Handouts:
4/28/08: Arrangements and Duality

Review duality of points and lines. I gave examples and defined the main duality the text uses, including its relationship to the parabola, etc.
Properties of duality: I go through in detail the 5 properties given in the text. We worked examples. I showed how the dual of the set of points that lie on a line segment is a family of lines that form a "double wedge" (and which one it forms).
Applications of Duality:
(1) Stabbing a set of line segments (dual of a segment, dual of a triangle): In the dual, we look at the arrangement of double wedges corresponding to each segment, and we label each face with the number of double wedges present there. Then, the problem of finding a line that stabs the most segments is equivalent to finding a point in a face that has the highest label.
(2) angular sorting (which applies to "visibility graphs", which allow us to compute shortest paths for a robot among polygonal obstacles): How to sort angularly points around each of n points. Naively this can be done in time O(n^2 log n), but the n sorts are not completely independent of each other. Using duality, and constructing the arrangement of the duals to the n points, I showed how to do the n sorts in total time O(n^2).
(3) Ham Sandwich Theorem: mentioned it only briefly with an example.
(4) Degeneracy testing: Determine if there are 3 points that are collinear among an input set of n points.