0，引入

前续：
Q79：怎么用三角形网格（Triangle Mesh）细分曲面
http://blog.csdn.net/libing_zeng/article/details/60600404
Q80：平坦着色（Flat Shading）和平滑着色（Smooth Shading）——“Q79：怎么用三角形网格（Triangle Mesh）细分曲面”（补充）
http://blog.csdn.net/libing_zeng/article/details/60760296
【修正】问题五十五：怎么用ray tracing画Utah teapot (bicubic bezier patches)
http://blog.csdn.net/libing_zeng/article/details/69258616

这一章节，我们以Utah Teapot（Bicubic Bezier Patches）为例来进行说明。

1，理论分析

我们知道Utah Teapot是由若干个Bicubic Bezier Patch拼接而成（我们这里用到的数据是32-patches、306-vertices版本）。
关于Bezier曲面的介绍，参考：“问题五十四：怎么用ray tracing画参数方程表示的曲面（2）—— bezier surface”
http://blog.csdn.net/libing_zeng/article/details/54565264

Bezier曲面的本质是：参数方程表示的曲面。这个“参数方程”就是两条Bezier曲线的的张量积。

在求得球面的参数方程之后，用三角形网格细分该曲面的方式可以参考：
“Q79：怎么用三角形网格（Triangle Mesh）细分曲面”
http://blog.csdn.net/libing_zeng/article/details/60600404

2，C++代码实现

2.1 读取teapot数据文件

本人是将文件读取函数移植到Vector3D.cpp中

// ----------------------------------------------------------  get_teapot_data
// read utah teapot with 32 patches and 306 vertices from external filebool get_teapot_data(int (&patches)[32][16], float (&vertices)[306][3]) {
    ifstream infile( ".\\teaset\\teapot");
    char str[100];
    char *token;
    int item_num;
    int patch_num = 0;
    int vertex_num = 0;
    int flag = 0;
    int tokens_i[16];
    float tokens_f[3];
    while (infile >> str) {        item_num = 0;
        token = strtok(str, ",");
        if ((flag == 0) || (flag == 1)) {            sscanf(token, "%d", &(tokens_i[item_num]));
            if (tokens_i[0] == 32) {                flag = 1;
            }
            else if (tokens_i[0] == 306) {                flag = 2;
            }
            else {                patches[patch_num][item_num] = tokens_i[item_num];
                item_num ++;
            }
            token = strtok(NULL, ",");
        }        while (token != NULL) {            if (flag == 1) {                sscanf(token, "%d", &(tokens_i[item_num]));
                patches[patch_num][item_num] = tokens_i[item_num];
            }
            if (flag == 2) {                sscanf(token, "%f", &(tokens_f[item_num]));
                vertices[vertex_num][item_num] = tokens_f[item_num];
            }
            item_num ++;
            token = strtok(NULL, ",");
        }
        if ((flag == 1) && (tokens_i[0] != 32)) {patch_num ++;}
        if (flag == 2) {            if ((tokens_i[0] == 306)) {tokens_i[0] = 0;}
            else {vertex_num ++;}
        }
    }
    infile.close();
    return true;
}// ----------------------------------------------------------  matrix_4_4_multiply_4_4void matrix_4_4_multiply_4_4(const float matrix1[4][4], const float matrix2[4][4], float (&result)[4][4]) {
    for (int k=0; k<4; k++) {        for (int i=0; i<4; i++) {            result[i][k] = 0.0;
            for (int j=0; j<4; j++) {                result[i][k] = result[i][k] + matrix1[i][j]*matrix2[j][k];
            }
        }
    }
}

