标签：06 glm OpenGL 0.0 float C++ vec3 pitch GL
1. 引言
本文基于C++语言，描述OpenGL的摄像机
前置知识可参考：
笔者这里不过多描述每个名词、函数和细节，更详细的文档可以参考：
2. 概述
OpenGL的坐标变换流程图如下：
有图可知：
- 摄像机的参数（如，位置、视点、方向）决定视图
根据变化的相对性，控制摄像机的参数可以看成物体的变化（如，摄像机后移相当于物体后移）
观察矩阵可由摄像机的位置、视点和方向计算，如下图：
计算公式：
\[LookAt = \begin{bmatrix} \color{red}{R_x} & \color{red}{R_y} & \color{red}{R_z} & 0 \\ \color{green}{U_x} & \color{green}{U_y} & \color{green}{U_z} & 0 \\ \color{blue}{D_x} & \color{blue}{D_y} & \color{blue}{D_z} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} * \begin{bmatrix} 1 & 0 & 0 & -\color{purple}{P_x} \\ 0 & 1 & 0 & -\color{purple}{P_y} \\ 0 & 0 & 1 & -\color{purple}{P_z} \\ 0 & 0 & 0 & 1 \end{bmatrix} \]其中R是右向量，U是上向量，D是方向向量，P是摄像机位置向量；
位置向量是相反的，因为我们最终希望把世界平移到与我们自身移动的相反方向
3. 编码
控制摄像机的参数实质就是控制观察矩阵（view）
生成一个观察矩阵需要位置、视点和方向向量，GLM的lookAt()函数可用于生成观察矩阵：
glm::mat4 view = glm::mat4(1.0f);
view = glm::lookAt(glm::vec3(0.0f, 0.0f, -3.0f),
glm::vec3(0.0f, 0.0f, 0.0f),
glm::vec3(0.0f, 1.0f, 0.0f));
可选项，让摄像机的位置绕圆转动，会形成物体转动的感觉
glm::mat4 view = glm::mat4(1.0f);
float radius = 10.0f;
float camX = sin(glfwGetTime()) * radius;
float camZ = cos(glfwGetTime()) * radius;
view = glm::lookAt(glm::vec3(camX, 0.0f, camZ),
glm::vec3(0.0f, 0.0f, 0.0f),
glm::vec3(0.0f, 1.0f, 0.0f));
运行一下，结果图如下：
4. 自由移动
控制摄像机的位置可以实现视角的前后左右上下移动
摄像机参数：
glm::vec3 cameraPos = glm::vec3(0.0f, 0.0f, 3.0f);
glm::vec3 cameraFront = glm::vec3(0.0f, 0.0f, -1.0f);
glm::vec3 cameraUp = glm::vec3(0.0f, 1.0f, 0.0f);
观察矩阵：
view = glm::lookAt(cameraPos, cameraPos + cameraFront, cameraUp);
使用按键WSAD实现前后左右移动：
void processInput(GLFWwindow *window)
{
...
float cameraSpeed = 0.05f; // adjust accordingly
if (glfwGetKey(window, GLFW_KEY_W) == GLFW_PRESS)
cameraPos += cameraSpeed * cameraFront;
if (glfwGetKey(window, GLFW_KEY_S) == GLFW_PRESS)
cameraPos -= cameraSpeed * cameraFront;
if (glfwGetKey(window, GLFW_KEY_A) == GLFW_PRESS)
cameraPos -= glm::normalize(glm::cross(cameraFront, cameraUp)) * cameraSpeed;
if (glfwGetKey(window, GLFW_KEY_D) == GLFW_PRESS)
cameraPos += glm::normalize(glm::cross(cameraFront, cameraUp)) * cameraSpeed;
}
5. 视角移动
根据鼠标变化计算视角变化
float lastX = 0.0f, lastY = 0.0f;
bool firstMouse = true;
// 鼠标变化
void mouse_callback(GLFWwindow* window, double xpos, double ypos)
{
if(firstMouse)
{
lastX = xpos;
lastY = ypos;
firstMouse = false;
}
float xoffset = xpos - lastX;
float yoffset = lastY - ypos;
lastX = xpos;
lastY = ypos;
float sensitivity = 0.05;
xoffset *= sensitivity;
yoffset *= sensitivity;
yaw += xoffset;
pitch += yoffset;
if(pitch > 89.0f)
pitch = 89.0f;
if(pitch < -89.0f)
pitch = -89.0f;
glm::vec3 front;
front.x = cos(glm::radians(yaw)) * cos(glm::radians(pitch));
front.y = sin(glm::radians(pitch));
front.z = sin(glm::radians(yaw)) * cos(glm::radians(pitch));
cameraFront = glm::normalize(front);
}
6. 滚轮缩放
根据透视投影的视角大小实现物体的缩放
float fov = 30.0f;
// 鼠标滚轮变化
void scroll_callback(GLFWwindow* window, double xoffset, double yoffset)
{
if(fov >= 1.0f && fov <= 45.0f)
fov -= yoffset;
if(fov <= 1.0f)
fov = 1.0f;
if(fov >= 45.0f)
fov = 45.0f;
}
...
