一、问题

已知两条直线 $l_1(x_1,y_1,x_2,y_2)$ 和 $l_2(x_3,y_3,x_4,y_4)$ ，现希望判断 $l_1$ 与 $l_2$ 间是否相交。若相交，计算出两条直线的交点和夹角。

二、分析

1、直线方程

$\frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1} \Rightarrow \begin{cases} ax+by+c=0 \\a=-(y_2-y_1) \\ b = x_2-x_1 \\ c = (y_2-y_1)x_1 - (x_2-x_1) y_1 \\ k = \frac{-a}{b}\end{cases}$

$l_1$ 的直线方程：
$a_1x+b_1y+c_1=0$

$l_2$ 的直线方程：
$a_2x+b_2y+c_2=0$

提示： $a$ 和 $b$ 不能同时为0。若 $a$ 和 $b$ 同时为0，起点和终点重合，该直线实际上是一个点。

2、判断相交

当 $l_1$ 垂直于 $x$ 轴， $l_2$ 倾斜于 $x$ 轴时， $l_1$ 与 $l_2$ 相交：

    b1==0 && b2!=0

当 $l_1$ 倾斜于 $x$ 轴， $l_2$ 垂直于 $x$ 轴时， $l_1$ 与 $l_2$ 相交：

    b1!=0 && b2==0

当 $l_1$ 与 $l_2$ 都倾斜于 $x$ 轴，且斜率不同时， $l_1$ 与 $l_2$ 相交：

    b1!=0 && b2!=0 && a1/b1!=a2/b2

3、计算交点

$\begin{cases} a_1x+b_1y+c_1=0 \\ a_2x+b_2y+c_2=0 \end{cases} \Rightarrow \begin{cases} x= \frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1} \\ y=\frac{a_1c_2-a_2c_1}{a_2b_1-a_1b_2} \end{cases}$

4、计算夹角

交点 $p_0(x_0,y_0)$ 、 $l_1$ 的终点 $p_2(x_2,y_2)$ 和 $l_2$ 的终点 $p_4(x_4,y_4)$ 组成一个三角形，则其各边边长为：

$a = \sqrt{(x_4-x_2)^2+(y_4-y_2)^2}$

$b = \sqrt{(x_4-x_0)^2+(y_4-y_0)^2}$

$c = \sqrt{(x_2-x_0)^2+(y_2-y_0)^2}$

由余弦定理得：
$a^2=b^2+c^2-2bc \cos \theta \Rightarrow \cos \theta = \frac{b^2+c^2-a^2}{2bc} \Rightarrow \theta = \arccos (\frac{b^2+c^2-a^2}{2bc})$

三、实现

#include <opencv2/opencv.hpp>
#include <iostream>
#include <string>
#include <cmath>
using namespace std;
using namespace cv;

Vec4d g_line1(200, 300, 600, 300), g_line2(400, 500, 400, 100); // 垂直的两条线

// 判断两条线是否相交，若相交则求出交点和夹角
Vec4d lines_intersection(const Vec4d l1, const Vec4d l2)
{
    double x1 = l1[0], y1 = l1[1], x2 = l1[2], y2 = l1[3];
    double a1 = -(y2 - y1), b1 = x2 - x1, c1 = (y2 - y1) * x1 - (x2 - x1) * y1; // 一般式：a1x+b1y1+c1=0
    double x3 = l2[0], y3 = l2[1], x4 = l2[2], y4 = l2[3];
    double a2 = -(y4 - y3), b2 = x4 - x3, c2 = (y4 - y3) * x3 - (x4 - x3) * y3; // 一般式：a2x+b2y1+c2=0
    bool r = false;                                                             // 判断结果
    double x0 = 0, y0 = 0;                                                      // 交点
    double angle = 0;                                                           // 夹角
    // 判断相交
    if (b1 == 0 && b2 != 0) // l1垂直于x轴，l2倾斜于x轴
        r = true;
    else if (b1 != 0 && b2 == 0) // l1倾斜于x轴，l2垂直于x轴
        r = true;
    else if (b1 != 0 && b2 != 0 && a1 / b1 != a2 / b2)
        r = true;
    if (r)
    {
        //计算交点
        x0 = (b1 * c2 - b2 * c1) / (a1 * b2 - a2 * b1);
        y0 = (a1 * c2 - a2 * c1) / (a2 * b1 - a1 * b2);
        // 计算夹角
        double a = sqrt(pow(x4 - x2, 2) + pow(y4 - y2, 2));
        double b = sqrt(pow(x4 - x0, 2) + pow(y4 - y0, 2));
        double c = sqrt(pow(x2 - x0, 2) + pow(y2 - y0, 2));
        angle = acos((b * b + c * c - a * a) / (2 * b * c)) * 180 / CV_PI;
    }
    return Vec4d(r, x0, y0, angle);
}

