1. UNC Chapel Hill M. C. Lin Point Location Chapter 6 of the Textbook –Review –Algorithm Analysis –Dealing with Degeneracies

2. UNC Chapel Hill M. C. Lin Voronoi Diagram Chapter 7 of the Textbook Driving Applications –The Post Office Problem –Robot Motion Planning –Finding Nearest Neighbors –Distance Computation by Tracking the Closest Features between 2 Polytopes

3. UNC Chapel Hill M. C. Lin Post Office Problem What is the trading area of certain cities? Where do people shop? Where is the nearest post office?

4. UNC Chapel Hill M. C. Lin Transform to a Geometric Problem The model where every point is assigned to the nearest site is called the Voronoi assignment. The subdivision induced by this model is the Voronoi diagram of the set of sites. Voronoi diagram has been used in robotics, astronomy, and many fields. It’s the dual of “Delaunay triangulation”.

5. UNC Chapel Hill M. C. Lin Definition Let P := {p 1, p 2, …, p n } be a set of n distinct points in the plane; these points are the sites. Definition: Voronoi diagram of P, Vor(P), as the sub-division of the plane into n cells, one for each site in P, with property that a point q lies in a cell corresponding to a site p i if and only if dist( q, p i ) < dist( q, p j ) for each p j  P with j  i. The cell that corresponds to a site is denoted V(p j ) or the Voronoi cell of p j.

6. UNC Chapel Hill M. C. Lin Observations For two points p and q in the plane we define the bisector of p and q as the perpendicular bisector of the line segment pq. This bisector splits the plane into 2 half-planes. We denote the open half-plane containing p&q by h(p, q) and h(q, p) respectively. r  h(p, q) if and only if dist( r, p ) < dist( r, q ) V(p i ) =  1  j  n, j  i h(p i, p j ). That is, V(p i ) is the intersection of n-1 half-planes and open convex polygonal region bounded by at most n-1 vertices and at most n-1 edges.

7. UNC Chapel Hill M. C. Lin Basic Properties Let P be a set of n point sites in the plane. If all sites are collinear then consists of n-1 parallel lines and n cells. Otherwise, Vor(P) is connected and its edges are either segments or half-lines. The number of vertices in the Voronoi diagram of a set of n point sites is at most 2n-5 and the number of the edges is at most 3n-6. –Use Euler’s formula and property of connected planar graph with one extra vertex at infinity

8. UNC Chapel Hill M. C. Lin Basic Properties For the Voronoi diagram Vor(P) of a set of points P, the following holds: –A point is a vertex of Vor(P) if and only if its largest empty circle C p (q) contains 3 or more sites on its boundary –The bisector between sites p i & p j defines an edge of Vor(P) if and only if there is a point q  E 2 s.t. C p (q) contains both p i and p j on its boundary but no other site.

9. UNC Chapel Hill M. C. Lin App: Tracking Closest Features See the transparencies in class.