Handouts: examples handout on duality
4/23/08: Arrangements of lines in the plane

Review: Arrangements of lines: vertices, edges, faces, sign vectors, etc. Basic Combinatorics: Number of vertices, edges, faces. "Simple" arrangements.
Complexity of one or more faces in an arrangement of n lines. Upper and lower bounds.
Examples of what a "zone" is and what its complexity is.

The Zone Theorem (6.2.2), and its proof using induction.
Corollary to the Zone Theorem: We give an incremental algorithm that takes worst-case time O(n^2) to compute a data structure (e.g., winged-edge, twin-edge, or quad-edge) for an arrangement of n lines in the plane. We went over this in some detail. Note that just computing the vertices of the arrangement is easy in O(n^2) time (naive for loops); using sorting one gets O(n^ log n) to build a data structure for the whole arrangement, but the Zone theorem gives us even better: O(n^2).

I mention results in higher dimensions: The Zone Theorem carries over to show that the zone of a hyperplane in d dimensions has complexity O(n^{d-1}), implying an incremental algorithm of complexity O(n^d). I discussed some history on this point, including the fact that the originally published proof had an error that went undetected for many years.

We introduce duality of points and lines. I gave examples and defined the main duality the text uses, including its relationship to the parabola, etc.