Affine plane (incidence geometry)

Using this definition, Playfair's axiom above can be replaced by:[2] Parallelism is an equivalence relation on the lines of an affine plane.

The lines in any parallel class form a partition the points of the affine plane.

The parallel class structure of an affine plane of order n may be used to construct a set of n − 1 mutually orthogonal latin squares.

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.

[5] 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 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:[6] An incidence structure more general than a finite affine plane is a k-net of order n. This consists of n2 points and nk lines such that: 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 F2n, any two of which intersect only in {0} (called a partial spread).

The members of Σ, and their cosets in F2n, form the lines of a translation net on the points of F2n.

How much information the code carries about the affine plane depends in part on the choice of field.

If the characteristic of the field does not divide the order of the plane, the code generated is the full space and does not carry any information.