Arrangement of Lines - Projective Arrangements and Projective Duality

Projective Arrangements and Projective Duality

It is often convenient to study line arrangements not in the Euclidean plane but in the projective plane, due to the fact that in projective geometry every pair of lines has a crossing point. In the projective plane, we may no longer define arrangements using sides of lines (a line in the projective plane does not separate the plane into two distinct sides), but we may still define the cells of an arrangement to be the connected components of the points not belonging to any line, the edges to be the connected components of sets of points belonging to a single line, and the vertices to be points where two or more lines cross. A line arrangement in the projective plane differs from its Euclidean counterpart in that the two Euclidean rays at either end of a line are replaced by a single edge in the projective plane that connects the leftmost and rightmost vertices on that line, and in that pairs of unbounded Euclidean cells are replaced in the projective plane by single cells that are crossed by the projective line at infinity.

Due to projective duality, many statements about the combinatorial properties of points in the plane may be more easily understood in an equivalent dual form about arrangements of lines. For instance, the Sylvester–Gallai theorem, stating that any non-collinear set of points in the plane has an ordinary line containing exactly two points, transforms under projective duality to the statement that any arrangement of lines with more than one vertex has an ordinary point, a vertex where only two lines cross. The earliest known proof of the Sylvester–Gallai theorem, by Melchior (1940), uses the Euler characteristic to show that such a vertex must always exist.