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Voronoi Diagram - Essay Example

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The use of computer simulations to a model a geographic location is already widespread especially in terrestrial systems. The need for modeling marine systems rose from the increasing active use of the said system especially in navigation and environmental management concerns…
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Voronoi Diagram
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Download file to see previous pages Traditional GIS methods have been found to inapplicable for marine mapping. This is primarily because they were built for two-dimensional land application making it hard to integrate marine features into the model. Marine objects are also likely to move over time which cannot be modeled using the traditional GIS. These limitations necessitated the development of a new modeling system that can accurately incorporate marine features while allowing modifications to the system which does not require an overhaul of the whole model.
Christopher Gold (1990) responded to the challenge by spearheading research and development of the Voronoi Diagram - a modeling system with a dual geometric structure. Most of the literature on the development of the VD, either in 2D or in 3D, was authored by him. Voronoi diagrams that were developed were able to solve most of the problems because of the following features:
All of these features are available in 2D and 3D Voronoi Diagrams. This paper aims to differentiate 2D Voronoi Diagrams from 3D Voronoi Diagrams delineating their differences, advantages and disadvantages over the other. This paper also aims at pointing out the strengths and weakness of the two diagrams such that a conclusion on which one is more advantageous can be made.
In 2D Voronoi Diagram, the cell surrounding a data point is a flat convex polygon having a defined number of neighbors (Gold and Ledoux, 1992). That is, its coordinates are only x and y with no z attribute. The analogy is the same as that of drawing figures on a piece of paper. When a plan view is done on the paper, one can see the shapes defined by the lines that were drawn. When the paper is leveled against one's eyesight, there are no figures which can be seen. This illustrates that no such elevation or depth attribute of the figures exist. The geometric dual structure of 2D Voronoi Diagrams are also "flat" in nature and are defined by Delaunay triangles.

In Figure 1, Delaunay Triangles are shown by the dashed lines while the solid lines defining a polygon represent the cells surrounding a data point p.

Figure 1. A 2D Voronoi Sample Output (Gold, 1991)
The vertices of the triangle generating each Voronoi cell must satisfy the empty circumcircle test. A circle is considered empty when there are no points in its interior but more than three points can be directly on the circle - i.e. the points are on its edges.
3D Voronoi Diagram Construct
3-Dimensional Voronoi Diagrams, as implied by its name, have 3 coordinates defining the space where the figure can be drawn. As opposed to 2D VDs', leveling the plane of the paper with one's eyesight provides a view of the sides of a figure. An appropriate analogy would be that of the viewing a cube held by the hand. When the figure is viewed from the top, one can see a square. When the hand is leveled against one's eyesight, one can still see the figure of a square. The figure is a volumetric object. The convex polygon in a 2D, thru a construction algorithm, generalizes to a convex polyhedron. The geometric dual becomes a Delaunay tetrahedron.

In Figure 2, the edges are the Delaunay edges joining the generator ...Download file to see next pagesRead More
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