Tutorial 3 – Computational Geometry Voronoi Diagrams Tutorial 3 – Computational Geometry

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.

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

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

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

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.

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.

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 :

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

Bibliography: Voronoi Applet: http://www.pi6.fernuni-hagen.de/GeomLab/voroadd/ Reitsma Reitsma, Stanislav Trubin, Saurabh Sethia, "Information Space Regionalization Using Adaptive Multiplicatively Weighted Voronoi Diagrams," iv, pp. 290-294,  Eighth International Conference on Information Visualisation (IV'04),  2004 Wikipedia – Voronoi Diagrams.