OpenGL数学库(GLM)使用以下算法来计算转换矩阵:
//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;
}
(这里向量v是3维向量,矩阵m是4X4矩阵,因为我们使用齐次坐标,向量v也是4维)。
以下来自线性代数理论:
让m具有以下条目:现在,假设矩阵m提供了一些线性变换,并且它也是一个变换矩阵,并且我们想分别在X,Y和Z维度上添加X,Y和Z的平移。没错,我们这样做的方法是形成一个复合矩阵:
并且添加了转换转换的矩阵,即m变为:现在,这两个矩阵不相等,因此它们将导致不同的转换,因此我对实际转换是哪个矩阵,哪个是正确的矩阵感到困惑,或者算法背后是否隐藏了其他任何想法?
非常感谢您的帮助。预先感谢。
编辑:用于执行转换的源代码:
#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;
}
GLSL Vector and Matrix Operations
注意,转换矩阵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 )
定义为:
t
[4x4矩阵的串联( 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
是R = m*t
的行和Dot product的列的m
,可以表示为:]
t
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] )
,dot(a, b) == a[0]*b[0] + a[1]*b[1] + a[2]*b[2] + a[3]*b[3]
是(m[0][j], m[1][j], m[2][j], m[3][j])
的第j行,m
是t[i]
的第i列。
对于t
,从glm::translate
,R[0]
和R[1]
复制R[2]
,m[0]
和m[1]
就足够了。例如对于(
m[2]
,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]
矩阵(作为OpenGL矩阵)以主列存储。如果您在调试器中调查矩阵,可能会导致混淆。
如果有矩阵
然后,一个4 * 4 OpenGL矩阵的内存图像如下所示:
c0 c1 c2 c3
-------------------
r0: Xx Yx Zx Tx
r1: Xy Yy Zy Ty
r2: Xz Yz Zz Tz
r3: 0 0 0 1
如果在调试器中进行调查,则可能类似于:
Xx, Xy, Xz, 0, Yx, Yy, Yz, 0, Zx, Zy, Zz, 0, Tx, Ty, Tz, 1