Affine plane (incidence geometry)
In geometry, an affine plane is a system of points and lines that satisfy the following axioms:[1]
For the algebraic definition of the same space, see Affine space.
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:[2]
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.[3]
If the number of points in an affine plane is finite, then if one line of the plane contains n points then:
The number n is called the order of the affine plane.
All known finite affine planes have orders that are prime or prime power integers. The smallest affine plane (of order 2) is obtained by removing a line and the three points on that line from the Fano plane. A similar construction, starting from the projective plane of order 3, produces the affine plane of order 3 sometimes called the Hesse configuration. An affine plane of order n exists if and only if a projective plane of order n exists (however, the definition of order in these two cases is not the same). Thus, there is no affine plane of order 6 or order 10 since there are no projective planes of those orders. The Bruck–Ryser–Chowla theorem provides further limitations on the order of a projective plane, and thus, the order of an affine plane.
The n2 + n lines of an affine plane of order n fall into n + 1 equivalence classes of n lines apiece under the equivalence relation of parallelism. These classes are called parallel classes of lines. The lines in any parallel class form a partition the points of the affine plane. Each of the n + 1 lines that pass through a single point lies in a different parallel class.
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. Only the incidence relations are needed for this construction.
Given the "line/point" incidence matrix of any finite incidence structure, M, and any field, F the row space of M over F is a linear code that we can denote by C = CF(M). Another related code that contains information about the incidence structure is the Hull of C which is defined as:[8]
where C⊥ is the orthogonal code to C.
Not much can be said about these codes at this level of generality, but if the incidence structure has some "regularity" the codes produced this way can be analyzed and information about the codes and the incidence structures can be gleaned from each other. When the incidence structure is a finite affine plane, the codes belong to a class of codes known as geometric codes. 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. On the other hand,[9]
Furthermore,[10]
When π = AG(2, q) the geometric code generated is the q-ary Reed-Muller Code.
Affine spaces[edit]
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.[11]