// 画夹角
void draw_angle(Mat img, const Point2d p0, const Point2d p1, const Point2d p2, const double radius, const Scalar color, const int thickness)
{
    // 计算直线的角度
    double angle1 = atan2(-(p1.y - p0.y), (p1.x - p0.x)) * 180 / CV_PI;
    double angle2 = atan2(-(p2.y - p0.y), (p2.x - p0.x)) * 180 / CV_PI;
    // 计算主轴的角度
    double angle = angle1 <= 0 ? -angle1 : 360 - angle1;
    // 计算圆弧的结束角度
    double end_angle = (angle2 < angle1) ? (angle1 - angle2) : (360 - (angle2 - angle1));
    if (end_angle > 180)
    {
        angle = angle2 <= 0 ? -angle2 : 360 - angle2;
        end_angle = 360 - end_angle;
    }
    // 画圆弧
    ellipse(img, p0, Size(radius, radius), angle, 0, end_angle, color, thickness);
}

// 画箭头
void draw_arrow(Mat img, const Point2d p1, const Point2d p2, const double angle, const double length, const Scalar color, const int thickness)
{
    double l1 = length * cos(angle * CV_PI / 180), l2 = length * sin(angle * CV_PI / 180);
    Point2d p3(0, 0), p4(0, 0);
    int i = (p2.x > p1.x) ? 1 : -1; // i,j代表p2、p3、p4相对于p0的正负
    int j = (p2.y > p1.y) ? 1 : -1;
    double a1 = abs(atan((p2.y - p1.y) / (p2.x - p1.x))); // 直线p1p2相对于x轴的角度，取正值
    double w1 = l1 * cos(a1), h1 = l1 * sin(a1);          // 用于计算p2相对于p0的宽高
    Point2d p0(p2.x - w1 * i, p2.y - h1 * j);
    double a2 = 90 * CV_PI / 180 - a1;           // 直线p3p4相对于x轴的角度
    double w2 = l2 * cos(a2), h2 = l2 * sin(a2); // 用于计算p3和p4相对于p0的宽高
    p3 = Point2d(p0.x - w2 * i, p0.y + h2 * j);
    p4 = Point2d(p0.x + w2 * i, p0.y - h2 * j);
    line(img, p2, p3, color, 2); //画箭头
    line(img, p2, p4, color, 2);
}

// 画虚线
void draw_dotted_line(Mat img, const Point2d p1, const Point2d p2, const Scalar color, const int thickness)
{
    double n = 15; // 小线段的长度
    double w = p2.x - p1.x, h = p2.y - p1.y;
    double l = sqrtl(w * w + h * h);
    // 矫正小线段长度，使小线段个数为奇数
    int m = l / n;
    m = m % 2 ? m : m + 1;
    n = l / m;

