Euclidean space

In mathematics, Euclidean space is initially the "space of our conception" as described in Euclid's Elements by axioms and postulates (cf. Euclidean geometry). Until the 19th century, it was assumed that this described the physical space surrounding us. The addition of "Euclidean" became necessary after more general concepts of space were developed in mathematics (e.g. hyperbolic space, Riemannian manifolds) and it became apparent in the context of special and general relativity that other spatial concepts were needed to describe space in physics (Minkowski space, Lorentz manifold).

In the course of time Euclid's geometry was made more precise and generalized in various ways:

axiomatically by Hilbert (see Hilbert's axiom system of Euclidean geometry),
as a Euclidean vector space (a vector space defined over $\mathbb {R}$ defined vector space with scalar product),
as a Euclidean point space (an affine space modeled over a Euclidean vector space),
As the coordinate space $\mathbb {R} ^{3}$ with the standard scalar product.

When talking about Euclidean space, any of these definitions can be meant or also a higher-dimensional generalization. The two-dimensional Euclidean space is also called Euclidean plane. In this two-dimensional case, the term is used somewhat more generally in synthetic geometry: Euclidean planes can there be defined as affine planes over a more general class of bodies, the Euclidean bodies. These bodies are (depending on the view) partial bodies or isomorphic to partial bodies of $\mathbb {R} .$

Euclidean space differs from affine space in that one can measure lengths and angles. One therefore draws out the mappings that preserve lengths and angles. These are traditionally called congruence mappings, other names are movements and isometries.

In non-Euclidean spaces, such as hyperbolic and elliptic space, the parallel axiom does not hold.

Euclidean vector spaces

From Euclidean visual space to Euclidean vector space

In analytic geometry, one associates a vector space with Euclidean space. One way to do this is to take the set of parallel displacements (translations), provided with the one behind the other as an addition. Each translation can be described by an arrow connecting a point to its image point. Two arrows, which are parallel in the same direction and have the same length, describe the same displacement. Two such arrows are called equivalent and the equivalence classes are called vectors.

If one chooses a point as reference point (origin) in the Euclidean space, one can assign to ${\vec {p}}={\overrightarrow {OP}}$ each point its location vector , the vector represented by an arrow from the origin to the point In this way, one gets a one-to-one relationship between the Euclidean space and the associated Euclidean vector space, and can thus identify the original Euclidean space with the Euclidean vector space. However, this identification is not canonical, but depends on the choice of the origin.

One can now also transfer the length and angle measurement from Euclidean space to vectors as the length of the associated arrows and angles between such. In this way one obtains a vector space with scalar product. The scalar product is characterized by the fact that the product ${\vec {a}}\cdot {\vec {a}}$ a vector ${\vec {a}}$ with itself $|{\vec {a}}|$ gives the square $|{\vec {a}}|^{2}$ its length From the laws of arithmetic for scalar products, the binomial formulas, and the cosine theorem (applied to a triangle whose sides correspond to vectors ${\vec {a}}$ , ${\vec {b}}$ and ${\vec {b}}-{\vec {a}}$ ), we obtain the formula

${\vec {a}}\cdot {\vec {b}}=|{\vec {a}}|\,|{\vec {b}}|\,\cos \sphericalangle ({\vec {a}},{\vec {b}})$ .

Here denotes $\sphericalangle ({\vec {a}},{\vec {b}})$ the angle between vectors ${\vec {a}}$ and ${\vec {b}}$ .

General term

Starting from this, any real vector space with scalar product (of arbitrary finite dimension ) is called a Euclidean vector space. One then uses the above formula to define length (norm) of a vector and angles between vectors. Two vectors are orthogonal if their scalar product is zero. Every three-dimensional Euclidean vector space is isometrically isomorphic to the vector space of arrow classes. Every -dimensional Euclidean vector space is isometrically isomorphic to the coordinate vector space $\mathbb {R} ^{n}$ (see below). Euclidean vector spaces of the same dimension are thus indistinguishable. This entitles one to call any such one the Euclidean vector space of dimension . Some authors use the term Euclidean space also for infinite-dimensional real vector spaces with scalar product, some also for complex vector spaces with scalar product, cf. scalar product space.

