Аффиндік жазықтықтар: Аксиомалық және алгебралық анықтамалар

Axiomatically defined geometrical space
the algebraic definition of the same space
In geometry, an affine plane is a system of points and lines that satisfy the following axioms:
Any two distinct points lie on a unique line. Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. (Playfair's axiom)
There exist four points such that no three are collinear (points not on a single line). In an affine plane, two lines are called parallel if they are equal or disjoint. Using this definition, Playfair's axiom above can be replaced by:
Given a point and a line, there is a unique line which contains the point and is parallel to the line. Parallelism is an equivalence relation on the lines of an affine plane. Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to incidence geometry. They are non degenerate linear spaces satisfying Playfair's axiom. The familiar Euclidean plane is an affine plane. There are many finite and infinite affine planes. As well as affine planes over fields (and division rings), there are also many non Desarguesian planes, not derived from coordinates in a division ring, satisfying these axioms. The Moulton plane is an example of one of these.

An affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane by adding a line at infinity, each of whose points is that point at infinity where an equivalence class of parallel lines meets. If the projective plane is non Desarguesian, the removal of different lines could result in non isomorphic affine planes. For instance, there are exactly four projective planes of order nine, and seven affine planes of order nine. There is only one affine plane corresponding to the Desarguesian plane of order nine since the collineation group of that projective plane acts transitively on the lines of the plane. Each of the three non Desarguesian planes of order nine have collineation groups having two orbits on the lines, producing two non isomorphic affine planes of order nine, depending on which orbit the line to be removed is selected from.

A line l in a projective plane Π is a translation line if the group of elations with axis l acts transitively on the points of the affine plane obtained by removing l from the plane Π. A projective plane with a translation line is called a translation plane and the affine plane obtained by removing the translation line is called an affine translation plane. While in general it is often easier to work with projective planes, in this context the affine planes are preferred and several authors simply use the term translation plane to mean affine translation plane. An alternate view of affine translation planes can be obtained as follows: Let V be a 2n dimensional vector space over a field F. A spread of V is a set S of n dimensional subspaces of V that partition the non zero vectors of V. The members of S are called the components of the spread and if Vi and Vj are distinct components then 1=Vi ⊕ Vj = V. Let A be the incidence structure whose points are the vectors of V and whose lines are the cosets of components, that is, sets of the form v + U where v is a vector of V and U is a component of the spread S. Then:
A is an affine plane and the group of translations x → x + w for a vector w is an automorphism group acting regularly on the points of this plane.

An incidence structure more general than a finite affine plane is a k net of order n. This consists of n^(2) points and nk lines such that:
Parallelism (as defined in affine planes) is an equivalence relation on the set of lines. Every line has exactly n points, and every parallel class has n lines (so each parallel class of lines partitions the point set). There are k parallel classes of lines. Each point lies on exactly k lines, one from each parallel class. An (n + 1) net of order n is precisely an affine plane of order n.

A k net of order n is equivalent to a set of k − 2 mutually orthogonal Latin squares of order n.

For an arbitrary field F, let Σ be a set of n dimensional subspaces of the vector space F^(2n), any two of which intersect only in {0} (called a partial spread). The members of Σ, and their cosets in F^(2n), form the lines of a translation net on the points of F^(2n). If this is a k net of order Starting with an affine translation plane, any subset of the parallel classes will form a translation net. Given a translation net, it is not always possible to add parallel classes to the net to form an affine plane. However, if F is an infinite field, any partial spread Σ with fewer than members can be extended and the translation net can be completed to an affine translation plane.

Affine spaces can be defined in an analogous manner to the construction of affine planes from projective planes. It is also possible to provide a system of axioms for the higher dimensional affine spaces which does not refer to the corresponding projective space.