/// <summary> /// 判断点与线的位置 /// </summary> /// <param name="P">P点</param> /// <param name="Q">Q点</param> /// <param name="R">R点</param> /// <returns>返回偏转方向</returns> /// <remarks> /// 假设L线的是 P->Q PQ为线的2个顶点 /// 返回值 大于 0 , 则PQ在R点拐向右侧后得到QR,等同于点R在PQ线段的右侧 /// 返回值 小于 0 , 则PQ在R点拐向左侧后得到QR,等同于点R在PQ线段的左侧 /// 返回值 等于 0 , 则P,Q,R三点共线。 /// </remarks> public static Int32 Position(PointI P, PointI Q, PointI R) { //PointI RP = PointAlgorithm.Substract(R, P);//R-P //PointI QP = PointAlgorithm.Substract(Q, P);//Q-P //return PointAlgorithm.Multiple(RP, QP); return(PointAlgorithm.Multiple(P, Q, R)); }
/// <summary> /// 计算(R-P)和(Q-P)的叉积 /// </summary> /// <param name="P">点P</param> /// <param name="Q">点Q</param> /// <param name="R">点R</param> /// <returns>返回叉积值</returns> /// <remarks> /// 返回值 大于0 R在矢量PQ的逆时针方向 /// 返回值 等于0 R,P,Q 三点共线 /// 返回值 小于0 R在矢量PQ的顺时针方向 /// </remarks> public static Int32 Multiple(PointI P, PointI Q, PointI R) { PointI RP = PointAlgorithm.Substract(R, P); //R-P PointI QP = PointAlgorithm.Substract(Q, P); //Q-P return(PointAlgorithm.Multiple(RP, QP)); }
/// <summary> /// 判断点与线的位置 /// </summary> /// <param name="P">P点</param> /// <param name="Q">Q点</param> /// <param name="R">R点</param> /// <returns>返回偏转方向</returns> /// <remarks> /// 假设L线的是 P->Q PQ为线的2个顶点 /// 返回值 大于 0 , 则PQ在R点拐向右侧后得到QR,等同于点R在PQ线段的右侧 /// 返回值 小于 0 , 则PQ在R点拐向左侧后得到QR,等同于点R在PQ线段的左侧 /// 返回值 等于 0 , 则P,Q,R三点共线。 /// </remarks> public static Double Position(PointD P, PointD Q, PointD R) { //PointD RP = PointAlgorithm.Substract(R, P);//R-P //PointD QP = PointAlgorithm.Substract(Q, P);//Q-P //return PointAlgorithm.Multiple(RP, QP); return(PointAlgorithm.Multiple(P, Q, R)); }
/// <summary> /// 计算(R-P)和(Q-P)的叉积 /// </summary> /// <param name="P">点P</param> /// <param name="Q">点Q</param> /// <param name="R">点R</param> /// <returns>返回叉积值</returns> /// <remarks> /// 返回值 大于0 R在矢量PQ的逆时针方向 /// 返回值 等于0 R,P,Q 三点共线 /// 返回值 小于0 R在矢量PQ的顺时针方向 /// </remarks> public static Double Multiple(PointD P, PointD Q, PointD R) { PointD RP = PointAlgorithm.Substract(R, P); //R-P PointD QP = PointAlgorithm.Substract(Q, P); //Q-P return(PointAlgorithm.Multiple(RP, QP)); }
/// <summary> /// 判断点P是否在圆内 /// </summary> /// <param name="C">圆C</param> /// <param name="P">点P</param> /// <returns>如果在圆内返回True,否则返回False。</returns> public static Boolean InCircle(CircleI C, PointI P) { //判断点是否在圆内: //计算圆心到该点的距离,如果小于等于半径则该点在圆内。 Double D = PointAlgorithm.Distance(P, C.Center); return((D < C.Radius) || DoubleAlgorithm.Equals(D, C.Radius)); }
