Delauney三角网剖分算法原理：分为三步：一、凸包生成：1）求出如下四点：min(x-y)、min(x+y)、max(x-y)、max(x+y)并顺次放入一个数组，组成初始凸包；2）对于凸包上的点I，设它的后续点为J，计算矢量线段IJ右侧的所有点到IJ的距离，求出距离最大的点K；3）将K插入I,J之间，并将K赋给J；4）重复2，3步，直到点集中没有在IJ右侧的点为止；5）将J赋给I，J取其后续点，重复2，3，4步，当遍历了一次凸包后，凸包生成完成。
二、环切边界法凸包三角剖分：在凸包数组中，每次寻找一个由相邻两条凸包边组成的三角形，在该三角形的内部和边界上都不包含凸包上的任何其他点，然后去掉该点得到新的凸包链表，重复这个过程，最终对凸包数组中的点进行三角剖分成功。
三、离散的内插：1）建立三角形的外接圆，找出外接圆包含待插入点的所有三角形，构成插入区域；2）删除插入区域内的三角形公共边，形成由影响三角形顶点构成的多边形；3）将插入点与多边形所有顶点相连，构成新的Delaunay三角形；4）重复1，2，3，直到所有非凸包上的离散点都插入完为止。
功能实现流程：1. 在绘图菜单栏下添加一个子菜单项为Delauney，并且在工具栏上添加一个工具项。
设置text为Delaunay三角剖分，name为delaunay等属性，添加单击事件，并为单击事件代码2.为事件函数添加如下代码Graphics gra = panel1.CreateGraphics();List<Point_T> pts = new List<Point_T>();foreach (Geometry_T geo in choosegeos.Geofeatures){if (geo.GetType() == typeof(Point_T)){Point_T pt = (Point_T)geo;pts.Add(pt);}}List<Tin> deltins = DelauneyTin(pts);//根据多点构建delauney三角网foreach (Tin tin in deltins){Point[] ctin = new Point[3];for (int i = 0; i < 3; i++){cp = new Point((int)tin.Pthree[i].X, (int)tin.Pthree[i].Y); ctin[i] = cp;}gra.DrawPolygon(Pens.Red, ctin);}3．三角形TIN的数据结构public class Tin{Point_T[] pthree = new Point_T[3];Line_T[] lthree = new Line_T[3];public Line_T[] Lthree{get { return lthree; }set { lthree = value; }}public Point_T[] Pthree{get { return pthree; }set { pthree = value; }}public Tin(){ }public Tin(Point_T p1, Point_T p2, Point_T p3){pthree[0] = p1;pthree[1] = p2;pthree[2] = p3;lthree[0] = new Line_T(p1, p2);lthree[1] = new Line_T(p2, p3);lthree[2] = new Line_T(p3, p1);}}4.圆的数据结构public class Circle_T:Geometry_T{private Point_T cpt;public Point_T Cpt{get { return cpt; }set { cpt = value; }}double radius;public double Radius{get { return radius; }set { radius = value; }}public Circle_T(){ }public Circle_T(Point_T pt, double r){cpt = pt;radius = r;}}5.实现Delaunay三角剖分算法1）public List<Tin> DelauneyTin(List<Point_T> pts)//根据多点构建delauney三角网；分三步：构建凸包；凸包剖分；离散点内插{Graphics gra = panel1.CreateGraphics();List<Tin> deltins = new List<Tin>();List<Point_T> envpts = EnvelopeTin(pts);//构建凸包//for (int i = 0; i < envpts.Count - 1; i++)//{// gra.DrawLine(Pens.Black, new Point((int)envpts[i].X,(int)envpts[i].Y), new Point((int)envpts[i + 1].X, (int)envpts[i + 1].Y));//}//gra.DrawLine(Pens.Black, new Point((int)envpts[0].X, (int)envpts[0].Y), new Point((int)envpts[envpts.Count - 1].X, (int)envpts[envpts.Count - 1].Y));List<Point_T> dispts = new List<Point_T>();//非凸包上的离散点foreach (Point_T pt in pts){if (!envpts.Contains(pt)){dispts.Add(pt);}}List<Tin> envtins = EnvelopeDivision(envpts);//凸包剖分//foreach (Tin tin in envtins)//{// Point[] ctin = new Point[3];// for (int i = 0; i < 3; i++)// {// cp = new