“判断一个点是否在一个多边形里”,一开始以为是个挺难的问题,但Google了一下之后发现其实蛮简单,所用到的算法叫做“Ray-casting Algorithm”,中文应该叫“光线投射算法”,这是维基百科的描述:[维基百科]
简单地说可以这么判断:从这个点引出一根“射线”,与多边形的任意若干条边相交,累计相交的边的数目,如果是奇数,那么点就在多边形内,否则点就在多边形外。
如图,A点引一条射线,与多边形3条边相交,奇数,所以A点在多边形内,而从B点引一条射线,与多边形的2条边相交,偶数,所以B点在多边形外。
我打算把这个算法用于判断地图上所在的位置是否在一个范围之内,我先用鼠标在地图上绘制出一个多边形区域,然后再用这个方法判断一个坐标是否在这个多边形范围内,我仍然拿五角星做试验品,在高德地图上描出一个五角星:
嗯?怎么五角星居然中间没被镂空?这是怎么回事?经过研究,我发现高德地图的鼠标工具的多边形填充用的是另外一套规则,叫做“None Zero Mode”,判断一个点是否在多边形内的规则就变成了:从这个点引出一根“射线”,与多边形的任意若干条边相交,计数初始化为0,若相交处被多边形的边从左到右切过,计数+1,若相交处被多边形的边从右到左切过,计数-1,最后检查计数,如果是0,点在多边形外,如果非0,点在多边形内。回到五角星的例子,这次要注意多边形线条描绘的方向:
从C点引出一条射线,与这条射线相交的两条多边形的边均是从左向右切过,总计数是2,因此C点在多边形内。用个更形象点的方式描述就是:从C点出发,一直朝一个方向走,遇到两条单行道,都是从自己的左边切至右边的方向,计数+1,计数+1,总计数所以是2。
算法实现起来居然很简单,几行代码即可,真的是几行代码,我用的是C#,大家可以轻轻松松改成别的。
public static class RayCastingAlgorithm { public static bool IsWithin(Point pt, IList<Point> polygon, bool noneZeroMode) { int ptNum = polygon.Count(); if (ptNum < 3) { return false; } int j = ptNum - 1; bool oddNodes = false; int zeroState = 0; for (int k = 0; k < ptNum; k++) { Point ptK = polygon[k]; Point ptJ = polygon[j]; if (((ptK.Y > pt.Y) != (ptJ.Y > pt.Y)) && (pt.X < (ptJ.X - ptK.X) * (pt.Y - ptK.Y) / (ptJ.Y - ptK.Y) + ptK.X)) { oddNodes = !oddNodes; if (ptK.Y > ptJ.Y) { zeroState++; } else { zeroState--; } } j = k; } return noneZeroMode?zeroState!=0:oddNodes; } }
我用WPF写了个demo,如图:
给懒得敲打吗的同学玩玩。(源码:VS2015)
原文地址: https://www.cnblogs.com/guogangj/p/5127527.html
考虑到double值有误差,以及有时在线的边上一点就需要识别为区域内,所以改了下加了个容差值(当然没考虑算法速度的问题,只是判断点是否在线上,有其他好的思路的可以提供下啊)
算法方法:
public static bool IsWithin(Point pt, IList<Point> polygon, bool noneZeroMode, double dTol = 0) { int ptNum = polygon.Count(); if (ptNum < 3) { return false; } int j = ptNum - 1; bool oddNodes = false; int zeroState = 0; for (int k = 0; k < ptNum; k++) { Point ptK = polygon[k]; Point ptJ = polygon[j]; if (((ptK.Y > pt.Y) != (ptJ.Y > pt.Y)) && (pt.X < (ptJ.X - ptK.X) * (pt.Y - ptK.Y) / (ptJ.Y - ptK.Y) + ptK.X)) { oddNodes = !oddNodes; if (ptK.Y > ptJ.Y) { zeroState++; } else { zeroState--; } } j = k; } bool flag = noneZeroMode ? zeroState != 0 : oddNodes; if (flag || dTol.IsAlmostEqualTo(0)) return flag; //增加判断点在直线附近容差范围内划分为区域内 { int len = polygon.Count; for (int i = 0; i < len; ++i) { var p1 = polygon[i]; var p2 = i == len - 1 ? polygon[0] : polygon[i + 1]; var d = pt.GetDropPoint(p1, p2); if (d.DistanceTo(pt).IsLessThanEqualTo(0, dTol)) { if (d.IsInLine(p1, p2, dTol)) return true; } } return false; } }
用到的其他一些容差计算的方法:
public static class CommonExpand { /// <summary> /// 判断两个double是否相同 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <param name="dTol"></param> /// <returns></returns> public static bool IsAlmostEqualTo(this double src, double dst, double dTol = 0.00328) { var dis = Math.Abs(src - dst); if (dis <= dTol) return true; else return false; } /// <summary> /// 小于 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <param name="dTol"></param> /// <returns></returns> public static bool IsLessThan(this double src, double dst, double dTol = 0.00328) { if (src < dst && !src.IsAlmostEqualTo(dst, dTol)) return true; else return false; } /// <summary> /// 小于等于 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <param name="dTol"></param> /// <returns></returns> public static bool IsLessThanEqualTo(this double src, double dst, double dTol = 0.00328) { if (src < dst || src.IsAlmostEqualTo(dst, dTol)) return true; else return false; } /// <summary> /// 大于 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <param name="dTol"></param> /// <returns></returns> public static bool IsGreaterThan(this double src, double dst, double dTol = 0.00328) { if (src > dst && !src.IsAlmostEqualTo(dst, dTol)) return true; else return false; } /// <summary> /// 大于等于 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <param name="dTol"></param> /// <returns></returns> public static bool IsGreaterThanEqualTo(this double src, double dst, double dTol = 0.00328) { if (src > dst || src.IsAlmostEqualTo(dst, dTol)) return true; else return false; } /// <summary> /// 判断两个Point是否相同 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <param name="dTol"></param> /// <returns></returns> public static bool IsAlmostEqualTo(this Point src, Point dst, double dTol = 0.00328) { if (Math.Abs(src.X - dst.X) < dTol && Math.Abs(src.Y - dst.Y) < dTol) return true; else return false; } /// <summary> /// 判断两个向量是否相等 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <param name="dTol"></param> /// <returns></returns> public static bool IsAlmostEqualTo(this Vector src, Vector dst, double dTol = 0.00328) { if (src.X.IsAlmostEqualTo(dst.X) && src.Y.IsAlmostEqualTo(dst.Y)) return true; else return false; } /// <summary> /// 判断同一条直线上的点,是否在线段以内 /// </summary> /// <param name="src"></param> /// <param name="p1"></param> /// <param name="p2"></param> /// <param name="deviation"></param> /// <returns></returns> public static bool IsInLine(this Point src, Point p1, Point p2, double deviation = 0.00328) { if (src.IsAlmostEqualTo(p1, deviation) || src.IsAlmostEqualTo(p2, deviation)) return true; var vec1 = new Vector(p1.X - src.X, p1.Y - src.Y); vec1.Normalize(); var vec2 = new Vector(p2.X - src.X, p2.Y - src.Y); vec2.Normalize(); if (vec1.IsAlmostEqualTo(-vec2)) return true; else return false; } /// <summary> /// 点到直线的垂足 /// </summary> /// <param name="src"></param> /// <param name="p1"></param> /// <param name="p2"></param> /// <returns></returns> public static Point GetDropPoint(this Point src, Point p1, Point p2) { if (p1.X.IsAlmostEqualTo(p2.X)) { return new Point(p1.X, src.Y); } else if (p1.Y.IsAlmostEqualTo(p2.Y)) { return new Point(src.X, p1.Y); } else { double k = (p2.Y - p1.Y) / (p2.X - p1.X); double b1 = p1.Y - k * p1.X; double b2 = src.Y + src.X / k; double x = k * (b2 - b1) / (k * k + 1); double y = k * x + b1; return new Point(x, y); } } /// <summary> /// 点到点的距离 /// </summary> /// <param name="src"></param> /// <param name="dst"></param> /// <returns></returns> public static double DistanceTo(this Point src, Point dst) { return Math.Sqrt(Math.Pow(src.X - dst.X, 2) + Math.Pow(src.Y - dst.Y, 2)); } }