Definition
For any set A of lines in the Euclidean plane, one can define an equivalence relation on the points of the plane according to which two points p and q are equivalent if, for every line l of A, either p and q are both on l or both belong to the same open half-plane bounded by l. When A is finite or locally finite the equivalence classes of this relation are of three types:
- the interiors of bounded or unbounded convex polygons (the cells of the arrangement), the connected components of the subset of the plane not contained in any of the lines of A,
- open line segments and open infinite rays (the edges of the arrangement), the connected components of the points of a single line that do not belong to any other lines of A, and
- single points (the vertices of the arrangement), the intersections of two or more lines of A.
These three types of objects link together to form a cell complex covering the plane. Two arrangements are said to be isomorphic or combinatorially equivalent if there is a one-to-one adjacency-preserving correspondence between the objects in their associated cell complexes.
Read more about this topic: Arrangement Of Lines
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