Point((int)tin.Pthree[i].X, (int)tin.Pthree[i].Y);// ctin[i] = cp;// }// gra.DrawPolygon(Pens.Blue, ctin);//}deltins = TinInsert(envtins, dispts);//离散点内插return deltins;}2）public List<Point_T> EnvelopeTin(List<Point_T> pts)//构建凸包{List<Point_T> envpts = new List<Point_T>();List<Point_T> othpts = new List<Point_T>();foreach (Point_T pt in pts){othpts.Add(pt);}//构建以x-y，x+y最大最小值组成的初始矩形框CompareXaddY comxandy = new CompareXaddY();CompareXsubY comxsuby = new CompareXsubY();pts.Sort(comxsuby);envpts.Add(pts[0]);envpts.Add(pts[pts.Count - 1]);othpts.Remove(pts[0]);othpts.Remove(pts[pts.Count-1]);pts.Sort(comxandy);if(!envpts.Contains(pts[0])){envpts.Insert(1, pts[0]);}if (!envpts.Contains(pts[pts.Count - 1])){envpts.Add(pts[pts.Count - 1]);}othpts.Remove(pts[0]);othpts.Remove(pts[pts.Count-1]);//构建以x-y，x+y最大最小值组成的初始矩形框int i = 0;int tag = 0;bool over = true;while(i<envpts.Count){Line_T cline;if (i==envpts.Count-1){cline = new Line_T(envpts[i], envpts[0]);}else{cline = new Line_T(envpts[i], envpts[i + 1]);}double dismax=0;for (int j = 0; j < othpts.Count ;j++ ){if (IsLeftPoint(othpts[j], cline)){double distance = PointToLine(othpts[j], cline);if (distance > dismax){dismax = distance;tag = j;over = false;}}}if (over){i++;}else{//envpts.RemoveAt(i);envpts.Insert(i+1, othpts[tag]);over = true;}}return envpts;}public List<Tin> EnvelopeDivision(List<Point_T> pts)//凸包剖分{List<Tin> envtins = new List<Tin>();List<Point_T> cpts = new List<Point_T>();foreach (Point_T pt in pts){cpts.Add(pt);}while (cpts.Count > 2){int tag = 0;double minangle = 120;for (int i = 0; i < cpts.Count; i++){double angle;if (i == 0){angle = CalcuAngle(cpts[cpts.Count - 1], cpts[i], cpts[i + 1]);}else if (i == cpts.Count - 1){angle = CalcuAngle(cpts[i-1], cpts[i], cpts[0]);}else{angle = CalcuAngle(cpts[i-1], cpts[i], cpts[i + 1]);}if ((angle - 60) < minangle){minangle = angle - 60;tag = i;}}int btag=tag-1;int atag=tag+1;if (tag == 0){btag = cpts.Count - 1;}else if (tag == cpts.Count - 1){atag = 0;}Tin ctin = new Tin(cpts[btag], cpts[tag], cpts[atag]);envtins.Add(ctin);cpts.RemoveAt(tag);}return envtins;}public List<Tin> TinInsert(List<Tin> tins, List<Point_T> pts)//离散点内插 {List<Tin> deltins = new List<Tin>();List<Tin> ctins = new List<Tin>();//临时凸包foreach (Tin tin in tins){ctins.Add(tin);}foreach (Point_T pt in pts)//对离散点遍历，内插{List<Point_T> cpts = new List<Point_T>();//临时点集foreach (Tin tin in ctins)//找到外接圆包含离散点的三角形{Circle_T ccir = DelauneyCicle(tin);//构造外接圆if (IsPointInCircle(pt, ccir))//点是否包含在圆内{//for (int i = 0; i < 3; i++)//{// if (!cpts.Contains(tin.Pthree[i]))// {// cpts.Add(tin.Pthree[i]);//记录当前点// }//}deltins.Add(tin); //记录保存当前三角形}}//List<Point_T> ecpts = EnvelopeTin(cpts);//求点集（外接圆包含离散的三角形）的凸包？