Abstract in this paper with the planar scattered point set to point to insert Delaunay three diagonalization method for the foundation, in three during the process of keratinization adopt certain strategy ,the improvement becomes a simple and efficient method .
The method can be adapted to various boundary ,including many islands ,such as complex multiply connected domain ,to generate body-fitted triangular mesh ,mesh & # 26684 ;to ensure compliance with the rules of Delaunay .
Key words Delaunay triangulation triangulation and other & # 20540 ;line triangulation is one of the main topics in the field of computational geometry ,and has a broad application prospects .
In computer graphics ,visualization in scientific computing ,natural science ,especially in the field of Geosciences ,often need to deal with a large number of distribution in one area within the discrete data .
Because these data distribution inhomogeneity ,resulted in a reasonable and effective use of these valuable data problem .Voronoi and Delaunay triangulation is generally accepted and widely used for the analysis of the study area are powerful tools for discrete data .
Of course ,it is not only suitable for the school,Headphones Online, and is active in all with simple three-dimensional analysis of related fields .But the classical method in planar scattered point set can be constructed on the convex set ,and the efficiency of state difference ,as the increase sharply increased number time .
In order to adapt to the need of engineering ,this paper improves the insertion point by point method ,make become simple with fast and efficient method ,and can adapt to various boundary ,including multiple Island ,multiple connected domain ,such as complex concave boundary .
1 Voronoi Delaunay triangulation polygon and the definitions and properties of x set of points on the plane ,then the regional V ( I ) = {x E2d ( x ,VI ) = D ( x ,Cheap Vibram,VJ ) ,j = 1, 2 ,.
.. ,N ,J and I } called the Voronoi polygon (V polygon ) .The V polygon composed of Voronoi - diagram ( referred to as V - diagram ) .Delaunay triangulation definition :public side of the V polygon is called adjacent V polygon .
Connect all the adjacent V - polygon growth center of the triangle formed by the network called Delaunay triangulation (D - triangle mesh.) .D triangulation of the outer boundary is a convex polygon ,it consists of nodes concentrated convex form ,often referred to as the convex hull .
D triangulation has two important properties .( 1) :empty circumcircle property from the point set V formed by D triangulation ,each of its circumcircle of a triangle are not included in the set of points V other arbitrary point .
( 2) the largest to smallest angle in nature :from point V to form triangular net ,D triangulation of minimal triangle angle is the largest .As a result of these two properties ,determines the D triangulation is of great application value and # 20540 ;.
At the same time ,it is also a two-dimensional plane triangle net only ,the best .2 improved methods in data structure would first need to triangulation points scattered points and boundary point ,boundary point according to a certain order ( clockwise or counterclockwise ) arrangement .
Some of the data structure for the one-way linked list ,defined as follows .Typedef struct Vertex {int ID ;/ / ID double x ,y ,Z ;/ / coordinate & # 20540 ;char mark ;/ / material ,boundary markers ,distinguish the scatter ,outer boundary point ,boundary point struct Vertex * next ;/ / the next node } * PVERTEX ,VERTEX ;triangle unit is defined by triangulation based ,must be well defined structure expression of triangle topological structure ,to improve the efficiency of the algorithm ,and must be convenient for triangular element in river network from topology .
This article is defined as follows .Typedef struct Triangle {struct Edge * PE ;/ / pointing triangle three sides of the struct Vertex * PV ;/ / at the three apexes of the triangle int ID ;/ / ID struct Triangle * next ,* pre ;/ / pointing and triangle } * PTRIANGLE ,TRIANGLE ;triangle unit topology but also through edge to edge ,defined as :typedef struct Edge {struct Vertex * PV ;/ / pointer to the side of the two vertices struct Triangle * Pt ;/ / pointer to the edge belongs to the two triangle struct Edge * next ,* pre ;/ / pointing and edge } * PEDGE ,EDGE ;boundary list is defined as follows :typedef struct BndChain {struct Vertex chain ;/ / a boundary char mark ;/ / boundary markers ,distinguish the scatter ,outer boundary point ,boundary point within the boundary points of struct BndChain * next ;/ / the next boundary } * PBNDCHAIN ,BNDCHAIN ;3 to determine whether a point on an arbitrary polygon method generally speaking ,judging whether a point is in a polygon of arbitrary shape is difficult .
