Delaunay Triangulation And Voronoi Graphs Pdf Free

The delaunayTriangulation classsupports creating Delaunay triangulations in 2-D and 3-D. It providesmany methods that are useful for developing triangulation-based algorithms.These class methods are like functions, but they are restricted towork with triangulations created using delaunayTriangulation. The delaunayTriangulation classalso supports the creation of related constructs such as the convexhull and Voronoi diagram. It also supports the creation of constrainedDelaunay triangulations.

The delaunayTriangulation classoffers more functionality for developing triangulation-based applications.It is useful when you require the triangulation and you want to performany of these operations:

MATLAB provides the delaunayn functionto support the creation of Delaunay triangulations in dimension 4-Dand higher. Two complementary functions tsearchn and dsearchn are also provided to supportspatial searching for N-D triangulations. See Spatial Searching for more information on triangulation-basedsearch.

The delaunayTriangulation classprovides another way to create Delaunay triangulations in MATLAB.While delaunay and delaunayTriangulation use thesame underlying algorithm and produce the same triangulation, delaunayTriangulation providescomplementary methods that are useful for developing Delaunay-basedalgorithms. These methods are like functions that are packaged togetherwith the triangulation data into a container called a class. Keepingeverything together in a class provides a more organized setup thatimproves ease of use. It also improves the performance of triangulation-basedsearches such as point-location and nearest-neighbor. delaunayTriangulation supportsincremental editing of the Delaunay triangulation. You also can imposeedge constraints in 2-D.

Triangulation Representations introducesthe triangulation class,which supports topological and geometric queries for 2-D and 3-D triangulations.A delaunayTriangulation isa special kind of triangulation.This means you can perform any triangulation queryon a delaunayTriangulation inaddition to the Delaunay-specific queries. In more formal MATLAB languageterms, delaunayTriangulation isa subclass of triangulation.

Indexing into the delaunayTriangulation output, DT, works like indexing into the triangulation array output from delaunay. The difference between the two are the extra methods that you can call on DT (for example, nearestNeighbor and pointLocation).

Use the delaunayTriangulation method, convexHull, to compute the convex hull and add it to the plot. Since you already have a Delaunay triangulation, this method allows you to derive the convex hull more efficiently than a full computation using convhull.

You can incrementally edit the delaunayTriangulation to add or remove points. If you need to add points to an existing triangulation, then an incremental addition is faster than a complete retriangulation of the augmented point set. Incremental removal of points is more efficient when the number of points to be removed is small relative to the existing number of points.

The tetramesh function plots both the internal and external faces of the triangulation. For large 3-D triangulations, plotting the internal faces might be an unnecessary use of resources. A plot of the boundary might be more appropriate. You can use the freeBoundary method to get the boundary triangulation in matrix format. Then pass the result to trimesh or trisurf.

The constraint between vertices (V1, V3) was honored, however, the Delaunay criterion was invalidated. This also invalidates the nearest-neighbor relation that is inherent in a Delaunay triangulation. This means the nearestNeighbor search method provided by delaunayTriangulation cannot be supported if the triangulation has constraints.

Constrained triangulations are generally used to triangulate a nonconvex polygon. The constraints give us a correspondence between the polygon edges and the triangulation edges. This relationship enables you to extract a triangulation that represents the region. The following example shows how to use a constrained delaunayTriangulation to triangulate a nonconvex polygon.

The plot shows that the edges of the triangulation respect the boundary of the polygon. However, the triangulation fills the concavities. What is needed is a triangulation that represents the polygonal domain. You can extract the triangles within the polygon using the delaunayTriangulation method, isInterior. This method returns a logical array whose true and false values that indicate whether the triangles are inside a bounded geometric domain. The analysis is based on the Jordan Curve theorem, and the boundaries are defined by the edge constraints. The ith triangle in the triangulation is considered to be inside the domain if the ith logical flag is true, otherwise it is outside.

The Delaunay algorithms in MATLAB construct a triangulationfrom a unique set of points. If the points passed to the triangulationfunction, or class, are not unique, the duplicate locations are detectedand the duplicate point is ignored. This produces a triangulationthat does not reference some points in the original input, namelythe duplicate points. When you work with the delaunay and delaunayn functions, the presence ofduplicates may be of little consequence. However, since many of thequeries provided by the delaunayTriangulation class are index based,it is important to understand that delaunayTriangulation triangulatesand works with the unique data set. Therefore, indexing based on theunique point set is the convention. This data is maintained by the Points propertyof delaunayTriangulation.

Because of its relative simplicity and ease of implementation, the Delaunay provides an attractive alternative to the Voronoi diagram. The property that allows it to be used as an alternative is the fact that the planar forms of the Delaunay triangulation and the Voronoi diagram are dual graphs of each other. Informally speaking, the fact that these two structures are dual graphs means that any set of sample points that produces a unique Delaunay triangulation will also produce a corresponding Voronoi diagram.

If a triangulation is properly Delaunay, then the circumcenters defined by its members correspond directly to the vertices that form the Voronoi diagram. The images below illustrate the relationships between two representative graphs. 2b1af7f3a8

Delaunay Triangulation And Voronoi Graphs Pdf Free

Inscrivez-vous à notre newsletter pour découvrir toutes nous nouveautés