/// <summary> /// 计算多边形PG面积 /// </summary> /// <param name="PG">多边形PG</param> /// <returns>返回面积。</returns> public static Double Area(PolygonI PG) { //formula (1/2) *( (Xi*Yi+1 -Xi+1*Yi) +...) Double result = 0; for (Int32 i = 0; i < PG.Vertex.Count; ++i) { result += PointAlgorithm.Multiple(PG.Vertex[i % PG.Vertex.Count], PG.Vertex[(i + 1) % PG.Vertex.Count]); } return(System.Math.Abs(0.5 * result)); }
/// <summary> /// 判断折线的偏转方向/线段拐向 /// </summary> /// <param name="P">P点</param> /// <param name="Q">Q点</param> /// <param name="R">R点</param> /// <returns>返回偏转方向</returns> /// <remarks> /// 返回值 大于 0 , 则PQ在R点拐向右侧后得到QR,等同于点R在PQ线段的右侧 /// 返回值 小于 0 , 则PQ在R点拐向左侧后得到QR,等同于点R在PQ线段的左侧 /// 返回值 等于 0 , 则P,Q,R三点共线。 /// </remarks> public static Int32 DeflectingDirection(PointI P, PointI Q, PointI R) { //formular //折线段的拐向判断方法可以直接由矢量叉积的性质推出。对于有公共端点的线段PQ和QR,通过计算(R - P) * (Q - P)的符号便可以确定折线段的拐向 //基本算法是(R-P)计算相对于P点的R点坐标,(Q-P)计算相对于P点的Q点坐标。 //(R-P) * (Q-P)的计算结果是计算以P为相对原点,点R与点Q的顺时针,逆时针方向。 PointI RP = PointAlgorithm.Substract(R, P); //R-P PointI QP = PointAlgorithm.Substract(Q, P); //Q-P return(PointAlgorithm.Multiple(RP, QP)); }
/// <summary> /// 判断折线PL是否在圆内 /// </summary> /// <param name="C">圆C</param> /// <param name="R">矩形R</param> /// <returns>如果在圆内返回True,否则返回False。</returns> public static Boolean InCircle(CircleI C, RectangleI R) { if (PointAlgorithm.Distance(new PointI(R.Left, R.Top), C.Center) > C.Radius) { return(false); } if (PointAlgorithm.Distance(new PointI(R.Right, R.Bottom), C.Center) > C.Radius) { return(false); } return(true); }
/// <summary> /// 获取线段L与圆C的交点集合 /// </summary> /// <param name="L">线段L</param> /// <param name="C">圆C</param> /// <returns>返回交点集合.</returns> public static PointD[] Intersection(LineI L, CircleI C) { List <PointD> result = new List <PointD>(); Int32? has = HasIntersection(L, C); if (has == 0 || has == null) { return(result.ToArray()); } //Points P (x,y) on a line defined by two points P1 (x1,y1,z1) and P2 (x2,y2,z2) is described by //P = P1 + u (P2 - P1) //or in each coordinate //x = x1 + u (x2 - x1) //y = y1 + u (y2 - y1) //z = z1 + u (z2 - z1) //A sphere centered at P3 (x3,y3,z3) with radius r is described by //(x - x3)2 + (y - y3)2 + (z - z3)2 = r2 //Substituting the equation of the line into the sphere gives a quadratic equation of the form //a u2 + b u + c = 0 //where: //a = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2 //b = 2[ (x2 - x1) (x1 - x3) + (y2 - y1) (y1 - y3) + (z2 - z1) (z1 - z3) ] //c = x32 + y32 + z32 + x12 + y12 + z12 - 2[x3 x1 + y3 y1 + z3 z1] - r2 //The solutions to this quadratic are described by PointD PD = PointAlgorithm.Substract(L.Starting, L.End); Double a = PD.X * PD.X + PD.Y * PD.Y; Double b = 2 * ((L.End.X - L.Starting.X) * (L.Starting.X - C.Center.X) + (L.End.Y - L.Starting.Y) * (L.Starting.Y - C.Center.Y)); Double c = C.Center.X * C.Center.X + C.Center.Y * C.Center.Y + L.Starting.X * L.Starting.X + L.Starting.Y * L.Starting.Y - 2 * (C.Center.X * L.Starting.X + C.Center.Y * L.Starting.Y) - C.Radius * C.Radius; Double u1 = ((-1) * b + System.Math.Sqrt(b * b - 4 * a * c)) / (2 * a); Double u2 = ((-1) * b - System.Math.Sqrt(b * b - 4 * a * c)) / (2 * a); //交点 PointD P1 = new PointD(L.Starting.X + u1 * (L.End.X - L.Starting.X), L.Starting.Y + u1 * (L.End.Y - L.Starting.Y)); PointD P2 = new PointD(L.Starting.X + u2 * (L.End.X - L.Starting.X), L.Starting.Y + u2 * (L.End.Y - L.Starting.Y)); if (LineAlgorithm.OnLine(L, P1) == true) { result.Add(P1); } if (LineAlgorithm.OnLine(L, P2) == true && P1.Equals(P2) == false) { result.Add(P2); } return(result.ToArray()); }
/// <summary> /// 判断折线的偏转方向/线段拐向 /// </summary> /// <param name="L">L线段</param> /// <param name="R">R点</param> /// <returns>返回偏转方向</returns> /// <remarks> /// 假设L线的是 P->Q PQ为线的2个顶点 /// 返回值 大于 0 , 则PQ在R点拐向右侧后得到QR,等同于点R在PQ线段的右侧 /// 返回值 小于 0 , 则PQ在R点拐向左侧后得到QR,等同于点R在PQ线段的左侧 /// 返回值 等于 0 , 则P,Q,R三点共线。 /// </remarks> public static Int32 Position(LineI L, PointI R) { //formular //折线段的拐向判断方法可以直接由矢量叉积的性质推出。对于有公共端点的线段PQ和QR,通过计算(R - P) * (Q - P)的符号便可以确定折线段的拐向 //基本算法是(R-P)计算相对于P点的R点坐标,(Q-P)计算相对于P点的Q点坐标。 //(R-P) * (Q-P)的计算结果是计算以P为相对原点,点R与点Q的顺时针,逆时针方向。 //PointI P = L.Starting; //PointI Q = L.End; //PointI RP = PointAlgorithm.Substract(R, P);//R-P //PointI QP = PointAlgorithm.Substract(Q, P);//Q-P //return PointAlgorithm.Multiple(RP, QP); return(PointAlgorithm.Multiple(L.Starting, L.End, R)); }
/// <summary> /// 判断线段L是否在圆内 /// </summary> /// <param name="C">圆C</param> /// <param name="L">线段L</param> /// <returns>如果在圆内返回True,否则返回False。</returns> public static Boolean InCircle(CircleD C, LineD L) { //判断点是否在圆内: //计算圆心到该点的距离,如果小于等于半径则该点在圆内。 if (PointAlgorithm.Distance(L.Starting, C.Center) > C.Radius) { return(false); } if (PointAlgorithm.Distance(L.End, C.Center) > C.Radius) { return(false); } return(true); }
/// <summary> /// 判断圆C是否在多边形PG内 /// </summary> /// <param name="PG">PG多边形</param> /// <param name="C">圆C</param> /// <returns>如果圆C在区域内返回True,否则返回False.</returns> public static Boolean InPolygon(PolygonI PG, CircleI C) { //如果圆心不在多边形内则返回不在多边形内 if (false == InPolygon(PG, C.Center)) { return(false); } Double D = PointAlgorithm.ClosestDistance(C.Center, PG); if (D > C.Radius || DoubleAlgorithm.Equals(D, C.Radius)) { return(true); } return(false); }
/// <summary> /// 判断折线PL是否在圆内 /// </summary> /// <param name="C">圆C</param> /// <param name="PL">折线PL</param> /// <returns>如果在圆内返回True,否则返回False。</returns> public static Boolean InCircle(CircleI C, PolylineI PL) { if (PL.Points == null) { return(false); } for (Int32 i = 0; i < PL.Points.Count; ++i) { if (PointAlgorithm.Distance(PL.Points[i], C.Center) > C.Radius) { return(false); } } return(true); }
