Euclidean space
Euclidean space is the fundamental space of
Ancient
After the introduction at the end of the 19th century of
There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore it is usually possible to work with a specific Euclidean space, denoted or , which can be represented using
Definition
History of the definition
Euclidean space was introduced by
In 1637,
Euclidean geometry was not applied in spaces of dimension more than three until the 19th century. Ludwig Schläfli generalized Euclidean geometry to spaces of dimension n, 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 dimension.[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 a 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 a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a
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, especially 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.[6]
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 affine spaces to distinguish them from Euclidean vector spaces.[6]
If E is a Euclidean space, its associated vector space (Euclidean vector space) is often denoted The dimension of a Euclidean space is the dimension of its associated vector space.
The elements of E are called points, and are commonly denoted by capital letters. The elements of are called
The action of a translation v on a point P provides a point that is denoted P + v. This action satisfies
Note: The second + in the left-hand side is a vector addition; each other + denotes an action of a vector on a point. This notation is not ambiguous, as, to distinguish between the two meanings of +, it suffices to look at the nature of its left argument.
The fact that the action is free and transitive means that, for every pair of points (P, Q), there is exactly one displacement vector v such that P + v = Q. This vector v is denoted Q − P or
As previously explained, some of the basic properties of Euclidean spaces result from the structure of affine space. They are described in § Affine structure and its subsections. The properties resulting from the inner product are explained in § Metric structure and its subsections.
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
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 on the fact that a Euclidean space is an
Subspaces
Let E be 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
as the associated vector space of F is a linear subspace (vector subspace) of A Euclidean subspace F is a Euclidean space with as the associated vector space. This linear subspace is also called the direction of F.
If P is a point of F then
Conversely, if P is a point of E 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 P and Q 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 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 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 (i.e., the same associated vector space).[a] Equivalently, they are parallel, if there is a translation vector v 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
The inner product of a Euclidean space is often called dot product and denoted x ⋅ y. 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 x ⋅ y in the remainder of this article.
The Euclidean norm of a vector x is
The inner product and the norm allows expressing and proving metric and topological properties of Euclidean geometry. The next subsection describe the most fundamental ones. In these subsections, E 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 PQ is the distance d(P, Q) between its endpoints P and Q. It is often denoted .
The distance is a
Moreover, the equality is true if and only if a point 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 triangle inequality.
With the Euclidean distance, every Euclidean space is a complete metric space.
Orthogonality
Two nonzero vectors u and v of (the associated vector space of a Euclidean space E) 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 AB and AC that share a common endpoint A 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:
Here, is used since these two vectors are orthogonal.
Angle
The (non-oriented) angle θ between two nonzero vectors x and y in is
where arccos is the
Angles are not useful in a Euclidean line, as they can be only 0 or π.
In an
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
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 π − θ. 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
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 of are the coefficients of v 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 i-th coefficient is equal to the dot product
The Cartesian coordinates of a point P of E are the Cartesian coordinates of the vector
Other coordinates
As a Euclidean space is an
An
Many other coordinates systems can be defined on a Euclidean space E of dimension n, in the following way. Let f be a
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
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,[b] 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
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, O ∈ E, O′ ∈ F, and is an isometry, then the map defined by
is an isometry of Euclidean spaces.
It follows from the preceding results that an isometry of Euclidean spaces maps lines to lines, and, more generally Euclidean subspaces to Euclidean subspaces of the same dimension, and that the restriction of the isometry on these subspaces are isometries of these subspaces.
Isometry with prototypical examples
If E is a Euclidean space, its associated vector space can be considered as a Euclidean space. Every point O ∈ E 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 Q – P 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
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
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 or 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
The
Euclidean spaces are
Axiomatic definitions
The definition of Euclidean spaces that has been described in this article differs fundamentally of
Two different approaches have been used.
On the other hand,
In Geometric Algebra, Emil Artin has proved that all these definitions of a Euclidean space are equivalent.[9] It is rather easy to prove that all definitions of Euclidean spaces satisfy Hilbert's axioms, and that those involving real numbers (including the above given definition) are equivalent. The difficult part of Artin's proof is the following. In Hilbert's axioms, congruence is 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 the ancient Greeks, Euclidean space has been used for modeling shapes in the physical world. It is thus used in many sciences, such as physics, mechanics, and 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, or technical drawing.
Space of dimensions higher than three occurs in several modern theories of physics; see
Beside
Other geometric spaces
It has been suggested that this section be Geometric space. (Discuss ) (March 2023) |
Since the introduction, at the end of 19th century, of
Affine space
A Euclidean space is an affine space equipped with a
As soon as non-linear questions are considered, it is generally useful to consider affine spaces over the
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
Projective space
Originally, projective spaces have been introduced by adding "
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
Curved spaces
A
Distances and angles can be defined on a smooth manifold by providing a
Euclidean spaces are trivially Riemannian manifolds. An example illustrating this well is the surface of a
Pseudo-Euclidean space
An
A fundamental example of such a space is the
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 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
- Position space, an application in physics
Footnotes
- ^ It may depend on the context or the author whether a subspace is parallel to itself
- ^ If the condition of being a bijection is removed, a function preserving the distance is necessarily injective, and is an isometry from its domain to its image.
- ^ Proof: one must prove that . For that, it suffices to prove that the square of the norm of the left-hand side is zero. Using the bilinearity of the inner product, this squared norm can be expanded into a linear combination of and As f is an isometry, this gives a linear combination of and which simplifies to zero.
References
- ^ a b Solomentsev 2001.
- ^ Ball 1960, pp. 50–62.
- ^ Berger 1987.
- ^ Coxeter 1973.
- ^ a b Berger 1987, Section 9.1.
- ^ a b Berger 1987, Chapter 9.
- ^ Anton (1987, pp. 209–215)
- ^ Berger 1987, Proposition 9.1.3.
- ^ Artin 1988.
- Anton, Howard (1987), Elementary Linear Algebra (5th ed.), New York: ISBN 0-471-84819-0
- Artin, Emil (1988) [1957], MR 1009557
- ISBN 0-486-20630-0.
- ISBN 3-540-11658-3
- Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover.
Schläfli ... discovered them before 1853 -- a time when Cayley, Grassman and Möbius were the only other people who had ever conceived of the possibility of geometry in more than three dimensions.
- Solomentsev, E.D. (2001) [1994], "Euclidean space", Encyclopedia of Mathematics, EMS Press