This paper uses a simple method to judge .Fig 1 input according to a certain order ( clockwise or counterclockwise ) arranged the boundary set P1 ,p2 ,... PN ,and the P coordinate .The judgment is based :if the points in polygon ,point and directed line segment corresponding to the directed arc angle equals 0 ;if it fell on a polygon ,then directed arc angle to the sum of the absolute & # 20540 ;equal to 2 pi ,as shown in figure 1 .
4 core algorithm Tsai based on the realization of the process, the generated D triangulation algorithms are divided into three categories :divide-and-conquer algorithm ,Moncler Jackets,point by point insertion method ,triangulation network growth method ( 2 ) .
This paper adopts the insertion point by point method ,the core idea of is :when adding a new point ,find out contains the point in the triangle ( included in the triangular edge) .If falling in triangle ,the triangle point and three point connection ,and the three sides of a triangle to the optimization of the queue ,in accordance with the Delaunay triangulation two properties were optimized ;if falling in triangle ,delete the edge ,to build two new edges ,and the remaining two sides or four edges ( a male of the adjacent triangles ) into the optimization of the queue optimization .
5 optimization algorithm ( 1 ) from the priority queue took out a side ,began to optimize this side .( 2 ) if the edge belongs to the edge ,the edge without optimization .( 3) if the edge is being optimized the circumcircle of a triangle contains the public this side of the triangle in addition to the outside edge of another vertex ,the current optimization edge from optimizing the queue is removed, the optimized triangle removed,Beats By Dre, while the other two sides of the triangle is added to the optimization of the queue in addition ;otherwise, will from the optimal queue is removed into the optimized queue ,to facilitate the establishment of new triangle unit .
( 4) repetition ( 1) ,( 2) ,( 3) the optimization step ,until the queue is empty .The nearest point of Voronoi linear dual graph is Delaunay triangulation, so it can be put on the paper describes the structure Voronoi graph partition algorithm for Delaunay triangulation .
Incremental algorithm :easy realization ,widely used ,suitable for small scale set three keratosis .Specific process is as follows :1 traversal of all scatter ,generates a contains all scatter the big triangle ( apex not in focus ) 2 untreated P ,insert ,or end algorithm ,exit 3 in split good triangulation find contains the P triangles T, P with t three vertices are connected, forming three triangle 4 according to the optimization criterion for locally generated triangle optimization ( interchangeable diagonal etc.
) 5 return second step local transformation method :according to the Delaunay triangulation nature 2 ,first constructs a does not satisfy the Delaunay triangulation conditions of the river network ,then on the two a total of triangular form a convex quadrilateral iterative switch so as to meet the Delaunay triangulation conditions (mainly exchange diagonal method ) Delaunay triangulation has the following properties :( 1) Delaunay Triangle branch forming a triangle ,minimal angle is the largest all triangulation .
The Delaunay triangulation of the triangle formed by the most close to equilateral triangles ,in many applications with optimal properties .This property is equivalent to Delaunay Triangle branch formed triangle circumcircle does not contain other points .
( 2) if any four points not all round ,the four points can only form the only Delaunay triangle ,otherwise not only .So that ,on the Delaunay triangulation local ensure that ensures that the whole to ensure that meet the Delaunay triangulation .
( 3) in Delaunay three hyperkeratosis of the network & # 26684 ;adding a little P ,only need to remove all the circumscribed circle contains the point in the triangle ,and the connection of P with all visible point (i.
e., after the connection not and other sides ) ,the net & # 26684 ;still meet the Delaunay triangulation condition .In 1 ,the incremental algorithm this algorithm is based on nature ( 3) ,the algorithm is simple, the time complexity is O ( nlogn ) ,widely used ,basic steps are as follows :( 1) generates a contains all points of the triangle ( the point is not in focus ) ;( 2 ) on the concentration of each point ,Louboutin Sale,depending on the nature of ( 3) for processing ( without removing the big triangle ) ;( 3) to remove all with big triangle side .
The algorithm is mainly used for the circle of the search and the vertex connectivity .Among them, the former can use the compass algorithm optimization ;the latter can be on point set are sorted ,to join network & # 26684 ;points uniformly distributed to reduce complexity .
In 2 ,local transform method the algorithm is based on nature ( 2) ,first constructs a does not satisfy the Delaunay triangulation triangulation & # conditions ,26684 ;for a total of two side of the triangle form convex quadrilateral iterative switch so as to meet the Delaunay triangulation conditions (mainly exchange diagonal method ) .
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