// 投影矩阵
projection = glm::perspective(glm::radians(fov), 800.0f / 600.0f, 0.1f, 100.0f);
7. 完整代码
按照上述步骤，如果顺利的话，已经实现了按键WSAD的移动、鼠标的视角移动和滚轮缩放
主要文件test.cpp：
#include <glad/glad.h>
#include <GLFW/glfw3.h>
#include <iostream>
#include <math.h>
#include "Shader.hpp"
#define STB_IMAGE_IMPLEMENTATION
#include "stb_image.h"
#include <glm/glm.hpp>
#include <glm/ext/matrix_transform.hpp> // glm::translate, glm::rotate, glm::scale
#include <glm/ext/matrix_clip_space.hpp> // glm::perspective
#include <glm/gtc/type_ptr.hpp>
//全局变量
glm::vec3 cameraPos = glm::vec3(0.0f, 0.0f, 3.0f);
glm::vec3 cameraFront = glm::vec3(0.0f, 0.0f, -1.0f);
glm::vec3 cameraUp = glm::vec3(0.0f, 1.0f, 0.0f);
float lastX = 400.0f, lastY = 300.0f, yaw = -90.0f, pitch = 0.0f, fov = 30.0f;
bool firstMouse = true;
// 函数声明
void framebuffer_size_callback(GLFWwindow *window, int width, int height);
void process_input(GLFWwindow *window);
unsigned int *renderInit();
void render(unsigned int shaderProgram, unsigned int VAO, unsigned int texture1, unsigned int texture2);
bool checkCompile(unsigned int shader);
bool checkProgram(unsigned int shaderProgram);
void mouse_callback(GLFWwindow* window, double xpos, double ypos);
void scroll_callback(GLFWwindow* window, double xoffset, double yoffset);
int main()
{
glfwInit();
glfwWindowHint(GLFW_CONTEXT_VERSION_MAJOR, 3);
glfwWindowHint(GLFW_CONTEXT_VERSION_MINOR, 3);
GLFWwindow *window = glfwCreateWindow(800, 600, "CoordinateSystem", nullptr, nullptr);
if (window == nullptr)
{
std::cout << "Faild to create window" << std::endl;
glfwTerminate();
}
glfwMakeContextCurrent(window);
if (!gladLoadGLLoader((GLADloadproc)glfwGetProcAddress))
{
std::cout << "Faild to initialize glad" << std::endl;
return -1;
}
glad_glViewport(0, 0, 800, 600);
glfwSetFramebufferSizeCallback(window, framebuffer_size_callback);
glfwSetCursorPosCallback(window, mouse_callback);
glfwSetScrollCallback(window, scroll_callback);
unsigned int *arr = renderInit();
while (!glfwWindowShouldClose(window))
{
process_input(window);
// render
std::cout << arr[0] << " " << arr[1] << " " << arr[2] << " " << arr[3] << " " << arr[4] << std::endl;
render(arr[0], arr[1], arr[3], arr[4]);
glfwSwapBuffers(window);
glfwPollEvents();
}
glDeleteProgram(arr[0]);
glDeleteVertexArrays(1, &arr[1]);
glDeleteBuffers(1, &arr[2]);
glfwTerminate();
return 0;
}
void framebuffer_size_callback(GLFWwindow *window, int width, int height)
{
glViewport(0, 0, width, height);
}
void process_input(GLFWwindow *window)
{
if (glfwGetKey(window, GLFW_KEY_ESCAPE) == GLFW_PRESS)
{
glfwSetWindowShouldClose(window, true);
}
float cameraSpeed = 0.05f; // adjust accordingly
if (glfwGetKey(window, GLFW_KEY_W) == GLFW_PRESS)
cameraPos += cameraSpeed * cameraFront;
if (glfwGetKey(window, GLFW_KEY_S) == GLFW_PRESS)
cameraPos -= cameraSpeed * cameraFront;
