剪辑中的 3D 透视投影坐标 space

Question

我正在尝试 3D 投影几个点。据我所知，在将投影矩阵应用于顶点后，该顶点最终出现在剪辑 space 中。此时如果 -w < x,y,z < w 顶点可见，否则它在可见区域之外，需要被裁剪。我遇到的问题是我无法将顶点设置为 -w < x,y,z < w。我一定是做错了什么，但我不知道是什么。我尽量坚持 openGL 约定（坐标系等）。请看看这个SSCCE。

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define DEG2RAD 0.01745329f

// multiplies two matrices
void m3d_matrixMul(const float matrix1[], const float matrix2[], float destination[])
{
    int i, row, col;

    for (i = 15; i >= 0; i--)
    {
        row = i % 4;
        col = i / 4 * 4;
        destination[i] = matrix1[row] * matrix2[col] + matrix1[row + 4] * matrix2[col + 1] + matrix1[row + 8] * matrix2[col + 2] + matrix1[row + 12] * matrix2[col + 3];
    }
}

// multiplies matrix with vector/vertex
void m3d_matVecMul(const float matrix[], const float vector[], float destination[])
{
    float x = vector[0], y = vector[1], z = vector[2], w = vector[3];

    destination[0] = x * matrix[0] + y * matrix[4] + z * matrix[8] + w * matrix[12];
    destination[1] = x * matrix[1] + y * matrix[5] + z * matrix[9] + w * matrix[13];
    destination[2] = x * matrix[2] + y * matrix[6] + z * matrix[10] + w * matrix[14];
    destination[3] = x * matrix[3] + y * matrix[7] + z * matrix[11] + w * matrix[15];
}

// creates the projection matrix (column major)
void m3d_getFrustum(float left, float right, float bottom, float top, float near, float far, float matrix[])
{
    matrix[0] = 2.0f * near / (right - left);
    matrix[1] = 0.0f;
    matrix[2] = 0.0f;
    matrix[3] = 0.0f;
    matrix[4] = 0.0f;
    matrix[5] = 2.0f * near / (top - bottom);
    matrix[6] = 0.0f;
    matrix[7] = 0.0f;
    matrix[8] = (right + left) / (right - left);
    matrix[9] = (top + bottom) / (top - bottom);
    matrix[10] = -(far + near) / (far - near);
    matrix[11] = -1.0f;
    matrix[12] = 0.0f;
    matrix[13] = 0.0f;
    matrix[14] = -2.0f * far * near / (far - near);
    matrix[15] = 0.0f;
}

void m3d_getPerspective(float vfov, float aspect, float near, float far, float matrix[])
{
    float height = tanf(vfov * 0.5f * DEG2RAD) * near;
    float width = aspect * height;
    m3d_getFrustum(-width, width, -height, height, near, far, matrix);
}

void m3d_getIdentity(float matrix[])
{
    matrix[0] = 1.0f;
    matrix[1] = 0.0f;
    matrix[2] = 0.0f;
    matrix[3] = 0.0f;
    matrix[4] = 0.0f;
    matrix[5] = 1.0f;
    matrix[6] = 0.0f;
    matrix[7] = 0.0f;
    matrix[8] = 0.0f;
    matrix[9] = 0.0f;
    matrix[10] = 1.0f;
    matrix[11] = 0.0f;
    matrix[12] = 0.0f;
    matrix[13] = 0.0f;
    matrix[14] = 0.0f;
    matrix[15] = 1.0f;
}

void m3d_translate(float x, float y, float z, float matrix[])
{
    matrix[12] += x;
    matrix[13] += y;
    matrix[14] += z;
}

int main(void)
{
    float vertex[4] = {0.5f, 0.5f, 0.5f, 1.0f};
    float modelView[16];
    float projection[16];
    float mvp[16];
    char buffer[4];

    // initialize model-view matrix with identity matrix
    m3d_getIdentity(modelView);
    // add z-translation to model-view matrix
    m3d_translate(0.0f, 0.0f, -2.0f, modelView);
    // get perspective projection matrix
    m3d_getPerspective(45.0f, 800.0f / 600.0f, -0.5f, -1000.0f, projection);
    // fuse projection and model-view matrix into one matrix
    m3d_matrixMul(projection, modelView, mvp);
    // apply model-view-projection matrix to vertex
    m3d_matVecMul(mvp, vertex, vertex);

    printf("Projected vertex: %f, %f, %f, %f\n", vertex[0], vertex[1], vertex[2], vertex[3]);
    printf("Press ENTER to quit.");
    fgets(buffer, 4, stdin);
    return EXIT_SUCCESS;

    // Projected vertex: 0.905330, 1.207107, 2.502001, 1.500000
}

提前致谢。

Answer 1

好的，我又看了一遍你的代码，我认为你的投影矩阵计算有问题。我已将 m3d_getPerspective 方法替换为 gluPerspective 中的定义，并添加了 printMatrix (pm) 方法。对于某些示例点，结果现在似乎是合理的（下面的代码）。

我没有具体发现你的错误，因为我不确定你为什么要这样做 - 解决方法。一个提示可能是您似乎使用了 tanf 而我现在使用了余切线，但这不是我认为的完整问题。此外，将 near 和 far 设置为负值并不直观，因为它们应该代表距离。你的乘法似乎没问题，我没有意识到你一开始是将顶点复制到 x,y,z,w。

