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);
}
}
GeometryPoint 定义
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);
}
}
SimplePolygon定义
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();
}
}
SimplePolygon 控制台调用方法
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，请参照另一篇文章“轮廓算法”。
完毕。
作者：Andy Zeng
欢迎任何形式的转载，但请务必注明出处。
http://www.cnblogs.com/andyzeng/p/3754005.html
几何算法：点集合构造简单多边形,布布扣,bubuko.com