    circle(img, p1, 1, color, thickness); // 画起点
    circle(img, p2, 1, color, thickness); // 画终点
    // 画中间的小线段
    if (p1.y == p2.y) //水平线：y = m
    {
        double x1 = min(p1.x, p2.x);
        double x2 = max(p1.x, p2.x);
        for (double x = x1, n1 = 2 * n; x < x2; x = x + n1)
            line(img, Point2d(x, p1.y), Point2d(x + n, p1.y), color, thickness);
    }
    else if (p1.x == p2.x) //垂直线, x = m
    {
        double y1 = min(p1.y, p2.y);
        double y2 = max(p1.y, p2.y);
        for (double y = y1, n1 = 2 * n; y < y2; y = y + n1)
            line(img, Point2d(p1.x, y), Point2d(p1.x, y + n), color, thickness);
    }
    else
    {
        // 直线方程的两点式：(y-y1)/(y2-y1)=(x-x1)/(x2-x1) -> y = (y2-y1)*(x-x1)/(x2-x1)+y1
        double n1 = n * abs(w) / l;
        double k = h / w;
        double x1 = min(p1.x, p2.x);
        double x2 = max(p1.x, p2.x);
        for (double x = x1, n2 = 2 * n1; x < x2; x = x + n2)
        {
            Point p3 = Point2d(x, k * (x - p1.x) + p1.y);
            Point p4 = Point2d(x + n1, k * (x + n1 - p1.x) + p1.y);
            line(img, p3, p4, color, thickness);
        }
    }
}

// 画延长线
void draw_extension_line(Mat img, const Vec4d l, Scalar color)
{
    double x1 = l[0], y1 = l[1], x2 = l[2], y2 = l[3];
    double a = -(y2 - y1), b = x2 - x1, c = (y2 - y1) * x1 - (x2 - x1) * y1;
    Point2d p1(0, 0), p2(0, 0);
    if (b != 0)
    {
        p1 = Point2d(0, -c / b);
        p2 = Point2d(img.cols, ((-a * img.cols - c) / b));
    }
    else
    {
        p1 = Point2d(-c / a, 0);
        p2 = Point2d(-c / a, img.rows);
    }
    draw_dotted_line(img, p1, p2, color, 1);
}

// 画图
void draw(Mat img, const Vec4d l1, const Vec4d l2)
{
    Vec4d v = lines_intersection(l1, l2); // 判断是否相交，并计算交点和夹角

    line(img, Point2d(l1[0], l1[1]), Point2d(l1[2], l1[3]), Scalar(255, 0, 0), 2);               // 画蓝线
    draw_arrow(img, Point2d(l1[0], l1[1]), Point2d(l1[2], l1[3]), 20, 20, Scalar(255, 0, 0), 2); // 画箭头

    line(img, Point2d(l2[0], l2[1]), Point2d(l2[2], l2[3]), Scalar(0, 255, 0), 2);               // 画绿线
    draw_arrow(img, Point2d(l2[0], l2[1]), Point2d(l2[2], l2[3]), 20, 20, Scalar(0, 255, 0), 2); // 画箭头

    draw_extension_line(img, l1, Scalar(255, 0, 0)); // 画延长线
    draw_extension_line(img, l2, Scalar(0, 255, 0)); // 画延长线

    draw_angle(img, Point2d(v[1], v[2]), Point2d(l1[2], l1[3]), Point2d(l2[2], l2[3]), 15, Scalar(0, 0, 255), 1); // 画夹角

    if (v[0])
    {
        string s = "(" + to_string(v[1]) + ", " + to_string(v[2]) + ") " + to_string(v[3]);
        putText(img, s, Point2d(10, 25), cv::FONT_HERSHEY_COMPLEX, 0.5, Scalar(0, 255, 0));
        circle(img, Point2d(v[1], v[2]), 2, Scalar(0, 0, 255), 2); // 画交点
    }
    else
    {
        putText(img, "no", Point2d(10, 25), cv::FONT_HERSHEY_COMPLEX, 0.5, Scalar(0, 0, 255));
    }
}

// 确定鼠标左键点击在两条直线的那个点上
Vec6d define_area(const Vec4d l1, const Vec4d l2, const Point2d p)
{
    Vec6d v(-1, 0, 0, 0, 0, 0);
    double w = 20, h = 20;
    double x1 = l1[0], y1 = l1[1], x2 = l1[2], y2 = l1[3];
    double x3 = l2[0], y3 = l2[1], x4 = l2[2], y4 = l2[3];
    Rect r0(x1 - w, y1 - h, 2 * w, 2 * h);    // l1的起点
    Point2d p1((x2 + x1) / 2, (y1 + y2) / 2); // l1的中间点
    Rect r1(p1.x - w, p1.y - h, 2 * w, 2 * h);
    Rect r2(x2 - w, y2 - h, 2 * w, 2 * h); // l1的终点