2.2 矩阵运算

计算过程会用到“矩阵运算”（没有使用“矩阵类”，仍旧使用二维数组表示矩阵）。矩阵运算的相关函数也是在Vector3D.cpp中实现：

// ----------------------------------------------------------  matrix_4_4_multiply_4_4void matrix_4_4_multiply_4_4(const float matrix1[4][4], const float matrix2[4][4], float (&result)[4][4]) {
    for (int k=0; k<4; k++) {        for (int i=0; i<4; i++) {            result[i][k] = 0.0;
            for (int j=0; j<4; j++) {                result[i][k] = result[i][k] + matrix1[i][j]*matrix2[j][k];
            }
        }
    }
}// ----------------------------------------------------------  matrix_1_4_multiply_4_4void matrix_1_4_multiply_4_4(const float matrix1[1][4], const float matrix2[4][4], float (&result)[1][4]) {
    for (int k=0; k<4; k++) {        result[0][k] = 0.0;
        for (int j=0; j<4; j++) {            result[0][k] = result[0][k] + matrix1[0][j]*matrix2[j][k];
        }
    }
}// ----------------------------------------------------------  matrix_1_4_multiply_4_1void matrix_1_4_multiply_4_1(const float matrix1[1][4], const float matrix2[4][1], float &result) {
    result = 0.0;
    for (int j=0; j<4; j++) {        result = result + matrix1[0][j]*matrix2[j][0];
    }
}

2.3 细分Bezier曲面

像之前一样，细分函数在Grid.cpp中实现。

// ------------------------------------  tessellate_flat_bezier_patches  ---------------------------------------
// tesselate a cubic bezier patch into flat triangles that are stored directly in the gridvoid
Grid::tessellate_flat_bezier_patches(const int horizontal_steps, const int vertical_steps,float vertices[306][3], int patches[32][16], const int patches_num) {Vector3D patches_vertices[patches_num][16];float matrix_c_x[4][4], matrix_c_y[4][4], matrix_c_z[4][4];float   points_x[4][4],   points_y[4][4],   points_z[4][4];float points_x_t[4][4], points_y_t[4][4], points_z_t[4][4];float        uuu[1][4],        vvv[4][1];float       uuu1[1][4],       vvv1[4][1];float      xxx_t[1][4],      yyy_t[1][4],      zzz_t[1][4];float              xxx,              yyy,              zzz;float             xxx1,             yyy1,             zzz1;float             xxx2,             yyy2,             zzz2;float             xxx3,             yyy3,             zzz3;float matrix_t[4][4] = {{ 1,  0,  0, 0},{-3,  3,  0, 0},{ 3, -6,  3, 0},{-1,  3, -3, 1}};float matrix[4][4]   = {{1, -3,  3, -1},{0,  3, -6,  3},{0,  0,  3, -3},{0,  0,  0,  1}};int ip1, ip2, ipv;for (int i=0; i<patches_num; i++) {for (int j=0; j<16; j++) {ip1 = int(j/4);ip2 = int(j%4);ipv = patches[i][j] - 1;points_x[ip1][ip2] = vertices[ipv][0];points_y[ip1][ip2] = vertices[ipv][1];points_z[ip1][ip2] = vertices[ipv][2];}matrix_4_4_multiply_4_4(matrix_t, points_x, points_x_t);matrix_4_4_multiply_4_4(points_x_t, matrix, matrix_c_x);matrix_4_4_multiply_4_4(matrix_t, points_y, points_y_t);matrix_4_4_multiply_4_4(points_y_t, matrix, matrix_c_y);matrix_4_4_multiply_4_4(matrix_t, points_z, points_z_t);matrix_4_4_multiply_4_4(points_z_t, matrix, matrix_c_z);for (int k = 0; k <= vertical_steps - 1; k++) {for (int m = 0; m <= horizontal_steps - 1; m++) {uuu[0][0] = 1.0;uuu[0][1] = float(m) / float(horizontal_steps);uuu[0][2] = uuu[0][1] * uuu[0][1];uuu[0][3] = uuu[0][1] * uuu[0][2];vvv[0][0] = 1.0;vvv[1][0] = float(k) / float(vertical_steps);vvv[2][0] = vvv[0][1] * vvv[0][1];vvv[3][0] = vvv[0][1] * vvv[0][2];uuu1[0][0] = 1.0;uuu1[0][1] = float(m+1) / float(horizontal_steps);uuu1[0][2] = uuu1[0][1] * uuu1[0][1];uuu1[0][3] = uuu1[0][1] * uuu1[0][2];vvv1[0][0] = 1.0;vvv1[1][0] = float(k+1) / float(vertical_steps);vvv1[2][0] = vvv1[0][1] * vvv1[0][1];vvv1[3][0] = vvv1[0][1] * vvv1[0][2];matrix_1_4_multiply_4_4(uuu, matrix_c_x, xxx_t);matrix_1_4_multiply_4_1(xxx_t, vvv, xxx);matrix_1_4_multiply_4_4(uuu, matrix_c_y, yyy_t);matrix_1_4_multiply_4_1(yyy_t, vvv, yyy);matrix_1_4_multiply_4_4(uuu, matrix_c_z, zzz_t);matrix_1_4_multiply_4_1(zzz_t, vvv, zzz);matrix_1_4_multiply_4_4(uuu, matrix_c_x, xxx_t);matrix_1_4_multiply_4_1(xxx_t, vvv1, xxx1);matrix_1_4_multiply_4_4(uuu, matrix_c_y, yyy_t);matrix_1_4_multiply_4_1(yyy_t, vvv1, yyy1);matrix_1_4_multiply_4_4(uuu, matrix_c_z, zzz_t);matrix_1_4_multiply_4_1(zzz_t, vvv1, zzz1);matrix_1_4_multiply_4_4(uuu1, matrix_c_x, xxx_t);matrix_1_4_multiply_4_1(xxx_t, vvv, xxx2);matrix_1_4_multiply_4_4(uuu1, matrix_c_y, yyy_t);matrix_1_4_multiply_4_1(yyy_t, vvv, yyy2);matrix_1_4_multiply_4_4(uuu1, matrix_c_z, zzz_t);matrix_1_4_multiply_4_1(zzz_t, vvv, zzz2);matrix_1_4_multiply_4_4(uuu1, matrix_c_x, xxx_t);matrix_1_4_multiply_4_1(xxx_t, vvv1, xxx3);matrix_1_4_multiply_4_4(uuu1, matrix_c_y, yyy_t);matrix_1_4_multiply_4_1(yyy_t, vvv1, yyy3);matrix_1_4_multiply_4_4(uuu1, matrix_c_z, zzz_t);matrix_1_4_multiply_4_1(zzz_t, vvv1, zzz3);// define the first trianglePoint3D v0(xxx2, yyy2, zzz2);Point3D v1(xxx3, yyy3, zzz3);Point3D v2( xxx,  yyy,  zzz);Triangle* triangle_ptr1 = new Triangle(v0, v1, v2);objects.push_back(triangle_ptr1);// define the second trianglev0 = Point3D(xxx1, yyy1, zzz1);v1 = Point3D( xxx,  yyy,  zzz);v2 = Point3D(xxx3, yyy3, zzz3);Triangle* triangle_ptr2 = new Triangle(v0, v1, v2);objects.push_back(triangle_ptr2);}}}}

如上代码的截图分析如下：

3，测试图形

3.1 测试代码

3.2 输出图形

teapot_32_0,0,100_rx-90,ts(0,-2,0)_light,directional(10,-20,-20)_20,20

teapot_32_0,0,100_rx-90,ts(0,-2,0)_light,directional(10,-20,-20)_40,40_190s_ns1

teapot_32_0,0,100_rx-90,ts(0,-2,0)_light,directional(10,-20,-20)_40,40_207s_ns16

teapot_32_0,0,100_rx-90,ts(0,-2,0)_light,directional(10,-20,-20)_40,40_176s_ns16_marble,p0.1

teapot_32_0,0,100_rx-90,ts(0,-2,0)_light,directional(10,-20,-20)_40,40_183s_ns16_marble,p4

teapot_32_0,0,100_rx-90,ts(0,-2,0)_light,directional(10,-20,-20)_40,40_226s_ns16_sandstone,p0.1

teapot_32_0,0,100_rx-90,ts(0,-2,0)_light,directional(10,-20,-20)_40,40_234s_ns16_wood

4，其他说明

完整代码链接：http://download.csdn.net/detail/libing_zeng/9804597

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