学习OpenCV3：判断两条直线平行，并计算平行距离

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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$l_2$间是否平行。若平行，计算出两条直线间的平行距离。

二、分析

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

  $l_1$的直线方程：
$$a_{1}x+b_{1}y+c_1=0$$

  $l_2$的直线方程：
$$a_{2}x+b_{2}y+c_2=0$$

1、排除$l_1$与$l_2$重合

  $l_2$的起点$p_3(x_3,y_3)$和终点$p_4(x_4,y_4)$不能同时在直线$l_1$上：

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

2、$l_1$与$l_2$都垂直$x$轴

  当$l_1$与$l_2$都垂直$x$轴时，$l_1$与$l_2$平行：

    b2==0 && b1==0

  $l_1$与$l_2$的直线方程：
$$l_1:a_{1}x+c_1=0l_2:a_{2}x+c_2=0$$

  $l_1$与$l_2$间的距离为：
$$d=|x_2-x_1|=|\frac{c_1}{a_1}-\frac{c_2}{a_2}|$$

3、$l_1$与$l_2$都倾斜于$x$轴，且斜率相同

  当$l_1$与$l_2$斜率相同时，$l_1$与$l_2$平行：

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

  $l_1$与$l_2$的直线方程：
$$l_1:a_{1}x+b_{1}y+c_1=0l_2:a_{2}x+b_{2}y+c_2=0$$

  $l_2$与$y$轴的交点$p_0$：
$$(x_0,y_0)=(0,\frac{-c_2}{b_2})$$

  $p_0$到$l_1$距离：
$$d=\frac{|a_1{x}_0+b_1{y}_0+c_1|}{\sqrt{a_1^2+b_1^2}}=\frac{|c_1-\frac{b_1{c}_2}{b_2}|}{\sqrt{a_1^2+b_1^2}}$$

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：

两条直线都与$x$轴水平（模式1）时：

两条直线都与$x$轴垂直（模式2）时：

两条直线都与$x$轴右倾斜（模式3）时：

两条直线都与$x$轴左倾斜（模式4）时：