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In geometry, an arrangement of lines is the subdivision of the Euclidean plane formed by a finite set of lines. An arrangement consists of bounded and unbounded convex polygons , the cells of the arrangement, line segments and rays , the edges of the arrangement, and points where two or more lines cross, the vertices of the arrangement.
In the projective plane C, it can be shown that there exist four lines, no three of which are concurrent. Dualizing this theorem and the first two axioms in the definition of a projective plane shows that the plane dual structure C* is also a projective plane, called the dual plane of C. If C and C* are isomorphic, then C is called self-dual.
As any line in this extension of σ corresponds to a plane through O, and since any pair of such planes intersects in a line through O, one can conclude that any pair of lines in the extension intersect: the point of intersection lies where the plane intersection meets σ or the line at infinity. Thus the axiom of projective geometry, requiring ...
Configurations (4 3 6 2) (a complete quadrangle, at left) and (6 2 4 3) (a complete quadrilateral, at right).. In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.
Steinitz (1894) [8] has shown that n 3-configurations (incidence structures with n points and n lines, three points per line and three lines through each point) are either realizable or require the use of only one curved line in their representations. [9] The Fano plane is the unique (7 3) and the Möbius–Kantor configuration is the unique (8 3).
[1]: 300 In two dimensions (i.e., the Euclidean plane), two lines that do not intersect are called parallel. In higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. On a Euclidean plane, a line can be represented as a boundary between two regions.
The lines in the plane can also be represented by homogeneous coordinates. A projective line corresponding to the plane ax + by + cz = 0 in R 3 has the homogeneous coordinates (a : b : c). Thus, these coordinates have the equivalence relation (a : b : c) = (da : db : dc) for all nonzero values of d.
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. [9] Such a drawing is called a plane graph or planar embedding of the graph.