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
Famous quotes containing the word definition:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)