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 arrangment, line segments and rays, the edges of the arrangement, and points where two or more lines cross, the vertices of the arrangement. When considered in the projective plane rather than in the Euclidean plane, every two lines cross, and an arrangement is the projective dual to a finite set of points. Arrangements of lines have also been considered in the hyperbolic plane, and generalized to pseudolines, curves that have similar topological properties to lines. The initial study of arrangements has been attributed to an 1826 paper by Jakob Steiner.
An arrangement is said to be simple when at most two lines cross at each vertex, and simplicial when all cells are triangles (including the unbounded cells, as subsets of the projective plane). There are three known infinite families of simplicial arrangements, as well as many sporadic simplicial arrangements that do not fit into any known family. Arrangements have also been considered for infinite but locally finite systems of lines. Certain infinite arrangements of parallel lines can form simplicial arrangements, and one way of constructing the aperiodic Penrose tiling involves finding the dual graph of an arrangement of lines forming five parallel subsets.
The maximum numbers of cells, edges, and vertices, for arrangements with a given number of lines, are quadratic functions of the number of lines. These maxima are attained by simple arrangements. The complexity of other features of arrangements have been studied in discrete geometry; these include zones, the cells touching a single line, and levels, the polygonal chains having a given number of lines passing below them. Roberts's triangle theorem and the Kobon triangle problem concern the minimum and maximum number of triangular cells in a Euclidean arrangement, respectively.
Algorithms in computational geometry are known for constructing the features of an arrangement in time proportional to the number of features, and space linear in the number of lines. As well, researchers have studied efficient algorithms for constructing smaller portions of an arrangement, and for problems such as the shortest path problem on the vertices and edges of an arrangement.