if (glfwGetKey(window, GLFW_KEY_A) == GLFW_PRESS)
cameraPos -= glm::normalize(glm::cross(cameraFront, cameraUp)) * cameraSpeed;
if (glfwGetKey(window, GLFW_KEY_D) == GLFW_PRESS)
cameraPos += glm::normalize(glm::cross(cameraFront, cameraUp)) * cameraSpeed;
}
// 鼠标变化
void mouse_callback(GLFWwindow* window, double xpos, double ypos)
{
if(firstMouse)
{
lastX = xpos;
lastY = ypos;
firstMouse = false;
}
float xoffset = xpos - lastX;
float yoffset = lastY - ypos;
lastX = xpos;
lastY = ypos;
float sensitivity = 0.05;
xoffset *= sensitivity;
yoffset *= sensitivity;
yaw += xoffset;
pitch += yoffset;
if(pitch > 89.0f)
pitch = 89.0f;
if(pitch < -89.0f)
pitch = -89.0f;
glm::vec3 front;
front.x = cos(glm::radians(yaw)) * cos(glm::radians(pitch));
front.y = sin(glm::radians(pitch));
front.z = sin(glm::radians(yaw)) * cos(glm::radians(pitch));
cameraFront = glm::normalize(front);
}
// 鼠标滚轮变化
void scroll_callback(GLFWwindow* window, double xoffset, double yoffset)
{
if(fov >= 1.0f && fov <= 45.0f)
fov -= yoffset;
if(fov <= 1.0f)
fov = 1.0f;
if(fov >= 45.0f)
fov = 45.0f;
}
unsigned int *renderInit()
{
//配置项
glEnable(GL_DEPTH_TEST);
unsigned int VAO;
glGenVertexArrays(1, &VAO);
glBindVertexArray(VAO);
unsigned int texture1;
glGenTextures(1, &texture1);
glBindTexture(GL_TEXTURE_2D, texture1);
// 为当前绑定的纹理对象设置环绕、过滤方式
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_S, GL_REPEAT);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_T, GL_REPEAT);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MIN_FILTER, GL_LINEAR);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MAG_FILTER, GL_LINEAR);
// 加载并生成纹理
int width, height, nrChannels;
unsigned char *data = stbi_load("../container.jpg", &width, &height, &nrChannels, 0);
if (data)
{
glTexImage2D(GL_TEXTURE_2D, 0, GL_RGB, width, height, 0, GL_RGB, GL_UNSIGNED_BYTE, data);
glGenerateMipmap(GL_TEXTURE_2D);
}
else
{
std::cout << "Failed to load texture" << std::endl;
}
stbi_image_free(data);
unsigned int texture2;
glGenTextures(1, &texture2);
glBindTexture(GL_TEXTURE_2D, texture2);
// 为当前绑定的纹理对象设置环绕、过滤方式
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_S, GL_REPEAT);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_WRAP_T, GL_REPEAT);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MIN_FILTER, GL_LINEAR);
glTexParameteri(GL_TEXTURE_2D, GL_TEXTURE_MAG_FILTER, GL_LINEAR);
// // 加载并生成纹理
int width2, height2, nrChannels2;
stbi_set_flip_vertically_on_load(true);
unsigned char *data2 = stbi_load("../awesomeface.png", &width2, &height2, &nrChannels2, 0);
if (data2)
{
glTexImage2D(GL_TEXTURE_2D, 0, GL_RGBA, width2, height2, 0, GL_RGBA, GL_UNSIGNED_BYTE, data2);
glGenerateMipmap(GL_TEXTURE_2D);
}
else
{
std::cout << "Failed to load texture" << std::endl;
}
stbi_image_free(data2);
float vertices[] = {
