Handouts:
4/16/08: More on Voronoi diagrams; Start of Arrangements of Lines in the Plane

More discussion of properties of Voronoi diagrams (see handout and text). In particular, I describe in detail the "lifting map" that takes points in the plane to points on a 3D paraboloid of revolution (z=x^2+y^2). Then, I argue that the Delaunay diagram is in fact the projection of the lower convex hull of the lifted points. One consequence is that the divide-and-conquer algorithm that runs in time O(n log n) for computing comvex hulls in 3D for point sets actually gives an O(n log n) algorithm for computing Delaunay diagrams in 2D (and thus also for Voronoi diagrams in 2D).
Summary of Algorithms: (1) brute force (O(n^4)), which just checks the empty circle property for all triples of points; (2) cell-by-cell construction: for each site, intersect the n-1 halfspaces that are defined with respect to each other site (using the bisector), doing this in O(n log n) per site for an overall O(n^2 log n); (3) incremental insertion (worst-case O(n^2), but O(n log n) on average if the points are added in random order); (4) divide and conquer; (5) plane sweep (Fortune's algorithm, which uses a wavefront consisting of parabolic arcs); (6) Lawson's edge flip (to complete next time).
I go through the edge flip algorithm in some detail, explaining how edge flips relate to 3D convex hull of the lifted points, etc. We argue that there can be at most O(n^2) flips in total (I used a folded piece of paper as a prop!), each taking time O(1).
Brief discussion of (1) medial axis (Voronoi diagram of sites that are line segments), (2) Delaunay/Voronoi in higher dimension, (3) other metrics
Introduction to arrangements of lines: vertices, edges, faces, sign vectors, etc. Basic Combinatorics: Number of vertices, edges, faces. "Simple" arrangements.