Lecture # 4

Instructor: Scot Drysdale June 27, 1996

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Continuations from yesterday

Convex hull problem W(n log n)
Given a convex hull algorithm, I can use it to sort .
Sort the numbers

Graph the points, you get a parabola - all points are on the convex hull.
Note that this works because of the function chosen. For example this will not work on a sin curve.

Homework review
Ceiling function (round up) ...

Floor function (round down) ...

Voronoi Diagrams

Voronoi Diagrams
The concept is more than a century old, discussed in 1850 by Dirichlet and in 1908 in a paper of Voronoi. A Voronoi Diagram records everything one would want to know about proximity to a set of points (or more general objects).

The post office problem
Where pizza parlors are located
Air rescue stations

Given the coordinates of a point quickly tell which station should respond. The closest one should respond. Each site should get a portion of the plane to cover.
One site - covers the whole world
Two sites - each sites cover half plane


Voronoi Diagrams:
Applications from Archeology through Zoology

Delaunay Triangulations - see diagram page 174--176
A triangulation of a set of points: connect any 2 points, connect another point using any edge as long as it doesn't cross an already existing edge.
Whenever you triangulate ....to be continued

Supplemental Notes:

Leonard Euler in 1758 noted that the sum of the number of vertices and faces is always two more than the number of edges in all polyhedra.
Euler's formula: Let V, E, and F be the number of vertices, edges, and faces respectively of a polyhedron then V - E + F = 2
A formal proof of Euler's formula is given on page 119.

Given a graph with 5 regions, as in figure 1 below, a triangulation is found by:
place a vertex (point) in each region.
connecting two regions if they have a common edge as shown in figure 2.
the complete triangulation is shown in figure 3.

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Last modified: October 3, 1996