-0.5f, -0.5f, -0.5f, 0.0f, 0.0f,
0.5f, -0.5f, -0.5f, 1.0f, 0.0f,
0.5f, 0.5f, -0.5f, 1.0f, 1.0f,
0.5f, 0.5f, -0.5f, 1.0f, 1.0f,
-0.5f, 0.5f, -0.5f, 0.0f, 1.0f,
-0.5f, -0.5f, -0.5f, 0.0f, 0.0f,
-0.5f, -0.5f, 0.5f, 0.0f, 0.0f,
0.5f, -0.5f, 0.5f, 1.0f, 0.0f,
0.5f, 0.5f, 0.5f, 1.0f, 1.0f,
0.5f, 0.5f, 0.5f, 1.0f, 1.0f,
-0.5f, 0.5f, 0.5f, 0.0f, 1.0f,
-0.5f, -0.5f, 0.5f, 0.0f, 0.0f,
-0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
-0.5f, 0.5f, -0.5f, 1.0f, 1.0f,
-0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
-0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
-0.5f, -0.5f, 0.5f, 0.0f, 0.0f,
-0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
0.5f, 0.5f, -0.5f, 1.0f, 1.0f,
0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
0.5f, -0.5f, 0.5f, 0.0f, 0.0f,
0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
-0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
0.5f, -0.5f, -0.5f, 1.0f, 1.0f,
0.5f, -0.5f, 0.5f, 1.0f, 0.0f,
0.5f, -0.5f, 0.5f, 1.0f, 0.0f,
-0.5f, -0.5f, 0.5f, 0.0f, 0.0f,
-0.5f, -0.5f, -0.5f, 0.0f, 1.0f,
-0.5f, 0.5f, -0.5f, 0.0f, 1.0f,
0.5f, 0.5f, -0.5f, 1.0f, 1.0f,
0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
0.5f, 0.5f, 0.5f, 1.0f, 0.0f,
-0.5f, 0.5f, 0.5f, 0.0f, 0.0f,
-0.5f, 0.5f, -0.5f, 0.0f, 1.0f};
unsigned int VBO;
glGenBuffers(1, &VBO);
glBindBuffer(GL_ARRAY_BUFFER, VBO);
glBufferData(GL_ARRAY_BUFFER, sizeof(vertices), vertices, GL_STATIC_DRAW);
glVertexAttribPointer(0, 3, GL_FLOAT, GL_FALSE, 5 * sizeof(float), (void *)0);
glEnableVertexAttribArray(0);
glVertexAttribPointer(1, 2, GL_FLOAT, GL_FALSE, 5 * sizeof(float), (void *)(3 * sizeof(float)));
glEnableVertexAttribArray(1);
Shader shaderProgram = Shader("../test.vs.glsl", "../test.fs.glsl");
shaderProgram.use();
glUniform1i(glGetUniformLocation(shaderProgram.ID, "texture1"), 0);
glUniform1i(glGetUniformLocation(shaderProgram.ID, "texture2"), 1);
return new unsigned int[5]{shaderProgram.ID, VAO, VBO, texture1, texture2};
}
void render(unsigned int shaderProgram, unsigned int VAO, unsigned int texture1, unsigned int texture2)
{
glClearColor(0.2, 0.3, 0.3, 1.0);
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glActiveTexture(GL_TEXTURE0);
glBindTexture(GL_TEXTURE_2D, texture1);
glActiveTexture(GL_TEXTURE1);
glBindTexture(GL_TEXTURE_2D, texture2);
glUseProgram(shaderProgram);
glm::vec3 cubePositions[] = {
glm::vec3(0.0f, 0.0f, 0.0f),
glm::vec3(2.0f, 5.0f, -15.0f),
glm::vec3(-1.5f, -2.2f, -2.5f),
glm::vec3(-3.8f, -2.0f, -12.3f),
glm::vec3(2.4f, -0.4f, -3.5f),
glm::vec3(-1.7f, 3.0f, -7.5f),
glm::vec3(1.3f, -2.0f, -2.5f),
glm::vec3(1.5f, 2.0f, -2.5f),
glm::vec3(1.5f, 0.2f, -1.5f),
glm::vec3(-1.3f, 1.0f, -1.5f)};
glm::mat4 view = glm::mat4(1.0f);
// 注意，我们将矩阵向我们要进行移动场景的反方向移动。
// view = glm::translate(view, glm::vec3(0.0f, 0.0f, -3.0f));
// float radius = 10.0f;
