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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. * Each line has at least two points. * 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 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 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. ==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. A similar construction, starting from the projective plane of order three, produces the affine plane of order three sometimes called the Hesse configuration. An affine plane of order exists if and only if a projective plane 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Affine plane (incidence geometry)」の詳細全文を読む スポンサード リンク
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