GLM 如何处理翻译

OpenGL 数学库 (GLM) 使用以下算法来计算平移矩阵：

//taken from source code
template<typename T, qualifier Q>
GLM_FUNC_QUALIFIER mat<4, 4, T, Q> translate(mat<4, 4, T, Q> const& m, vec<3, T, Q> const& v)
{
    mat<4, 4, T, Q> Result(m);
    Result[3] = m[0] * v[0] + m[1] * v[1] + m[2] * v[2] + m[3];
    return Result;
}

（这里的向量v是一个 3 维向量，矩阵 m 是一个 4X4 矩阵，因为我们使用齐次坐标向量v也是 4 维的）。

以下来自线性代数理论：

Let m有条目：

现在，假设矩阵m给出一些线性变换，也是一个变换矩阵，我们想分别在 X、Y 和 Z 维度上添加 X、Y 和 Z 的平移，如果我没记错的话，我们的方式这是通过形成一个复合矩阵来实现的：

这给出了类似的东西：

现在，我不明白 Translate 的 GLM 函数的作用，因为它的作用如下：

添加平移变换的矩阵，即 m 变为：

现在，这两个矩阵不相等，因此它们会导致不同的变换，所以我很困惑哪个矩阵进行实际翻译，哪个矩阵是正确的，或者算法背后是否隐藏着任何其他想法？

注意：在阅读答案之前，请注意在矩阵的列主表示中，您可以使用以下方式访问矩阵的条目：matrix[column-index][row-index].

Edit

我用来执行转换的源代码：

#include <iostream>
#include <GL/glew.h>
#include <GLFW/glfw3.h>
#include <cmath>
#include <string.h>

#include "glm/glm.hpp"
#include "glm/gtc/matrix_transform.hpp"
#include "glm/gtc/type_ptr.hpp"


// Window Dimensions
const GLint WIDTH=800, HEIGHT=600;
GLuint VAO, VBO, shader;
GLint uniformModel {};
GLint uniformModelRot {};
GLfloat triOffset {};
float triMaxOffset = 0.7f;
bool direction = true;
const float toRadians =  3.14159265f/180.0f;


// vertex shader
static const char* vShader = 
"#version 330\n"
"layout (location = 0) in vec3 pos;\n"
"uniform mat4 model;\n"
"void main(){\n"
"   gl_Position = model * vec4(0.5*pos, 1.0);\n"
"}\n";

// fragment shader
static const char* fShader = ""
"#version 330\n"
"out vec4 color;\n"
"uniform mat4 model;\n"
"void main(){\n"
"   color = model *vec4(1.0, 1.0, 0.0, 1.0);\n"
"}\n";

void AddShader(GLuint theProgram, const char* ShaderCode, GLenum shaderType, std::string info){
    std::cerr <<"INFO: Adding "<<info<<" Shader"<<std::endl;
    GLuint theShader = glCreateShader(shaderType);

    const GLchar* theCode[1];
    theCode[0] = ShaderCode;

    GLint codeLength[1];
    codeLength[0] = strlen(ShaderCode);

    glShaderSource(theShader, 1, theCode, codeLength);
    glCompileShader(theShader);

    GLint result =0;
    GLchar eLog[1024] ={0};

    glGetShaderiv(theShader, GL_COMPILE_STATUS, &result);
    if(!result){
        glGetShaderInfoLog(shader, sizeof(eLog), NULL, eLog);
        std::cerr<<"Error compiling program"<<std::endl;
        return;
    }
    glAttachShader(theProgram, theShader);

}

void CompileShader(){
    shader = glCreateProgram();
    if(!shader){
        std::cerr<<"Error creating shader"<<std::endl;
        return;
    }

    AddShader(shader, vShader, GL_VERTEX_SHADER, "vertex");
    AddShader(shader, fShader, GL_FRAGMENT_SHADER, "fragment");