Lengths, angles, orthogonality and orthonormal bases

Isometrics

If and are two -dimensional Euclidean vector spaces, then a linear mapping is called $f\colon V\to W$ a (linear) isometry if it preserves the scalar product, i.e. if

$f({\vec {a}})\cdot f({\vec {b}})={\vec {a}}\cdot {\vec {b}}$

${\vec {a}},{\vec {b}}\in V$ holds for all Such a mapping is also called an orthogonal mapping. In particular, an isometry preserves lengths

$|f({\vec {a}})|=|{\vec {a}}|$

and angles, i.e. in particular orthogonality

$f({\vec {a}})\perp f({\vec {b}})\Longleftrightarrow {\vec {a}}\perp {\vec {b}}.$

Conversely, any linear mapping that preserves lengths is an isometry.

An isometry maps every orthonormal basis back to an orthonormal basis. Conversely, if ${\vec {e}}_{1},\dotsc ,{\vec {e}}_{n}$ is an orthonormal basis of and ${\vec {e}}_{1}{}',\dotsc ,{\vec {e}}_{n}{}'$ an orthonormal basis of , then there is exactly one isometry that ${\vec {e}}_{i}{}'$ maps ${\vec {e}}_{i}$ to

From this it follows that two Euclidean vector spaces of the same dimension are isometric, that is, they are indistinguishable as Euclidean vector spaces.

Two points and their position vectors

Angle between two vectors

The Euclidean point space

Motivation

Euclidean vector spaces often serve as models for Euclidean space. The elements of the vector space are then called points or vectors, depending on the context. No distinction is made between points and their location vectors. Computationally, this can be advantageous. Conceptually, however, it is unsatisfactory:

From a geometric point of view, points and vectors should be distinguished conceptually.

Vectors can be added and multiplied by numbers, but points cannot.
Points are connected or merged by vectors.

In the vector space there is a distinguished element, the zero vector. But in Euclidean geometry all points are equal.

Description

The remedy is the concept of a Euclidean point space. This is an affine space over a Euclidean vector space. Here one distinguishes between points and vectors.

The totality of points forms the Euclidean point space. This is usually denoted by , $E_{n}$ , $E^{n}$ or ^{n}} (The superscript $\mathbb {E} ^{n}$ is not an exponent, but an index denoting the dimension. So $E^{n}$ is not a Cartesian product).
The set of all vectors forms a Euclidean vector space .

For every two points and $Q\in E$
there exists exactly one connection vector, denoted by ${\overrightarrow {PQ}}$
. The connection vector of a point with itself is the zero vector:
${\overrightarrow {PP}}={\vec {0}}$
A point can be uniquely transformed into a point ${\vec {v}}$ by a vector This is often denoted by $P+{\vec {v}}$ . (This is a purely formal notation. The plus sign does not denote vector space addition, nor addition on the point space.)
The zero vector merges every vector into itself: $P+{\vec {0}}=P$
If the vector ${\vec {v}}$ passes the point to the point and the vector ${\vec {w}}$ passes the point to the point , then ${\vec {v}}+{\vec {w}}$ transits the point into the point This can be expressed as follows:

$(P+{\vec {v}})+{\vec {w}}=P+({\vec {v}}+{\vec {w}})$

${\overrightarrow {PQ}}+{\overrightarrow {QR}}={\overrightarrow {PR}}$

In the language of algebra, these properties mean: The additive group of the vector space operates freely and transitively on the set .

Lengths, distances and angles

Line lengths, distances between points, angles and orthogonality can now be defined using the scalar product of vectors:

The length ${\overline {PQ}}$ of the line [PQ] and the distance d(P,Q) the points and is defined as the length of the vector ${\overrightarrow {PQ}}$ :

$d(P,Q)={\overline {PQ}}=|{\overrightarrow {PQ}}|$

The size of the angle $\sphericalangle QPR$ is defined as the angle between the vectors ${\overrightarrow {PQ}}$ and ${\overrightarrow {PR}}$ :

$\sphericalangle QPR=\sphericalangle ({\overrightarrow {PQ}},{\overrightarrow {PR}})$

Two distances [PQ] and [RS] are orthogonal if and only if the associated vectors ${\overrightarrow {PQ}}$ and ${\overrightarrow {RS}}$ orthogonal.

Images

Length-preserving mappings of a Euclidean point space to itself are called isometries, congruence mappings (in plane geometry), or motions. They automatically preserve angles as well. If $f\colon E\to E$ a motion, then there exists an orthogonal mapping (linear isometry) ${\vec {f}}\colon V\to V$ , such that for all points and holds: