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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): ''In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the ...
) * There exist three non-collinear points (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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
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 In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''inciden ...
. They are non-degenerate linear spaces satisfying Playfair's axiom. The familiar
Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
is an affine plane. There are many finite and infinite affine planes. As well as affine planes over fields (and
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
s), there are also many
non-Desarguesian plane In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective ...
s, not derived from coordinates in a division ring, satisfying these axioms. The Moulton plane is an example of one of these.


Finite affine planes

If the number of points in an affine plane is finite, then if one line of the plane contains points then: * each line contains points, * each point is contained in lines, * there are points in all, and * there is a total of lines. The number 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 In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines ...
. A similar construction, starting from the projective plane of order 3, produces the affine plane of order 3 sometimes called the
Hesse configuration In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by , is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as t ...
. An affine plane of order exists if and only if a
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
of order 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 lines of an affine plane of order fall into equivalence classes of 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 lines that pass through a single point lies in a different parallel class. The parallel class structure of an affine plane of order may be used to construct a set of mutually orthogonal latin squares. Only the incidence relations are needed for this construction.


Relation with projective planes

An affine plane can be obtained from any
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
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 In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The ...
, each of whose points is that
point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
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 In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
of order nine since the
collineation group In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is ...
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.


Affine translation planes

A line in a projective plane is a translation line if the group of elations with axis
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
transitively on the points of the affine plane obtained by removing from the plane . 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 be a -dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over a field . A spread of is a set of -dimensional subspaces of that partition the non-zero vectors of . The members of are called the components of the spread and if and are distinct components then . Let be the
incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
whose points are the vectors of and whose lines are the cosets of components, that is, sets of the form where is a vector of and is a component of the spread . Then: : is an affine plane and the group of
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
for a vector is an automorphism group acting regularly on the points of this plane.


Generalization: -nets

An incidence structure more general than a finite affine plane is a -''net of order'' . This consists of points and lines such that: * Parallelism (as defined in affine planes) is an equivalence relation on the set of lines. * Every line has exactly points, and every parallel class has lines (so each parallel class of lines partitions the point set). * There are parallel classes of lines. Each point lies on exactly lines, one from each parallel class. An -net of order is precisely an affine plane of order . A -''net of order'' is equivalent to a set of mutually orthogonal Latin squares of order .


Example: translation nets

For an arbitrary field , let be a set of -dimensional subspaces of the vector space , any two of which intersect only in (called a partial spread). The members of , and their cosets in , form the lines of a translation net on the points of . If this is a -net of order . Starting with an affine
translation plane In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desar ...
, 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 is an infinite field, any partial spread with fewer than members can be extended and the translation net can be completed to an affine translation plane.


Geometric codes

Given the "line/point"
incidence matrix In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is ''X'' and the second is ''Y'', the matrix has one row for each element ...
of any finite
incidence structure In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore al ...
, , and any field, the row space of over is a linear code that we can denote by . Another related code that contains information about the incidence structure is the Hull of which is defined as: :\operatorname(C) = C \cap C^, where is the orthogonal code to . 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, * If is an affine plane of order and is a field of characteristic , where divides , then the minimum weight of the code is and all the minimum weight vectors are constant multiples of vectors whose entries are either zero or one. Furthermore, * If is an affine plane of order and is a field of characteristic , then and the minimum weight vectors are precisely the scalar multiples of the (incidence vectors of) lines of . When the geometric code generated is the -ary Reed-Muller Code.


Affine spaces

Affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
s 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 In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
., but see also


Notes


References

* * * * * *


Further reading

* * * * * *{{Citation , last = Stevenson , first = Frederick W. , title = Projective Planes , publisher = W.H. Freeman and Company , place = San Francisco , year = 1972 , isbn = 0-7167-0443-9 Incidence geometry Geometry