1. Tutorial 3 – Computational Geometry
Voronoi Diagrams
Tutorial 3 – Computational Geometry

2. Simple Voronoi diagrams
A tessellation of the planes into cells. Each cell is the set of points closed to a point site (Euclidean metric).
All edges are parts of bisectors between the sites.

3. Additive Weighted Voronoi Diagrams
Every site begins to grow in a different point in time.
New distance function:
Bisectors are usually Hyperbolic arcs

4. Multiplicatively-Weighted Diagrams
Each site grows at different rate.
Distance function:
Bisectors are usually circular arcs.
Regions can be surrounded.

5. Different Metrics - Metric is defined as:
Also called “Manhattan Distances” because it measures distances through axis-aligned streets.

6. Centroidal Voronoi Diagrams
Each site is also the center-of-mass of each cell.
Points are distributed evenly.
Uses: data compression, quantization, optimal mesh generation, etc.

7. CVD Computation: Lloyd’s Algorithm
Given a set of sites :
Compute VD.
While (tolerance value not reached):
Move sites to center of respective cells.
Recompute VD.
Tolerance value is a function of distance from the sites to the respective centers of cells.

8. Lloyd’s Algorithm – cont’d
Converges to a centroidal Voronoi diagram – slowly.
Simple to apply.
Because of slow convergence, the algorithm stops at a tolerance value.
Computing of centroid of a polygon (CCW order of the vertices :

9. Higher Dimensions Voronoi Diagrams
Cells are convex polytopes.
Bisectors are (d-1)-halfplanes.
Complexity:

10. Bibliography:
Voronoi Applet:
Reitsma Reitsma, Stanislav Trubin, Saurabh Sethia, "Information Space Regionalization Using Adaptive Multiplicatively Weighted Voronoi Diagrams," iv, pp ,  Eighth International Conference on Information Visualisation (IV'04),  2004
Wikipedia – Voronoi Diagrams.