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间是否重合。
二、分析
l 1 l_1 l1的直线方程:
y − y 1 y 2 − y 1 = x − x 1 x 2 − x 1 ⇒ { a 1 x + b 1 y + c 1 = 0 a 1 = − ( y 2 − y 1 ) b 1 = x 2 − x 1 c 1 = ( y 2 − y 1 ) x 1 − ( x 2 − x 1 ) y 1 \frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1} \Rightarrow \begin{cases} a_1x+b_1y+c_1=0 \\a_1=-(y_2-y_1) \\b_1 = x_2-x_1 \\c_1 = (y_2-y_1)x_1 - (x_2-x_1) y_1\end{cases} y2−y1y−y1=x2−x1x−x1⇒⎩⎪⎪⎪⎨⎪⎪⎪⎧a1x+b1y+c1=0a1=−(y2−y1)b1=x2−x1c1=(y2−y1)x1−(x2−x1)y1
若 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)不是同一个点,且都在直线 l 1 l_1 l1上,则 l 2 l_2 l2与 l 1 l_1 l1重合。
判断条件:p3!=p4 && a1*x3+b1*y3+c1==0 && a1*x4+b1*y4+c1==0
三、实现
#include <opencv2/opencv.hpp>
#include <iostream>
#include <string>
#include <cmath>
using namespace std;
using namespace cv;
Vec4d g_line1(100, 300, 300, 300), g_line2(500, 300, 700, 300); // 重合的两条线
// 判断两条线是否重合
bool lines_coincidence(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];
Point2d p3(x3, y3), p4(x4, y4);
if (p3 != p4 && a1 * x3 + b1 * y3 + c1 == 0 && a1 * x4 + b1 * y4 + c1 == 0) // 判断条件
return true;
else
return false;
}
// 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) //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(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
if (lines_coincidence(l1, l2)) // 重合
{
putText(img, "yes", 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
}
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(100, 300, 300, 300);
g_line2 = Vec4d(500, 300, 700, 300);
}
else if (c == '2') // l1,l2与x轴垂直
{
g_line1 = Vec4d(400, 50, 400, 250);
g_line2 = Vec4d(400, 350, 400, 550);
}
else if (c == '3') // l1,l2与x轴右倾斜
{
g_line1 = Vec4d(100, 500, 300, 400);
g_line2 = Vec4d(500, 300, 700, 200);
}
else if (c == '4') // l1,l2与x轴左倾斜
{
g_line1 = Vec4d(100, 200, 300, 300);
g_line2 = Vec4d(500, 400, 700, 500);
}
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重合的情况。键盘按其它的键会退出程序。
两条直线不重合时:
模式1:
模式2:
模式3:
模式4: