几何算法：点集合构造简单多边形

问题：给定平面中n个点所组成的集合，将它们连接起来形成一条简单的封闭路径。所谓简单路径，是指边与边无交叉。

如下图所示10个点组成的简单轮廓：

思路：取x坐标最大的点A（如果最大x坐标的点不止一个，则取Y坐标最小的点），依次计算A点与其余各点的连线与水平线之间夹角的正切值，然后按照正切值排序，依次连接排序后的各点即组成一个简单图形。

原理：其它所有点都在A点的左侧，所有夹角的范围为-Pi/2~Pi/2，单调递增函数。

举一个例子如下：

 

各点坐标与A点的角度斜率如下（已经排序好）：

x:426.192518536091,y:30.5668629242884,slope:-2.21036105157629
x:132.904271903869,y:111.805767306036,slope:0.0233827696146631
x:209.153583263584,y:158.396180071121,slope:0.216615047225945
x:51.2625493860163,y:271.425922467106,slope:0.409713066051227
x:172.80558813494,y:320.363658168522,slope:0.754116336162768
x:174.841647802313,y:361.474091434606,slope:0.903935084923323
x:262.993097888768,y:306.679940091763,slope:1.03059799172764
x:405.520514378101,y:212.478244240618,slope:2.00680658499766
x:410.405247491042,y:324.597360433357,slope:4.49064367657446
x:459.491329337233,y:104.169257382941,slope:1.79769313486232E+308

其中A点为：x:459.491329337233,y:104.169257382941,slope:1.79769313486232E+308

下面给出具体算法（C#实现）：

几何点定义，实现IComparable<T>接口，按照正切值排序要用到：

public struct GeometryPoint : IComparable<GeometryPoint>
    {        
        public GeometryPoint(double x, double y, double slope = double.NaN)
        {
            this.x = x;
            this.y = y;
            this.slope = slope;
        }
        private double x;
        public double X
        {
            get { return x; }
            set { x = value; }
        }
        private double y;
        public double Y
        {
            get { return y; }
            set { y = value; }
        }
        private double slope;
        public double SLOPE
        {
            get { return slope; }
            set { slope = value; }
        }
      
        public int CompareTo(GeometryPoint p)
        {
            if (this.slope < p.slope)
            {
                return -1;
            }
            else if (this.slope > p.slope)
            {
                return 1;
            }
            else
            {
                if (this.x == p.x && this.SLOPE == p.SLOPE && this.SLOPE == double.MaxValue)
                {
                    if (this.y == p.y)
                    {
                        return 0;
                    }
                    else if (this.y < p.y)
                    {
                        return 1;
                    }
                    else//(this.y > p.y)
                    {
                        return -1;
                    }
                }
                return 0;
            }
        }
        public override string ToString()
        {
            return string.Format("x:{0},y:{1},slope:{2}", x, y, slope);
        }
    }

简单封闭图形定义，并定义初始化简单封闭图形的方法，该方法随机产生多边形的顶点：

public class SimplePolygon
    {
        private GeometryPoint[] geometrypoints;

        public GeometryPoint[] GeometryPoints
        {
            get { return geometrypoints; }
            set { geometrypoints = value; }
        }


        public SimplePolygon()
        {
        }
        public void Initialize(int size, double minX, double maxX, double minY, double maxY)
        {
            if (size <= 0) throw new ArgumentOutOfRangeException();
            geometrypoints = new GeometryPoint[size];
            Random rnd = new Random(DateTime.Now.Millisecond);
            double xRange = maxX - minX;
            double yRange = maxY - minY;
            int MaxXPointIndex = 0;//选取x坐标最大的点
            for (int i = 0; i < size; i++)
            {
                GeometryPoint gp = new GeometryPoint(minX + xRange * rnd.NextDouble(), minY + yRange * rnd.NextDouble());
                geometrypoints[i] = gp;
                if (geometrypoints[MaxXPointIndex].X < gp.X)////选取x坐标最大的点
                {
                    MaxXPointIndex = i;
                }
                else if (geometrypoints[MaxXPointIndex].X < gp.X && geometrypoints[MaxXPointIndex].Y > gp.Y)//选取x坐标最大的点，如果最大x坐标点有多个，去y最小者
                {
                    MaxXPointIndex = i;
                }
            }
            //计算斜率
            for (int i = 0; i < size; i++)
            {
                if (i == MaxXPointIndex)
                {
                    geometrypoints[MaxXPointIndex].SLOPE = double.MaxValue;
                }
                else
                {
                    if (geometrypoints[i].X == geometrypoints[MaxXPointIndex].X)//与最大x坐标的x相同的点,因为x坐标之差为零，所以取SLOPE最大值
                    {
                        geometrypoints[i].SLOPE = double.MaxValue;
                    }
                    else//计算斜率，注意正切函数在-0.5Pi和0.5Pi之间是单调递增的
                    {
                        geometrypoints[i].SLOPE = (geometrypoints[i].Y - geometrypoints[MaxXPointIndex].Y) / (geometrypoints[MaxXPointIndex].X - geometrypoints[i].X);
                    }
                }
            }
            //按照斜率slope排序，取稳定排序方法的堆排序。
            HeapSort<GeometryPoint> heapsort = new HeapSort<GeometryPoint>();
            heapsort.Sort(this.geometrypoints,0,size-1);
        }
    }

控制台程序调用方法，按照连线顺序打印顶点：

class Program
    {
        static void Main(string[] args)
        {
            SimplePolygon sp = new SimplePolygon();
            sp.Initialize(10, -50, 50, -50, 50);
            for (int i = 0; i < sp.GeometryPoints.Length; i++)
            {
                Console.WriteLine(sp.GeometryPoints[i]);
            }
            Console.ReadKey();
        }
    }

如果用界面绘图，应用WPF几何绘图可实现如下效果，红线为计算正切值的示例连线，绿色线为生成的简单多边形：

关于坐标系与绘图的方法

  完毕。