向量基础

1.向量的表示

2.向量运算

2.1.向量的内积（点乘）

向量的内积是一个标量，代表向量b在向量a反向上的投影


$$\overrightarrow{a}\cdot \overrightarrow{b}\{\begin{array}{ll}>0,& 同向(锐角)\\ =0,& 垂直（直角）\\ <0,& 反向（钝角）\end{array}$$

2.2.向量的外积（叉乘）

向量a和向量b的外积结果是一个向量，即垂直于a和b向量构成的平面的法向量

$$\overrightarrow{a}\times \overrightarrow{b}=|\overrightarrow{a}|\cdot |\overrightarrow{b}|\cdot sin\theta \{\begin{array}{ll}>0,& \overrightarrow{b}在\overrightarrow{a}的逆时针方向\\ =0,& 共线\\ <0,& \overrightarrow{b}在\overrightarrow{a}的顺时针方向\end{array}$$

折返线检查

思路：得到几何图形的折点，按顺时针方向进行遍历，若内角为锐角，则向量b在向量a的顺时针方向，若为钝角，则向量b在向量a的逆时针方向，可用向量的叉乘进行判断

/// <summary>
/// 获取角度限差内的节点
 /// </summary>
 /// <param name="geo"></param>
 /// <param name="torrance">角度上限</param>
 /// <returns></returns>
 public static IPointCollection GetPointsWithinTorrance(IGeometry geo, double torrance)
 {
    
      bool isPolygon = false;
      if (geo.GeometryType == esriGeometryType.esriGeometryPolygon)
          isPolygon = true;
      var points = geo as IPointCollection;
      IPointCollection result = new Multipoint();
      if (isPolygon)
      {
    
          ///面图形得到的点为顺时针，且起点会被记录两次形成闭环
          for (int i = 1; i < points.PointCount - 1; i++)
          {
    
              var point = points.Point[i];
              if (isSkipPoint(points.Point[i - 1], point, points.Point[i + 1]))
                  continue;
              var angel = CalAngel(points.Point[i - 1], point, points.Point[i + 1]);
              if (angel < torrance && angel > 0)
                  result.AddPoint(point);
          }
      }
      else
      {
    
          for (int i = 1; i < points.PointCount - 1; i++)
          {
    
              var point = points.Point[i];
              var angel = CalAngel(points.Point[i - 1], point, points.Point[i + 1]);
              if (angel < torrance && angel > 0)
                  result.AddPoint(point);
          }
      }
      return result;
  }
 
 /// <summary>
 /// 计算折角
 /// </summary>
 /// <param name="p1">起点</param>
 /// <param name="p2">顶点</param>
 /// <param name="p3">终点</param>
 /// <returns></returns>
 private static double CalAngel(IPoint p1, IPoint p2, IPoint p3)
 {
    
     var angel = Math.Atan2(p1.Y - p2.Y, p1.X - p2.X) - Math.Atan2(p3.Y - p2.Y, p3.X - p2.X);
     angel = Math.Abs(angel) / Math.PI * 180;
     return angel;
 }
 //钝角直接跳过
 private static bool isSkipPoint(IPoint p1, IPoint p2, IPoint p3)
 {
    
     var va = new PointClass()
     {
    
         X = p2.X - p1.X,
         Y = p2.Y - p1.Y
     };
     var vb = new PointClass()
     {
    
         X = p3.X - p2.X,
         Y = p3.Y - p2.Y
     };
     ///向量a×向量b=x1*y2-x2*y1,
     ///大于0：向量b在向量a的逆时针
     ///=0共线
     ///小于0：向量b在向量a的顺时针
     ///读出的多边形的点均按照顺时针排序，若叉乘大于0，则凹点，内角必然大于180
     var VaCrossVb = va.X * vb.Y - vb.X * va.Y;
     if (VaCrossVb > 0)
         return true;
     else
         return false;
 }

求两线交点

1.根据两点求解一般式的系数

设两个点为 (x1, y1) , (x2, y2)，则有：
A = y2 - y1
B = x1 - x2
C = x2y1-x1y2

首先设交点坐标为 (x, y)，两线段对应直线的一般式为：
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0

那么对 1 式乘 a2，对 2 式乘 a1 得：
a2a1x + a2b1y + a2c1 = 0
a1a2x + a1b2y + a1c2 = 0
两式相减得：
y = (c1 * a2 - c2 * a1) / (a1 * b2 - a2 * b1)
同样可以推得：
x = (c2 * b1 - c1 * b2) / (a1 * b2 - a2 * b1)
其中a1b2 - a2b1 == 0时，代表两条直线重合或平行
如果(x,y)在两线段上，则(x,y)即为答案，否则交点不存在。

2.代码求解

private SGPoint GetCrossPointOfTwoLine(SGPoint p1, SGPoint p2, SGPoint p3, SGPoint p4)
{
    
    SGPoint crossPoint = new SGPoint();
    var a1 = p2.y - p1.y;
    var b1 = p1.x - p2.x;
    var c1 = p1.x * p2.y - p2.x * p1.y;
    var a2 = p4.y - p3.y;
    var b2 = p3.x - p4.x;
    var c2 = p3.x * p4.y - p4.x * p3.y;
    var det = a1 * b2 - a2 * b1;
    //平行线
    if (det == 0) return null;

    crossPoint.x = (c1 * b2 - c2 * b1) / det;
    crossPoint.y = (a1 * c2 - a2 * c1) / det;
    return crossPoint;
}

点到几何图形的距离

/// <summary>
/// 点到几何的最短距离是否在容差内
/// </summary>
/// <param name="pGeometryA"></param>
/// <param name="pGeometryB"></param>
/// <returns></returns>
public static bool IfPointToGeometryWinthTor(IPoint point, IGeometry geo, double torrance)
{
    
    var segments = geo as ISegmentCollection;
    torrance = torrance * torrance;
    for (int i = 0; i < segments.SegmentCount; i++)
    {
    
        var line = segments.Segment[i];
        var dis = GetPointLineDis(line.FromPoint, line.ToPoint, point);
        if (dis < torrance)
            return true;
    }
    return false;
}
/// <summary>
/// 求点p到线段ab的最短距离的平方,避免过多开方运算
/// </summary>
/// <param name="pa"></param>
/// <param name="pb"></param>
/// <param name="p"></param>
/// <returns></returns>
private static double GetPointLineDis(IPoint a, IPoint b, IPoint p)
{
    
    double dis = 0;
    //向量AB·向量Ap
    var abCrossAp = (b.X - a.X) * (p.X - a.X) + (b.Y - a.Y) * (p.Y - a.Y);
    //∠PAB为钝角，最短距离为PA
    if (abCrossAp <= 0)
        dis = (p.X - a.X) * (p.X - a.X) + (p.Y - a.Y) * (p.Y - a.Y);
    else
    {
    
        var ab2 = (b.X - a.X) * (b.X - a.X) + (b.Y - a.Y) * (b.Y - a.Y);
        //p的垂线相交于AB向量的延长线上，最短距离为PB
        if (abCrossAp >= ab2)
            dis = (p.X - b.X) * (p.X - b.X) + (p.Y - b.Y) * (p.Y - b.Y);
        //p的垂线相交于AB向量上（交点为D），最短距离为垂线的值
        else
        {
    
            var r = abCrossAp / ab2;
            var Dx = a.X + (b.X - a.X) * r;
            var Dy = a.Y + (b.Y - a.Y) * r;
            dis = (p.X - Dx) * (p.X - Dx) + (p.Y - Dy) * (p.Y - Dy);
        }
    }
    return dis;
}