pexamples of axioms for four line geometry

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. For instance, there are exactly four projective planes of order nine, and seven affine planes of order nine. If the projective plane is non-Desarguesian, the removal of different lines could result in non-isomorphic affine planes. Only the incidence relations are needed for this construction.Īn 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. 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 n + 1 lines that pass through a single point lies in a different parallel class. The lines in any parallel class form a partition the points of the affine plane. These classes are called parallel classes of lines. The n 2 + 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. The Bruck–Ryser–Chowla theorem provides further limitations on the order of a projective plane, and thus, the order of an affine plane. Thus, there is no affine plane of order 6 or order 10 since there are no projective planes of those orders. 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). A similar construction, starting from the projective plane of order 3, produces the affine plane of order 3 sometimes called the Hesse configuration. The smallest affine plane (of order 2) is obtained by removing a line and the three points on that line from the Fano plane. The number n is called the order of the affine plane.Īll known finite affine planes have orders that are prime or prime power integers. each point is contained in n + 1 lines,.If the number of points in an affine plane is finite, then if one line of the plane contains n points then: The Moulton plane is an example of one of these. 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. There are many finite and infinite affine planes. The familiar Euclidean plane is an affine plane. They are non-degenerate linear spaces satisfying Playfair's axiom. 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. Parallelism is an equivalence relation on the lines of an affine plane. Given a point and a line, there is a unique line which contains the point and is parallel to the line.Using this definition, Playfair's axiom above can be replaced by: In an affine plane, two lines are called parallel if they are equal or disjoint. There exist three non-collinear points (points not on a single 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.Any two distinct points lie on a unique line.In geometry, an affine plane is a system of points and lines that satisfy the following axioms: The reason is that the point where these lines appear to meet is not a point of the geometry.For the algebraic definition of the same space, see Affine space. It does not look as if L1 and L1' are parallel in Figure above but these lines are parallel in this four point geometry. To help you understand the meaning of this diagram, note that if one chooses the point a1 which is not on line L1' (which consists of points a2 and a3), then the line L1 (consisting of points a1 and a0) contains a1 and is parallel to L1'. The geometry in Figure above has four points and six lines and obeys the 'Playfair' parallelism axiom. The figure given above is an example of this model because it satisfy all axioms. Each distinct line has exactly one line parallel to it If two distinct lines intersect, they contains exactly one pointģ. Any two distinct points have exactly one lineġ.

Pexamples of axioms for four line geometry