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define DEG2RAD 0.01745329f

void m3d_matrixMul(const float matrix1[], const float matrix2[], float destination[])
{
    int i, row, col;

    for (i = 15; i >= 0; i--)
    {
        row = i % 4;
        col = i / 4 * 4;
        destination[i] = matrix1[row] * matrix2[col] + matrix1[row + 4] * matrix2[col + 1] + matrix1[row + 8] * matrix2[col + 2] + matrix1[row + 12] * matrix2[col + 3];
    }
}

void m3d_matVecMul(const float matrix[], const float vector[], float destination[])
{
    float x = vector[0], y = vector[1], z = vector[2], w = vector[3];

    destination[0] = x * matrix[0] + y * matrix[4] + z * matrix[8] + w * matrix[12];
    destination[1] = x * matrix[1] + y * matrix[5] + z * matrix[9] + w * matrix[13];
    destination[2] = x * matrix[2] + y * matrix[6] + z * matrix[10] + w * matrix[14];
    destination[3] = x * matrix[3] + y * matrix[7] + z * matrix[11] + w * matrix[15];
}

//not used anymore
void m3d_getFrustum(float left, float right, float bottom, float top, float near, float far, float matrix[])
{
    matrix[0] = 2.0f * near / (right - left);
    matrix[1] = 0.0f;
    matrix[2] = 0.0f;
    matrix[3] = 0.0f;
    matrix[4] = 0.0f;
    matrix[5] = 2.0f * near / (top - bottom);
    matrix[6] = 0.0f;
    matrix[7] = 0.0f;
    matrix[8] = (right + left) / (right - left);
    matrix[9] = (top + bottom) / (top - bottom);
    matrix[10] = -(far + near) / (far - near);
    matrix[11] = -1.0f;
    matrix[12] = 0.0f;
    matrix[13] = 0.0f;
    matrix[14] = -2.0f * far * near / (far - near);
    matrix[15] = 0.0f;
}

void m3d_getPerspective(float vfov, float aspect, float near, float far, float matrix[])
{
    float f = 1.0f/tanf(vfov * 0.5f * DEG2RAD);

    matrix[0] = f/aspect;
    matrix[1] = 0.0f;
    matrix[2] = 0.0f;
    matrix[3] = 0.0f;

    matrix[4] = 0.0f;
    matrix[5] = f;
    matrix[6] = 0.0f;
    matrix[7] = 0.0f;

    matrix[8] = 0;
    matrix[9] = 0;
    matrix[10] = (near+far)/(near-far);
    matrix[11] = -1;

    matrix[12] = 0.0f;
    matrix[13] = 0.0f;
    matrix[14] = 2.0f * far * near / (near-far);
    matrix[15] = 0.0f;



}

void m3d_getIdentity(float matrix[])
{
    matrix[0] = 1.0f;
    matrix[1] = 0.0f;
    matrix[2] = 0.0f;
    matrix[3] = 0.0f;
    matrix[4] = 0.0f;
    matrix[5] = 1.0f;
    matrix[6] = 0.0f;
    matrix[7] = 0.0f;
    matrix[8] = 0.0f;
    matrix[9] = 0.0f;
    matrix[10] = 1.0f;
    matrix[11] = 0.0f;
    matrix[12] = 0.0f;
    matrix[13] = 0.0f;
    matrix[14] = 0.0f;
    matrix[15] = 1.0f;
}

void m3d_translate(float x, float y, float z, float matrix[])
{
    matrix[12] += x;
    matrix[13] += y;
    matrix[14] += z;
}

void pm(float* m) {
    printf("%f, %f, %f, %f\n", m[0], m[4], m[8], m[12]);
    printf("%f, %f, %f, %f\n", m[1], m[5], m[9], m[13]);
    printf("%f, %f, %f, %f\n", m[2], m[6], m[10], m[14]);
    printf("%f, %f, %f, %f\n", m[3], m[7], m[11], m[15]);
    printf("\n");
}

int main(void)
{
    //float vertex[4] = {0.0f, 0.0f, -1000.0f, 1.0f}; //point at center & far
    //float vertex[4] = {0.0f, 0.0f, -0.5f, 1.0f}; //point at center & near
    float vertex[4] = {(800.0f/600.0f)*1.0f, 1.0f, -2.0f, 1.0f}; //point at one quater of the x and y range
    float modelView[16];
    float projection[16];
    float mvp[16];
    char buffer[4];

    m3d_getIdentity(modelView);
    pm(modelView);
    m3d_translate(0.0f, 0.0f, 0.0f, modelView); 
    pm(modelView);
    m3d_getPerspective(90.0f, 800.0f / 600.0f, 0.5f, 1000.0f, projection);
    pm(projection);
    m3d_matrixMul(projection, modelView, mvp);
    pm(mvp);
    m3d_matVecMul(mvp, vertex, vertex);

    printf("Projected vertex: %f, %f, %f, %f\n", vertex[0], vertex[1], vertex[2], vertex[3]);
    printf("Press ENTER to quit.");
    fgets(buffer, 4, stdin);
    return EXIT_SUCCESS;
}

剪辑中的 3D 透视投影坐标 space

3D perspective projection coordinates in clip space

c

matrix

3d

projection