/// <summary> /// 判断多边形PG是否在圆内 /// </summary> /// <param name="C">圆C</param> /// <param name="PG">多边形PG</param> /// <returns>如果在圆内返回True,否则返回False。</returns> public static Boolean InCircle(CircleI C, PolygonI PG) { if (PG.Vertex == null) { return(false); } for (Int32 i = 0; i < PG.Vertex.Count; ++i) { if (PointAlgorithm.Distance(PG.Vertex[i], C.Center) > C.Radius) { return(false); } } return(true); }
/// <summary> /// 根据点P,Q,R三点确定一个圆,注意三点不能共线 /// </summary> /// <param name="P">点P</param> /// <param name="Q">点Q</param> /// <param name="R">点R</param> /// <returns>返回圆,如果圆不存在则返回null.</returns> public static CircleD?CreateCircle(PointI P, PointI Q, PointI R) { if (DoubleAlgorithm.Equals(LineAlgorithm.Position(P, Q, R), 0)) { return(null); //三点共线无法确定圆 } //formula //(x-a)^2+(y-b)^2=r^2 //f1:(x1-a)^2 + (y1-b)^2 = r^2 //f2:(x2-a)^2 + (y2-b)^2 = r^2 //f3:(x3-a)^2 + (y3-b)^2 = r^2 //f1=f2: x1^2-2ax1+y1^2-2by1= x2^2-2ax2+y2^2-2by2; //a=(((X(1)^2-X(2)^2+Y(1)^2-Y(2)^2)*(Y(2)-Y(3)))-((X(2)^2-X(3)^2+Y(2)^2-Y(3)^2)*(Y(1)-Y(2))))/(2*(X(1)-X(2))*(Y(2)-Y(3))-2*(X(2)-X(3))*(Y(1)-Y(2))) //b=(((X(1)^2-X(2)^2+Y(1)^2-Y(2)^2)*(X(2)-X(3)))-((X(2)^2-X(3)^2+Y(2)^2-Y(3)^2)*(X(1)-X(2))))/(2*(Y(1)-Y(2))*(X(2)-X(3))-2*(Y(2)-Y(3))*(X(1)-X(2))) Double a = (Double)(((P.X * P.X - Q.X * Q.X + P.Y * P.Y - Q.Y * Q.Y) * (Q.Y - R.Y)) - ((Q.X * Q.X - R.X * R.X + Q.Y * Q.Y - R.Y * R.Y) * (P.Y - Q.Y))) / (Double)(2 * (P.X - Q.X) * (Q.Y - R.Y) - 2 * (Q.X - R.X) * (P.Y - Q.Y)); Double b = (Double)(((P.X * P.X - Q.X * Q.X + P.Y * P.Y - Q.Y * Q.Y) * (Q.X - R.X)) - ((Q.X * Q.X - R.X * R.X + Q.Y * Q.Y - R.Y * R.Y) * (P.X - Q.X))) / (Double)(2 * (P.Y - Q.Y) * (Q.X - R.X) - 2 * (Q.Y - R.Y) * (P.X - Q.X)); Double r = PointAlgorithm.Distance(P, new PointD(a, b)); return(new CircleD(a, b, r)); }
/// <summary> /// 判断线段L与圆C的交点个数 /// </summary> /// <param name="L">线段L</param> /// <param name="C">圆形C</param> /// <returns>相交返回交点数目,否则返回0</returns> public static Int32?HasIntersection(LineI L, CircleI C) { Int32 count = 0; //如果和圆C有交点首先是L到圆心的距离小于或等于C的半径 if (DoubleAlgorithm.Equals(PointAlgorithm.ClosestDistance(C.Center, L), C.Radius)) { return(1); } else if (PointAlgorithm.ClosestDistance(C.Center, L) > C.Radius) { return(0); } if (PointAlgorithm.Distance(C.Center, L.Starting) >= C.Radius) { ++count; } if (PointAlgorithm.Distance(C.Center, L.End) >= C.Radius) { ++count; } return(count); }
