基于opencv计算二维直线交点

0.根据a（x1，y1），b（x2，y2）求直线方程:

根据两点式(x-x1)/(x2-x1)=(y-y1)/(y2-y1)，可知
A=y2-y1 ，B=x1-x2 ，
C=(y1-y2)*x1 + (x2-x1)*y1

//已知
double x1=10,y1=10;
double x2=15,y2=20;
//求直线方程
double A=y2-y1;
double B=x1-x2;
double C=(y1-y2)*x1 + (x2-x1)*y1

1.两条直线叉乘，计算交点

二维向量叉乘公式
$$a（x_1，y_1），b（x_2，y_2），则a\times b＝（x_1{y}_2－x_2{y}_1）$$
假设两直线方程为
$$\begin{gathered} F0(x)=a_0\ast x+b_0\ast y+c_0=0 \\ F1(x)=a_1\ast x+b_1\ast y+c_1=0 \end{gathered}$$
那么两条直线的交点应该满足
$$a_0\ast x+b_0\ast y+c_0=a_1\ast x+b_1\ast y+c_1$$
由此可推出
$$\begin{gathered} x=(b_0\ast c_1–b_1\ast c_0)/D \\ y=(a_1\ast c_0–a_0\ast c_1)/D \\ D=a_0\ast b_1–a_1\ast b_0 \end{gathered}$$ (D为0时，表示两直线平行)
实际上就是叉积应用

//得到直线方程
double A1,B1,C1;
double A2,B2,C2;
double D=(A1*B2-A2*B1);
//计算交点
cv::Point crossPoint;
if(D!=0)
{
crossPoint.x=(B1*C2-B2*C1)/D;
crossPoint.Y=(A2*C1-A1*C2)/D;
}

2. 根据两条直线的方向向量，以及直线上的两点计算交点

double deltaX_X1 = endPoint.x - orgin.x;
double deltaY_X1 = endPoint.y -orgin.y;
double deltaX_X2 = endPoint.x - orgin.x;
double deltaY_X2 = endPoint.y - orgin.y;
double lineLengthX1 = sqrt(deltaX_X1*deltaX_X1+deltaY_X1*deltaY_X1);
double lineLengthX2 = sqrt(deltaX_X2*deltaX_X2+deltaY_X2*deltaY_X2);
X1DirectionVector.push_back(deltaX_X1/lineLengthX1);
X1DirectionVector.push_back(deltaY_X1/lineLengthX1);
X2DirectionVector.push_back(deltaX_X2/lineLengthX2);
X2DirectionVector.push_back(deltaY_X2/lineLengthX2);
//交点计算函数
cv::Point intersectionPointCalculate(std::vector<double> X1DirectionVec,cv::Point point1, std::vector<double> X2DirectionVec,cv::Point point2)
{
cv::Point crossPoint(9999,9999);
if((X1DirectionVec.at(0)!=0) && X2DirectionVec.at(0)!=0)
{
double ka = X1DirectionVec.at(1)/X1DirectionVec.at(0);
double kb = X2DirectionVec.at(1)/X2DirectionVec.at(0);
double x = ((-1)*kb*double(point2.x)+double(point2.y)+ka*double(point1.x)-double(point1.y)) / (ka - kb);
crossPoint.x = int(x);
crossPoint.y = ka*(x-double(point1.x))+double(point1.y);
//std::cout<<"point1: "<<point1<<" point2: "<<point2<<" ka: "<<ka<<" kb "<<kb<<" crossPoint: "<<crossPoint<<std::endl;
}
if((X1DirectionVec.at(0)==0) && X2DirectionVec.at(0)==0)
{
//std::cout<<"tensioner coordinate calculate error!"<<std::endl;
}
if((X1DirectionVec.at(0)==0) && X2DirectionVec.at(0)!=0)
{
double kb = X2DirectionVec.at(1)/X2DirectionVec.at(0);
crossPoint.x = point1.x;
crossPoint.y = kb*(crossPoint.x-point2.x)+point2.y;
}
if((X1DirectionVec.at(0)!=0) && X2DirectionVec.at(0)==0)
{
double ka = X1DirectionVec.at(1)/X1DirectionVec.at(0);
crossPoint.x = point2.x;
crossPoint.y = ka*(crossPoint.x-point1.x)+point1.y;
}
return crossPoint;
}

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