学习OpenCV3：判断两条直线平行，并计算平行距离

我不是女神ヾ 2023-02-21 09:27 26阅读 0赞

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# 一、问题 #

已知两条直线 l 1 ( x 1 , y 1 , x 2 , y 2 ) l\_1(x\_1,y\_1,x\_2,y\_2) l1(x1,y1,x2,y2)和 l 2 ( x 3 , y 3 , x 4 , y 4 ) l\_2(x\_3,y\_3,x\_4,y\_4) l2(x3,y3,x4,y4)，现希望判断 l 1 l\_1 l1与 l 2 l\_2 l2间是否平行。若平行，计算出两条直线间的平行距离。

# 二、分析 #

y − y 1 y 2 − y 1 = x − x 1 x 2 − x 1 ⇒ \{ a x + b y + 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 = − a b \\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\} y2−y1y−y1=x2−x1x−x1⇒⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧ax\+by\+c=0a=−(y2−y1)b=x2−x1c=(y2−y1)x1−(x2−x1)y1k=b−a

l 1 l\_1 l1的直线方程：  
 a 1 x + b 1 y + c 1 = 0 a\_1x+b\_1y+c\_1=0 a1x\+b1y\+c1=0

l 2 l\_2 l2的直线方程：  
 a 2 x + b 2 y + c 2 = 0 a\_2x+b\_2y+c\_2=0 a2x\+b2y\+c2=0

## 1、排除 l 1 l\_1 l1与 l 2 l\_2 l2重合 ##

l 2 l\_2 l2的起点 p 3 ( x 3 , y 3 ) p\_\{3\}(x\_\{3\},y\_\{3\}) p3(x3,y3)和终点 p 4 ( x 4 , y 4 ) p\_\{4\}(x\_\{4\},y\_\{4\}) p4(x4,y4)不能同时在直线 l 1 l\_1 l1上：

!(p3!=p4 && a1*x3+b1*y3+c1==0 && a1*x4+b1*y4+c1==0)

## 2、 l 1 l\_1 l1与 l 2 l\_2 l2都垂直 x x x轴 ##

![format_png_pic_center][]

当 l 1 l\_1 l1与 l 2 l\_2 l2都垂直 x x x轴时， l 1 l\_1 l1与 l 2 l\_2 l2平行：

b2==0 && b1==0

l 1 l\_1 l1与 l 2 l\_2 l2的直线方程：  
 l 1 : a 1 x + c 1 = 0 l 2 : a 2 x + c 2 = 0 l\_1:a\_1x+c\_1=0 \\quad l\_2:a\_2x+c\_2=0 l1:a1x\+c1=0l2:a2x\+c2=0

l 1 l\_1 l1与 l 2 l\_2 l2间的距离为：  
 d = ∣ x 2 − x 1 ∣ = ∣ c 1 a 1 − c 2 a 2 ∣ d= |x\_2-x\_1|= |\\frac\{c\_1\}\{a\_1\} - \\frac\{c\_2\}\{a\_2\} | d=∣x2−x1∣=∣a1c1−a2c2∣

## 3、 l 1 l\_1 l1与 l 2 l\_2 l2都倾斜于 x x x轴，且斜率相同 ##

![format_png_pic_center 1][]

当 l 1 l\_1 l1与 l 2 l\_2 l2斜率相同时， l 1 l\_1 l1与 l 2 l\_2 l2平行：

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

l 1 l\_1 l1与 l 2 l\_2 l2的直线方程：  
 l 1 : a 1 x + b 1 y + c 1 = 0 l 2 : a 2 x + b 2 y + c 2 = 0 l\_1:a\_1x+b\_1y+c\_1=0 \\quad l\_2:a\_2x+b\_2y+c\_2=0 l1:a1x\+b1y\+c1=0l2:a2x\+b2y\+c2=0

l 2 l\_2 l2与 y y y轴的交点 p 0 p\_0 p0：  
 ( x 0 , y 0 ) = ( 0 , − c 2 b 2 ) (x\_0,y\_0)=(0,\\frac\{-c\_2\}\{b\_2\}) (x0,y0)=(0,b2−c2)

p 0 p\_0 p0到 l 1 l\_1 l1距离：  
 d = ∣ a 1 x 0 + b 1 y 0 + c 1 ∣ a 1 2 + b 1 2 = ∣ c 1 − b 1 c 2 b 2 ∣ a 1 2 + b 1 2 d= \\frac\{|a\_1x\_0+b\_1y\_0+c\_1|\}\{\\sqrt\{a\_1^2+b\_1^2\}\} = \\frac\{|c\_1-\\frac\{b\_1c\_2\}\{b\_2\}|\}\{\\sqrt\{a\_1^2+b\_1^2\}\} d=a12\+b12∣a1x0\+b1y0\+c1∣=a12\+b12∣c1−b2b1c2∣

# 三、实现 #

#include <opencv2/opencv.hpp>
    #include <iostream>
    #include <string>
    #include <cmath>
    using namespace std;
    using namespace cv;
    
