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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.
Although the most popular version is for arrangements of lines in the plane, there exist some generalizations of the zone theorem. For instance, in dimension , considering arrangements of hyperplanes, the complexity of the zone of a hyperplane is the number of facets (- dimensional faces) bounding the set of cells (-dimensional faces) intersected by .
Euclidean line and Euclidean geometry are terms introduced to avoid confusion with generalizations introduced ... this partition is known as an arrangement of lines.
Line arrangements. In discrete geometry, an arrangement is the decomposition of the d-dimensional linear, affine, or projective space into connected cells of different dimensions, induced by a finite collection of geometric objects, which are usually of dimension one less than the dimension of the space, and often of the same type as each other, such as hyperplanes or spheres.
Here, an arrangement is simple when no two of its lines are parallel and no three lines pass through the same point. A face is one of the polygons formed by the arrangement, not crossed by any of its lines. Faces may be bounded or infinite, but only the bounded faces with exactly three sides count as triangles for the purposes of the theorem. [1]
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In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set A of hyperplanes in a linear, affine, or projective space S.Questions about a hyperplane arrangement A generally concern geometrical, topological, or other properties of the complement, M(A), which is the set that remains when the hyperplanes are removed from the whole space.
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