/// <summary> /// 线段L是否在多边形区域内 /// </summary> /// <param name="PG">多边形PG</param> /// <param name="L">线段L</param> /// <returns>如果线段L在区域内返回True,否则返回False.</returns> public static Boolean InPolygon(PolygonI PG, LineI L) { //if 线端PQ的端点不都在多边形内 // then return false; //点集pointSet初始化为空; //for 多边形的每条边s // do if 线段的某个端点在s上 // then 将该端点加入pointSet; // else if s的某个端点在线段PQ上 // then 将该端点加入pointSet; // else if s和线段PQ相交 // 这时候已经可以肯定是内交了 // then return false; //将pointSet中的点按照X-Y坐标排序; //for pointSet中每两个相邻点 pointSet[i] , pointSet[ i+1] // do if pointSet[i] , pointSet[ i+1] 的中点不在多边形中 // then return false; //return true; List <PointI> PointList = new List <PointI>(); if (InPolygon(PG, L.Starting) == false) { return(false); } foreach (LineI S in PG) { if (LineAlgorithm.OnLine(S, L.Starting) == true) { PointList.Add(L.Starting); } else if (LineAlgorithm.OnLine(S, L.End) == true) { PointList.Add(L.End); } else if (LineAlgorithm.OnLine(L, S.Starting) == true) { PointList.Add(S.Starting); } else if (LineAlgorithm.OnLine(L, S.End) == true) { PointList.Add(S.End); } else if (LineAlgorithm.HasIntersection(L, S) > 0) { return(false); } } PointI[] OrderedPointList = PointList.ToArray(); for (Int32 i = 0; i < (OrderedPointList.Length - 1); ++i) { for (Int32 j = 0; j < (OrderedPointList.Length - i - 1); j++) { MinMax <PointI> MM = new MinMax <PointI>(OrderedPointList[j], OrderedPointList[j + 1]); OrderedPointList[j] = MM.Min; OrderedPointList[j + 1] = MM.Max; } } for (Int32 i = 0; i < (OrderedPointList.Length - 1); ++i) { if (false == InPolygon(PG, PointAlgorithm.MidPoint(OrderedPointList[i], OrderedPointList[i + 1]))) { return(false); } } return(true); }
/// <summary> /// 计算点到圆的距离 /// </summary> /// <param name="C">圆C</param> /// <param name="P">点P</param> /// <returns>返回点到圆周的距离。</returns> /// <remarks> /// 返回值小于0 表示点在圆内。 /// 返回值等于0 表示点在圆周上。 /// 返回值大于0 表示点在圆外。 /// </remarks> public static Double Distance(CircleI C, PointI P) { return(PointAlgorithm.Distance(C.Center, P) - C.Radius); }
/// <summary> /// 计算线段L与X轴的夹角 /// </summary> /// <param name="L">线段L</param> /// <returns>返回夹角弧度</returns> public static Double IncludedAngle(LineD L) { return(PointAlgorithm.Angle(L.Starting, L.End, new PointD(10000000, L.Starting.Y))); }
/// <summary> /// 判断圆C2是否在圆C1内 /// </summary> /// <param name="C1">圆C1</param> /// <param name="C2">圆C2</param> /// <returns>如果在圆内返回True,否则返回False。</returns> public static Boolean InCircle(CircleI C1, CircleI C2) { //formula //C2的中心点到C1中心点的距离 加上C2的半径小于C1的半径 return(((PointAlgorithm.Distance(C1.Center, C2.Center) + C2.Radius) > C1.Radius) ? false : true); }
/// <summary> /// 计算线L到圆C的距离 /// </summary> /// <param name="C">圆C</param> /// <param name="L">线L</param> /// <returns>返回线到圆周的距离。</returns> /// <remarks> /// 返回值小于0 表示线在圆内或与圆周相交。 /// 返回值等于0 表示线在圆周上与圆周相切。 /// 返回值大于0 表示线在圆外与圆周没有交点。 /// </remarks> public static Double Distance(CircleI C, LineI L) { return(PointAlgorithm.Distance(C.Center, L) - C.Radius); }