    GLint result =0;
    GLchar eLog[1024] ={0};

    glLinkProgram(shader);
    glGetProgramiv(shader, GL_LINK_STATUS, &result);
    if(!result){
        glGetProgramInfoLog(shader, sizeof(eLog), NULL, eLog);
        std::cerr<<"Error linking program"<<std::endl;
        return;
    }

    glValidateProgram(shader);
    glGetProgramiv(shader, GL_VALIDATE_STATUS, &result);
    if(!result){
        glGetProgramInfoLog(shader, sizeof(eLog), NULL, eLog);
        std::cerr<<"Error Validating program"<<std::endl;
        return;
    }

    uniformModel = glGetUniformLocation(shader,"model");

}

void CreateTriangles(){
    GLfloat vertices[]={
        -1.0f, -1.0f, 0.0f,
        1.0f, -1.0f, 0.0f,
        0.0f, 1.0f, 0.0f
    };

    glGenVertexArrays(1, &VAO);
    glBindVertexArray(VAO);

        glGenBuffers(1, &VBO);
        glBindBuffer(GL_ARRAY_BUFFER, VBO);
        glBufferData(GL_ARRAY_BUFFER, sizeof(GLfloat)*9,vertices, GL_STATIC_DRAW);
        glVertexAttribPointer(0,3,GL_FLOAT,GL_FALSE,0,0);
        glEnableVertexAttribArray(0);

    glBindBuffer(GL_ARRAY_BUFFER, 0);
    glBindVertexArray(0);
}


int main(){
    //initialize GLFW
    if(!glfwInit()){
        std::cerr << "GLFW initialization failed!" << std::endl;
        glfwTerminate();
        return 1;
    }

    //Setup GLFW window properties
    //openGL version
    glfwWindowHint(GLFW_CONTEXT_VERSION_MAJOR, 3);
    glfwWindowHint(GLFW_CONTEXT_VERSION_MINOR, 3);
    // core profile = no backward compatibility
    glfwWindowHint(GLFW_OPENGL_PROFILE, GLFW_OPENGL_CORE_PROFILE);
    //allow forward compatibility
    glfwWindowHint(GLFW_OPENGL_FORWARD_COMPAT, GL_TRUE);

    GLFWwindow *mainWindow = glfwCreateWindow(WIDTH, HEIGHT, "TEST WINDOW", NULL, NULL);

    if(!mainWindow){
        std::cerr << "GLFW Window creation failed" << std::endl;
        glfwTerminate();
        return 1;
    }

    // get Buffer size information
    int bufferWidth, bufferHeight;
    glfwGetFramebufferSize(mainWindow, &bufferWidth, &bufferHeight);
    // set context for GLEW to use
    glfwMakeContextCurrent(mainWindow);

    // allow modern extension features

    if(glewInit()!=GLEW_OK){
        std::cerr << "GLEW initialization failed" << std::endl;
        glfwDestroyWindow(mainWindow);
        glfwTerminate();
        return 1;
    }

    // setup viewport size
    glViewport(0, 0, bufferWidth, bufferHeight);
    CreateTriangles();
    CompileShader();




    while(!glfwWindowShouldClose(mainWindow)){
        // get and handle user input events
        glfwPollEvents();

        glClearColor(1.0f, 0.0f, 0.0f, 1.0);
        glClear(GL_COLOR_BUFFER_BIT);

        if(direction){
            triOffset += 0.05f;
        }else{
            triOffset -= 0.05f;
        }

        if(abs(triOffset) >= triMaxOffset){
            direction = !direction;
        }

        glUseProgram(shader);

        glm::mat4 modelMatrix(1.0f); 
        modelMatrix = glm::translate(modelMatrix, glm::vec3(triOffset, 0.0f, 0.0f));

        glUniformMatrix4fv(uniformModel, 1, GL_FALSE,glm::value_ptr(modelMatrix));
            glBindVertexArray(VAO);
                glDrawArrays(GL_TRIANGLES,0,3);
            glBindVertexArray(0);
        glUseProgram(0);
        // swap buffers
        glfwSwapBuffers(mainWindow);
    }

    return 0;
}

OpenGL 数学 (GLM)是基于OpenGL 着色语言 (GLSL). What glm::translate实际上就是建立一个平移矩阵并将输入矩阵乘以平移。它计算m*t的意思是GLSL 向量和矩阵运算:

mat<4, 4, T, Q> Result(m);
Result[3] = m[0] * v[0] + m[1] * v[1] + m[2] * v[2] + m[3];

（在下面的Result被替换为R)

Note, m[0] * v[0]将该列的每个分量相乘m[0]由标量v[0]。结果是向量(m[0][0]*v[0], m[0][1]*v[0], m[0][2]*v[0], m[0][3]*v[0]).

