ''origin'' and often denoted . A Cartesian frame
allows defining Cartesian coordinates for both and
in the following way.
The Cartesian coordinates of a vector of
are the coefficients of 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, the -th coefficient
is equal to the dot product
The Cartesian coordinates of a point of are the Cartesian coordinates of the vector
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 related ...
, 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 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 isomor ...
(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 ...
) 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 of to an open subset of
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 th ...
(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 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
manifolds.
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'' me ...
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 set ...
s is a bijection preserving the distance, that is
In the case of a Euclidean vector space, an isometry that maps the origin to the origin 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 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
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 and are Euclidean spaces, , , 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 is a Euclidean space, its associated vector space
can be considered as a Euclidean space. Every point defines an isometry of Euclidean spaces
which maps 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 .
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 ''transl ...
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 of a Euclidean space defines a linear isometry
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 mapping V \to W between two vector spaces that pr ...
) in the following way: denoting by the vector
, if is an arbitrary point of , one has
It is straightforward to prove that this is a linear map that does not depend from the choice of
The map
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)
w ...
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 form the
stabilizer subgroup 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
fixes . 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
handedness. It is a subgroup of
index 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 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,
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
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 groups 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.
Ma ...
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 set ...
, 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 po ...
. 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 metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot\, ...
. In the case of
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-s ...
.
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 su ...
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 defi ...
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
topologically invariant way.
Informal discussion
For ordinary Euclidean ...
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
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. 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, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
'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 ge ...
.
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 grou ...
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 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. 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, compass ...
, 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 ...
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 ...
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'',
Emil Artin 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 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 cult ...
, Euclidean space is used for modeling
shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie ...
s in the physical world. It is thus used in many
science
Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earli ...
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 object ...
, and
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
. 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,
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 sc ...
,
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, or
technical drawing
Technical drawing, drafting or drawing, is the act and discipline of composing drawings that visually communicate how something functions or is constructed.
Technical drawing is essential for communicating ideas in industry and engineering ...
.
Space of dimensions higher than three occurs in several modern theories of physics; see Higher dimension. They occur also in configuration space (physics), configuration spaces of physical systems.
Beside
Euclidean geometry
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 axioms ...
, Euclidean spaces are also widely used in other areas of mathematics. Tangent spaces of differentiable manifolds are Euclidean vector spaces. More generally, a
manifold 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 ge ...
can be modeled by a manifold, and embedding, 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 mathematical objects that are ''a priori'' not of a geometrical nature. An example among many is the usual representation of Graph (discrete mathematics), graphs.
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 ge ...
, 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
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 axioms ...
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 mathem ...
. Affine spaces have many other uses in mathematics. In particular, as they are defined over any field (mathematics), 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 (geometry), 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 variety, affine algebraic varieties.
Affine spaces over the rational numbers and more generally over algebraic number fields provide a link between (algebraic) geometry 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 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
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 (mathematics), field, and are fundamental spaces of algebraic geometry.
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 segmen ...
is false. They include elliptic geometry, where the sum of the angles of a triangle is more than 180°, and hyperbolic geometry, where this sum is less than 180°. Their introduction in the second half of 19th century, and the proof that their theory is consistency, 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 theory, axiomatic theories in mathematics.
Curved spaces
A
manifold 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 po ...
, such that each point has a neighborhood that is
homeomorphic to an
open subset of a Euclidean space. Manifolds can be classified 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 smooth function, smoothly varying Euclidean metric on the tangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a Riemannian manifold. Generally,
straight 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 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 space that has been bent.
Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a sphere. In this case, geodesics are great circle, 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 ge ...
can be realized as Riemannian manifolds.
Pseudo-Euclidean space
An
inner product
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 ...
of a real vector space is a positive definite bilinear form, and so characterized by a Bilinear form#Derived quadratic form, positive definite quadratic form. A pseudo-Euclidean space is an affine space with an associated real vector space equipped with a non-degenerate quadratic form (that may be indefinite quadratic form, indefinite).
A fundamental example of such a space is the Minkowski space, which is the space-time of Albert Einstein, Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form
where the last coordinate (''t'') is temporal, and the other three (''x'', ''y'', ''z'') are spatial.
To take gravity into account, general relativity uses a pseudo-Riemannian manifold that has Minkowski spaces as tangent spaces. The Curvature of Riemannian manifolds, curvature of this manifold at a point is a function of the value of the gravitational field at this point.
See also
* Hilbert space, a generalization to infinite dimension, used in functional analysis
Footnotes
References
*
*
*
*
*
*
{{Authority control
Euclidean geometry
Linear algebra
Homogeneous spaces
Norms (mathematics)