，接下来，插入点，构建新三角网//for (int j = 0; j < ecpts.Count;j++ )//{// Tin tin;// if (j == ecpts.Count-1)// {// tin = new Tin(ecpts[j], ecpts[0], pt);// }// else// {// tin=new Tin(ecpts[j],ecpts[j+1],pt);// }// ctins.Add(tin);//}List<Line_T> eli = BorderTin(deltins);foreach (Line_T line in eli){Tin tin = new Tin(line.Frompt, line.Topt, pt);ctins.Add(tin);}foreach (Tin tin in deltins)//改变临时三角网（删除deltins保存的三角网）{ctins.Remove(tin);}deltins.Clear();}return ctins;}3）public bool IsLeftPoint(Point_T pt, Line_T line)//点在线的左边;叉积大于{bool yes = false;if ((pt.X - line.Frompt.X) * line.ParaA + (pt.Y - line.Frompt.Y) * line.ParaB > 0){yes = true;}return yes;}public double CalcuAngle(Point_T fp, Point_T mp, Point_T tp)//首，中，尾三点构成的夹角{double angle = 0;Point_T vector1 = new Point_T(fp.X - mp.X, fp.Y - mp.Y);Point_T vector2 = new Point_T(tp.X - mp.X, tp.Y - mp.Y);angle = Math.Acos((vector1.X * vector2.X + vector1.Y * vector2.Y) /(Math.Sqrt(vector1.X * vector1.X + vector1.Y * vector1.Y) *Math.Sqrt(vector2.X * vector2.X + vector2.Y * vector2.Y)));return angle;}public Circle_T DelauneyCicle(Tin tin)//构建三角形的外接圆{double x1 = tin.Pthree[0].X;double x2 = tin.Pthree[1].X;double x3 = tin.Pthree[2].X;double y1 = tin.Pthree[0].Y;double y2 = tin.Pthree[1].Y;double y3 = tin.Pthree[2].Y;double x = ((y2 - y1) * (y3 * y3 - y1 * y1 + x3 * x3 - x1 * x1) - (y3 - y1) * (y2 * y2 - y1 * y1 + x2 * x2 - x1 * x1))/ (2 * (x3 - x1) * (y2 - y1) - 2 * ((x2 - x1) * (y3 - y1)));double y = ((x2 - x1) * (x3 * x3 - x1 * x1 + y3 * y3 - y1 * y1) - (x3 - x1) * (x2 * x2 - x1 * x1 + y2 * y2 - y1 * y1))/ (2 * (y3 - y1) * (x2 - x1) - 2 * ((y2 - y1) * (x3 - x1)));Point_T cpt = new Point_T(x, y);double radius=Math.Sqrt(Math.Pow((x1-x),2)+Math.Pow((y1-y),2));Circle_T cir = new Circle_T(cpt,radius);return cir;}public bool IsPointInCircle(Point_T pt, Circle_T cir){if(Math.Sqrt(Math.Pow((pt.X-cir.Cpt.X),2)+Math.Pow((pt.Y-cir.Cpt.Y),2))<cir.Radius) {return true;}elsereturn false;}public List<Line_T> BorderTin(List<Tin> tins){List<Line_T> borli = new List<Line_T>();for (int i = 0; i < tins.Count; i++){for (int t = 0; t < 3; t++){bool tag = false;Line_T cl = tins[i].Lthree[t];for (int j = 0; j < tins.Count; j++){if (j!=i&&IsContainByTin(cl, tins[j])){tag = true;}}if (!tag)borli.Add(cl);}}return borli;}public bool IsContainByTin(Line_T li, Tin tin){for (int i = 0; i < 3; i++){if ((li.Frompt == tin.Lthree[i].Frompt || li.Frompt ==tin.Lthree[i].Topt) && (li.Topt == tin.Lthree[i].Topt || li.Topt ==tin.Lthree[i].Frompt)){return true;}}return false;}6.实现两个排序类CompareXsubY（x-y排序）和CompareXaddY（x+y 排序），仿照CompareX写功能操作步骤：先在面板上绘制多个点；框选部分点；按下实现Delaunay三角网剖分工具，Delaunay三角网剖分成功。