So R[3] = m[0]*v[0] + m[1]*v[1] + m[2]*v[2] + m[3]是相同的

R[3][0] = m[0][0] * v[0] + m[1][0] * v[1] + m[2][0] * v[2] + m[3][0]
R[3][1] = m[0][1] * v[0] + m[1][1] * v[1] + m[2][1] * v[2] + m[3][1]
R[3][2] = m[0][2] * v[0] + m[1][2] * v[1] + m[2][2] * v[2] + m[3][2]
R[3][3] = m[0][3] * v[0] + m[1][3] * v[1] + m[2][3] * v[2] + m[3][3]

glm::translate实际上计算：

vh = (v[0], v[1], v[2], 1)
R = m
R[3][0] = dot( (m[0][0], m[1][0], m[2][0], m[3][0]), vh )
R[3][1] = dot( (m[0][1], m[1][1], m[2][1], m[3][1]), vh )
R[3][2] = dot( (m[0][2], m[1][2], m[2][2], m[3][2]), vh )
R[3][3] = dot( (m[0][3], m[1][3], m[2][3], m[3][3]), vh )

上面的代码计算点积的行数来自m, by vh. vh是翻译的第四列t。注意平移矩阵t定义为：

     c0  c1  c2  c3 
---------------------  
r0:   1   0   0  v[0]  
r1:   0   1   0  v[1]
r2:   0   0   0  v[2]
r3:   0   0   0  1  

4x4 矩阵的串联 (R = m*t） 是个点积的行数m和列t并可表示为： （看OpenGL 着色语言 4.60 规范 - 5.10。向量和矩阵运算)

for i from 0 to 3
    for j fro 0 to 3
        R[i][j] = dot( (m[0][j], m[1][j], m[2][j], m[3][j]), t[i] )

Where dot(a, b) == a[0]*b[0] + a[1]*b[1] + a[2]*b[2] + a[3]*b[3],
(m[0][j], m[1][j], m[2][j], m[3][j]) is the j第-行m and
t[i] is i第 列t.

For glm::translate复制就足够了R[0], R[1] and R[2] from m[0], m[1] and m[2].

例如为了 （i=0, j=0):

R[0][0] = dot( (m[0][0], m[1][0], m[2][0], m[3][0]), t[0] )
R[0][0] = dot( (m[0][0], m[1][0], m[2][0], m[3][0]), (1, 0, 0, 0) )
R[0][0] = m[0][0] * 1 + m[1][0] * 0 + m[2][0] * 0 + m[3][0]) * 0
R[0][0] = m[0][0]

GLM矩阵（如 OpenGL 矩阵）按列主序存储。如果您在调试器中研究矩阵，可能会导致混乱。

如果你有矩阵

     c0  c1  c2  c3 
-------------------  
r0:  Xx  Yx  Zx  Tx  
r1:  Xy  Yy  Zy  Ty 
r2:  Xz  Yz  Zz  Tz  
r3:   0   0   0   1  

那么 4*4 OpenGL 矩阵的内存图像如下所示：

Xx, Xy, Xz, 0, Yx, Yy, Yz, 0, Zx, Zy, Zz, 0, Tx, Ty, Tz, 1

如果您在调试器中调查它，它可能看起来像：

[ [ Xx, Xy, Xz, 0 ],
  [ Yx, Yy, Yz, 0 ],
  [ Zx, Zy, Zz, 0 ],
  [ Tx, Ty, Tz, 1 ] ]