    Rect r3(x3 - w, y3 - h, 2 * w, 2 * h);    // l2的起点
    Point2d p2((x3 + x4) / 2, (y3 + y4) / 2); // l2的中间点
    Rect r4(p2.x - w, p2.y - h, 2 * w, 2 * h);
    Rect r5(x4 - w, y4 - h, 2 * w, 2 * h); // l2的终点

    if (r0.contains(p)) // 判断点是否在矩形中
    {
        v = Vec6d(0, x1, y1, 0, 0, 0);
    }
    else if (r1.contains(p))
    {
        v = Vec6d(1, x1, y1, x2, y2, 0);
    }
    else if (r2.contains(p))
    {
        v = Vec6d(2, x2, y2, 0, 0, 0);
    }
    else if (r3.contains(p))
    {
        v = Vec6d(3, x3, y3, 0, 0, 0);
    }
    else if (r4.contains(p))
    {
        v = Vec6d(4, x3, y3, x4, y4, 0);
    }
    else if (r5.contains(p))
    {
        v = Vec6d(5, x4, y4, 0, 0, 0);
    }
    return v;
}

// 根据鼠标移动相应的修改直线的起点和终点
void modify_line(Vec4d &l1, Vec4d &l2, const Vec6d area, const double w, const double h)
{
    if (area[0] == 0)
    {
        l1[0] = area[1] + w;
        l1[1] = area[2] + h;
    }
    else if (area[0] == 1)
    {
        l1[0] = area[1] + w;
        l1[1] = area[2] + h;
        l1[2] = area[3] + w;
        l1[3] = area[4] + h;
    }
    else if (area[0] == 2)
    {
        l1[2] = area[1] + w;
        l1[3] = area[2] + h;
    }
    else if (area[0] == 3)
    {
        l2[0] = area[1] + w;
        l2[1] = area[2] + h;
    }
    else if (area[0] == 4)
    {
        l2[0] = area[1] + w;
        l2[1] = area[2] + h;
        l2[2] = area[3] + w;
        l2[3] = area[4] + h;
    }
    else if (area[0] == 5)
    {
        l2[2] = area[1] + w;
        l2[3] = area[2] + h;
    }
}

// 鼠标回调函数
void mouse_callback(int event, int x, int y, int flags, void *param)
{
    static Point2d p1(0, 0), p2(0, 0);
    static Vec6d area(-1, 0, 0, 0, 0, 0);
    switch (event)
    {
    case cv::EVENT_LBUTTONDOWN: // 鼠标左键点击
        p1 = Point2d(x, y);
        area = define_area(g_line1, g_line2, p1); // 确定鼠标所要移动的区域
        break;
    case cv::EVENT_MOUSEMOVE: // 鼠标移动
        if (area[0] > -1)     // 移动直线
        {
            p2 = Point2d(x, y);
            double w = p2.x - p1.x, h = p2.y - p1.y;
            modify_line(g_line1, g_line2, area, w, h); // 根据鼠标移动相应的修改直线的起点和终点
        }
        break;
    case cv::EVENT_LBUTTONUP: // 鼠标左键释放
        p1 = Point2d(0, 0);
        p2 = Point2d(0, 0);
        area = Vec6d(-1, 0, 0, 0, 0, 0);
        break;
    default:
        break;
    }
}

// 主函数
int main()
{
    string window_name = "image";
    namedWindow(window_name, WINDOW_AUTOSIZE);
    int w = 800, h = 600;
    Mat image_original = Mat(h, w, CV_8UC3, Scalar(255, 255, 255));
    cv::setMouseCallback(window_name, mouse_callback); // 调用鼠标回调函数
    while (true)
    {
        Mat img = image_original.clone(); // 拷贝空白图片，方便重复画图
        draw(img, g_line1, g_line2);      // 画图
        imshow(window_name, img);
        if (waitKey(3) > 0) // 退出循环
            break;
    }
    return 0;
}

操作方法：
鼠标点击直线起点或终点并按住移动，可改变直线。鼠标点击直线的中间点并按住移动，可平移直线。在键盘按任意的键可退出程序。

运行结果：

学习OpenCV3：判断两条直线相交，并计算交点和夹角