Affine space

The affine space , sometimes also called linear manifold , occupies a middle position between Euclidean space and projective space in the systematic structure of geometry .

Like Euclidean space, three-dimensional affine space is a mathematical model for the three-dimensional visual space that is familiar to us. The terms length, distance and angle are not used here.

In a broader sense, an affine space, like other mathematical spaces , can have any dimension : An affine space can also be a single point, the affine line , the affine plane and four- and higher-dimensional spaces. As a rule, these spaces are only finite-dimensional.

Different mathematical disciplines have found different clarifications of this term.

The affine space in linear algebra

definition

Triangle rule
Deductibility rule

Given a set whose elements are geometrically understood as points, a vector space over a body and a mapping from to , which assigns a connection vector to two points , so that the following two rules apply: ${\ displaystyle A \ neq \ emptyset}$ ${\ displaystyle V}$ ${\ displaystyle K}$${\ displaystyle A \ times A}$${\ displaystyle V}$${\ displaystyle P, Q \ in A}$ ${\ displaystyle {\ overrightarrow {PQ}} \ in V}$

• for every three points the following applies: ( triangle rule , Chasles relation ),${\ displaystyle P, Q, R \ in A}$${\ displaystyle {\ overrightarrow {PQ}} + {\ overrightarrow {QR}} = {\ overrightarrow {PR}}}$
• for every point and every vector there is a uniquely determined point , so that ( removability rule ).${\ displaystyle P \ in A}$${\ displaystyle {\ vec {v}} \ in V}$${\ displaystyle Q \ in A}$${\ displaystyle {\ vec {v}} = {\ overrightarrow {PQ}}}$

The triple is called affine space . If it is clear which vector space and which arrow mapping it is based, one speaks of affine space alone . The field is often the field of real numbers. ${\ displaystyle (A, V, {\ overrightarrow {}})}$${\ displaystyle V}$${\ displaystyle {\ overrightarrow {}}}$${\ displaystyle A}$${\ displaystyle K}$${\ displaystyle \ mathbb {R}}$

Translations

In affine space, an “addition” is defined as a mapping of by the fact that the point is uniquely determined via . For fixed , the corresponding mapping is called translation (displacement) or precise translation around the vector and is then called the corresponding translation vector . ${\ displaystyle A \ times V \ to A, \ (P, {\ vec {v}}) \ mapsto P + {\ vec {v}},}$${\ displaystyle P + {\ vec {v}}}$${\ displaystyle {\ vec {v}} = {\ overrightarrow {PQ}}}$${\ displaystyle Q}$${\ displaystyle {\ vec {v}} \ in V}$${\ displaystyle T _ {\ vec {v}} \ colon A \ to A, \ P \ mapsto P + {\ vec {v}},}$ ${\ displaystyle {\ vec {v}}}$${\ displaystyle {\ vec {v}}}$

Translations are always bijections . Together with the series connection as a group link, they form a subgroup of the automorphism group of , where and for always and apply. ${\ displaystyle \ operatorname {Aut} (A)}$${\ displaystyle A}$${\ displaystyle {T _ {\ vec {0}}} = \ operatorname {id} _ {A}}$${\ displaystyle {\ vec {v}}, {\ vec {w}} \ in V}$${\ displaystyle {T _ {\ vec {v}}} \ circ {T _ {\ vec {w}}} = T _ {{\ vec {v}} + {\ vec {w}}}}$${\ displaystyle {T _ {\ vec {v}}} ^ {- 1} = T _ {- {\ vec {v}}}}$

Affine subspace

If a fixed point is off and a subspace of , then an affine subspace is of . Instead of the term “affine subspace”, the equivalent term affine subspace is often used. The subspace belonging to an affine subspace is uniquely determined by. ${\ displaystyle P}$${\ displaystyle A}$${\ displaystyle U}$${\ displaystyle V}$${\ displaystyle B = P + U = \ {P + {\ vec {u}} \ mid {\ vec {u}} \ in U \}}$${\ displaystyle A}$ ${\ displaystyle B}$${\ displaystyle U}$${\ displaystyle B}$