// float camX = sin(glfwGetTime()) * radius;
// float camZ = cos(glfwGetTime()) * radius;
// view = glm::lookAt(glm::vec3(camX, 20.0f, camZ),
// glm::vec3(0.0f, 0.0f, 0.0f),
// glm::vec3(0.0f, 1.0f, 0.0f));
view = glm::lookAt(cameraPos, cameraPos + cameraFront, cameraUp);
glm::mat4 projection = glm::mat4(1.0f);
projection = glm::perspective(glm::radians(fov), 800.0f / 600.0f, 0.1f, 100.0f);
// 模型矩阵
int modelLoc = glGetUniformLocation(shaderProgram, "model");
// 观察矩阵
int viewLoc = glGetUniformLocation(shaderProgram, "view");
glUniformMatrix4fv(viewLoc, 1, GL_FALSE, glm::value_ptr(view));
// 投影矩阵
int projectionLoc = glGetUniformLocation(shaderProgram, "projection");
glUniformMatrix4fv(projectionLoc, 1, GL_FALSE, glm::value_ptr(projection));
glBindVertexArray(VAO);
for (unsigned int i = 0; i < 10; i++)
{
glm::mat4 model = glm::mat4(1.0f);
model = glm::translate(model, cubePositions[i]);
float angle = 20.0f * (i + 1);
model = glm::rotate(model, (float)glfwGetTime() * glm::radians(50.0f), glm::vec3(1.0f, 0.3f, 0.5f));
glUniformMatrix4fv(modelLoc, 1, GL_FALSE, glm::value_ptr(model));
glDrawArrays(GL_TRIANGLES, 0, 36);
}
}
顶点着色器test.vs.glsl：
#version 330 core
layout (location = 0) in vec3 aPos;
layout (location = 1) in vec2 aTexCoord;
out vec2 TexCoord;
uniform mat4 model;
uniform mat4 view;
uniform mat4 projection;
void main()
{
// 注意乘法要从右向左读
gl_Position = projection * view * model * vec4(aPos, 1.0);
TexCoord = aTexCoord;
}
片段着色器test.fs.glsl：
#version 330 core
out vec4 FragColor;
in vec2 TexCoord;
uniform sampler2D texture1;
uniform sampler2D texture2;
void main()
{
FragColor = mix(texture(texture1, TexCoord), texture(texture2, TexCoord), 0.2);
}
8. 封装摄像机类
camera.hpp文件：
#ifndef CAMERA_HPP
#define CAMERA_HPP
#include <glad/glad.h>
#include <glm/glm.hpp>
#include <glm/gtc/matrix_transform.hpp>
#include <vector>
// Defines several possible options for camera movement. Used as abstraction to stay away from window-system specific input methods
enum Camera_Movement {
FORWARD,
BACKWARD,
LEFT,
RIGHT
};
// Default camera values
const float YAW = -90.0f;
const float PITCH = 0.0f;
const float SPEED = 2.5f;
const float SENSITIVITY = 0.1f;
const float ZOOM = 45.0f;
// An abstract camera class that processes input and calculates the corresponding Euler Angles, Vectors and Matrices for use in OpenGL
class Camera
{
public:
// camera Attributes
glm::vec3 Position;
glm::vec3 Front;
glm::vec3 Up;
glm::vec3 Right;
glm::vec3 WorldUp;
// euler Angles
float Yaw;
float Pitch;
// camera options
float MovementSpeed;
float MouseSensitivity;
float Zoom;
// constructor with vectors
Camera(glm::vec3 position = glm::vec3(0.0f, 0.0f, 0.0f), glm::vec3 up = glm::vec3(0.0f, 1.0f, 0.0f), float yaw = YAW, float pitch = PITCH) : Front(glm::vec3(0.0f, 0.0f, -1.0f)), MovementSpeed(SPEED), MouseSensitivity(SENSITIVITY), Zoom(ZOOM)