    Vec4d g_line1(200, 200, 600, 200), g_line2(200, 400, 600, 400); // 平行的两条线
    
    // 判断两条线是否平行，若平行，则求出平行距离
    double lines_parellel(const Vec4d l1, const Vec4d l2)
    {
        double x1 = l1[0], y1 = l1[1], x2 = l1[2], y2 = l1[3];                      // 两点式：(y-y1)/(y2-y1)=(x-x1)/(x2-x1)
        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];                      // 两点式：(y-y3)/(y4-y3)=(x-x3)/(x4-x3)
        double a2 = -(y4 - y3), b2 = x4 - x3, c2 = (y4 - y3) * x3 - (x4 - x3) * y3; // 一般式：a2x+b2y1+c2=0
        Point2d p3(x3, y3), p4(x4, y4);
        double d = 0;
        if (!(p3 != p4 && a1 * x3 + b1 * y3 + c1 == 0 && a1 * x4 + b1 * y4 + c1 == 0)) // 排除重合的情况
        {
            if (b2 == 0 && b1 == 0) // l2，l1垂直x轴
            {
                d = abs(c1 / a1 - c2 / a2);
            }
            else if (b2 != 0 && b1 != 0 && a2 / b2 == a1 / b1) // l2，l1斜率相同
            {
                d = abs(c1 - b1 * c2 / b2) / sqrt(a1 * a1 + b1 * b1);
            }
        }
        return d;
    }
    
    // 计算垂足的坐标
    Point calculate_foot_point(Point2d p, Vec4d l)
    {
        double x0 = p.x, y0 = p.y;
        double x1 = l[0], y1 = l[1], x2 = l[2], y2 = l[3];
        Point2d p1(0, 0); // 垂足
        if (x1 == x2)     // 线与x轴垂直
        {
            p1.x = x1;
            p1.y = y0;
        }
        else if (y1 == y2) // 线与x轴水平
        {
            p1.x = x0;
            p1.y = y1;
        }
        else // 线与x轴倾斜
        {
            int a = -(y2 - y1);
            int b = x2 - x1;
            int c = (y2 - y1) * x1 - (x2 - x1) * y1;
            p1.x = (b * b * x0 - a * b * y0 - a * c) / (a * a + b * b);
            p1.y = (a * a * y0 - a * b * x0 - b * c) / (a * a + b * b);
        }
        return p1;
    }
    
    // 画虚线
    void draw_dotted_line(Mat img, const Point2d p1, const Point2d p2, const Scalar color, const int thickness)
    {
        double n = 15; // 小虚线的长度
        double w = p2.x - p1.x, h = p2.y - p1.y;
        double l = sqrtl(w * w + h * h);
        // 矫正线长度，使线个数为奇数
        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
        // 画中间点
        if (p1.y == p2.y) // 与x轴水平：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) // 与x轴垂直, 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 // 与x轴倾斜，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) // 与x轴倾斜
        {
            p1 = Point2d(0, -c / b);
            p2 = Point2d(img.cols, ((-a * img.cols - c) / b));
        }
        else // 与x轴垂直
        {
            p1 = Point2d(-c / a, 0);
            p2 = Point2d(-c / a, img.rows);
        }
        draw_dotted_line(img, p1, p2, color, 1);
    }
    
    // 画图
    void draw(const Mat img, const Vec4d l1, const Vec4d l2)
    {
        line(img, Point2d(l1[0], l1[1]), Point2d(l1[2], l1[3]), Scalar(0, 255, 0), 2); // 画绿线
        line(img, Point2d(l2[0], l2[1]), Point2d(l2[2], l2[3]), Scalar(0, 255, 0), 2); // 画绿线
        double d = lines_parellel(l1, l2);
        if (d > 0) // 重合
        {
            putText(img, "yes d = " + to_string(d), Point2d(10, 25), cv::FONT_HERSHEY_COMPLEX, 0.5, Scalar(0, 255, 0));
            draw_extension_line(img, l1, Scalar(0, 255, 0));      // 画延长线
            draw_extension_line(img, l2, Scalar(0, 255, 0));      // 画延长线
            Point2d p1((l1[2] + l1[0]) / 2, (l1[3] + l1[1]) / 2); // 获取线段的中间点
            Point p2 = calculate_foot_point(p1, l2);              // 获取垂足
            draw_dotted_line(img, p1, p2, Scalar(0, 255, 0), 1);  // 画垂线
        }
        else // 不重合
        {
            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);
            char c = waitKey(3);
            if (c == '1') // l1,l2与x轴平行
            {
                g_line1 = Vec4d(200, 200, 600, 200);
                g_line2 = Vec4d(200, 400, 600, 400);
            }
            else if (c == '2') // l1,l2与x轴垂直
            {
                g_line1 = Vec4d(300, 200, 300, 400);
                g_line2 = Vec4d(500, 200, 500, 400);
            }
            else if (c == '3') // l1,l2与x轴右倾斜
            {
                g_line1 = Vec4d(200, 450, 600, 150);
                g_line2 = Vec4d(200, 500, 600, 200);
            }
            else if (c == '4') // l1,l2与x轴左倾斜
            {
                g_line1 = Vec4d(100, 100, 500, 500);
                g_line2 = Vec4d(100, 150, 500, 550);
            }
            else if (c > 0 && (c < '1' || c > '4')) // 退出循环
            {
                break;
            }
        }
        return 0;
    }

**操作方法：**  
  鼠标点击两条直线的起点或终点并按住移动，由此可以修改直线。鼠标点击两条直线的中间点并按住移动，由此可以平移直线。  
  键盘按住`1`、`2`、`3`或`4`可选择对应的模式，画出不同角度下两条直线平行的情况。键盘按其它的键会退出程序。

**两条直线不平行时：**

![format_png_pic_center 2][]

**两条直线都与 x x x轴水平（模式1）时：**

![format_png_pic_center 3][]

**两条直线都与 x x x轴垂直（模式2）时：**

![format_png_pic_center 4][]

**两条直线都与 x x x轴右倾斜（模式3）时：**

![format_png_pic_center 5][]

**两条直线都与 x x x轴左倾斜（模式4）时：**

![format_png_pic_center 6][]

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