The dimension of an affine space to a vector space over a body is defined as the dimension of the vector space over . It is often convenient to view the empty set as an affine (partial) space. The dimension -1 is then assigned to this empty subspace. ${\ displaystyle A}$${\ displaystyle V}$${\ displaystyle K}$${\ displaystyle V}$${\ displaystyle K}$

The affine point space and the vector space assigned to it

If a point is firmly chosen as the origin in affine space , one has a one-to-one mapping between the affine space and its vector space of the displacements through the mapping that assigns the displacement , the position vector of , to each point . It should be noted that this assignment between points and position vectors depends on the choice of the origin ! ${\ displaystyle A}$${\ displaystyle O \ in A}$${\ displaystyle P \ in A}$${\ displaystyle {\ overrightarrow {OP}}}$${\ displaystyle P}$

Conversely, one can regard every vector space as an affine point space: with is the mapping that assigns their connection vector to two points . A point of the affine space is thus distinguished from the start , namely the zero vector of the vector space. ${\ displaystyle V}$${\ displaystyle V \ times V \ to V}$${\ displaystyle ({\ vec {v}}, {\ vec {w}}) \ mapsto {\ vec {w}} - {\ vec {v}}}$

In the first case, after the identification of a point with its position vector (depending on the choice of the origin!), In the second case the addition in the vector space can be understood from the outset in such a way that the group as a mapping group of the displacements on itself as a set of points operated on . ${\ displaystyle V}$ ${\ displaystyle (V, +)}$

For these reasons, a rigid distinction between the affine point space on the one hand and the vector space of the displacement vectors on the other hand is sometimes dispensed with.

Examples

• The -dimensional Euclidean space is the affine space over a -dimensional Euclidean vector space (i.e. a -dimensional vector space with a scalar product ).${\ displaystyle n}$ ${\ displaystyle E ^ {n}}$${\ displaystyle n}$${\ displaystyle n}$
• Every vector space can be understood as an affine space. As a result, every affine subspace of a vector space is also an affine space.

Use in algebraic geometry

• In classical algebraic geometry, the -dimensional affine space over an algebraically closed body is the algebraic variety .${\ displaystyle n}$${\ displaystyle A ^ {n}}$ ${\ displaystyle K}$ ${\ displaystyle K ^ {n}}$
• In modern algebraic geometry, the -dimensional affine space over a commutative ring with one element is defined as the spectrum of the polynomial ring in indefinite terms . For an - algebra , the -valent points of are equal .${\ displaystyle n}$${\ displaystyle A_ {A} ^ {n}}$ ${\ displaystyle A}$${\ displaystyle A [X_ {1}, \ dotsc, X_ {n}]}$${\ displaystyle n}$
${\ displaystyle A}$ ${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle A_ {A} ^ {n}}$${\ displaystyle B ^ {n}}$

Synthetic Geometry Definitions

An affine space in the sense of synthetic geometry consists of the following data:

• a lot of points,
• a set of straight lines,
• an incidence relation indicating which points are on which straight line and
• a parallelism relation, which indicates which lines are parallel,

so that certain axioms are fulfilled, which the view suggests (among others Euclid's famous axiom of parallels ).

The structures thus defined generalize the term affine space , which is defined in this article. The following applies:

1. Every two-dimensional affine space fulfills the requirements for an affine plane . An affine plane, which fulfills Desargues' theorem , defines a unique inclined body , so that it is geometrically isomorphic to the two-dimensional affine space above this inclined body.
2. Every affine space fulfills the requirements of an affine geometry . An affine geometry that is at least three-dimensional (i.e. that contains an affine plane as a true subspace) satisfies Desargues' theorem and determines a unique oblique body so that it is geometrically isomorphic to an at least three-dimensional space above this oblique body.
3. Every affine space is a weakly affine space
4. Every finite, at least two-dimensional affine space is a block diagram .

→ For further details, see the articles mentioned, in which the generalized structures are described. How the term “affine space” (as a space with displacements that form a vector space) can be distinguished from the axiomatic terms of synthetic geometry is explained in more detail in the article Affine Geometry .