{
Position = position;
WorldUp = up;
Yaw = yaw;
Pitch = pitch;
updateCameraVectors();
}
// constructor with scalar values
Camera(float posX, float posY, float posZ, float upX, float upY, float upZ, float yaw, float pitch) : Front(glm::vec3(0.0f, 0.0f, -1.0f)), MovementSpeed(SPEED), MouseSensitivity(SENSITIVITY), Zoom(ZOOM)
{
Position = glm::vec3(posX, posY, posZ);
WorldUp = glm::vec3(upX, upY, upZ);
Yaw = yaw;
Pitch = pitch;
updateCameraVectors();
}
// returns the view matrix calculated using Euler Angles and the LookAt Matrix
glm::mat4 GetViewMatrix()
{
return glm::lookAt(Position, Position + Front, Up);
}
// processes input received from any keyboard-like input system. Accepts input parameter in the form of camera defined ENUM (to abstract it from windowing systems)
void ProcessKeyboard(Camera_Movement direction, float deltaTime)
{
float velocity = MovementSpeed * deltaTime;
if (direction == FORWARD)
Position += Front * velocity;
if (direction == BACKWARD)
Position -= Front * velocity;
if (direction == LEFT)
Position -= Right * velocity;
if (direction == RIGHT)
Position += Right * velocity;
}
// processes input received from a mouse input system. Expects the offset value in both the x and y direction.
void ProcessMouseMovement(float xoffset, float yoffset, GLboolean constrainPitch = true)
{
xoffset *= MouseSensitivity;
yoffset *= MouseSensitivity;
Yaw += xoffset;
Pitch += yoffset;
// make sure that when pitch is out of bounds, screen doesn't get flipped
if (constrainPitch)
{
if (Pitch > 89.0f)
Pitch = 89.0f;
if (Pitch < -89.0f)
Pitch = -89.0f;
}
// update Front, Right and Up Vectors using the updated Euler angles
updateCameraVectors();
}
// processes input received from a mouse scroll-wheel event. Only requires input on the vertical wheel-axis
void ProcessMouseScroll(float yoffset)
{
Zoom -= (float)yoffset;
if (Zoom < 1.0f)
Zoom = 1.0f;
if (Zoom > 45.0f)
Zoom = 45.0f;
}
private:
// calculates the front vector from the Camera's (updated) Euler Angles
void updateCameraVectors()
{
// calculate the new Front vector
glm::vec3 front;
front.x = cos(glm::radians(Yaw)) * cos(glm::radians(Pitch));
front.y = sin(glm::radians(Pitch));
front.z = sin(glm::radians(Yaw)) * cos(glm::radians(Pitch));
Front = glm::normalize(front);
// also re-calculate the Right and Up vector
Right = glm::normalize(glm::cross(Front, WorldUp)); // normalize the vectors, because their length gets closer to 0 the more you look up or down which results in slower movement.
Up = glm::normalize(glm::cross(Right, Front));
}
};
#endif
9. 参考资料
[1]摄像机 - LearnOpenGL CN (learnopengl-cn.github.io)
[2]OpenGL学习笔记（八）摄像机 - 知乎 (zhihu.com)
标签：06,glm,OpenGL,0.0,float,C++,vec3,pitch,GL 来源： https://www.cnblogs.com/jiujiubashiyi/p/16550836.html
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