Q97：怎么用三角形网格细分Bezier曲面——以Utah Teapot为例

0，引入

前续：
Q79：怎么用三角形网格（Triangle Mesh）细分曲面
http://blog.youkuaiyun.com/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.youkuaiyun.com/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.youkuaiyun.com/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.youkuaiyun.com/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.youkuaiyun.com/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 file

bool 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] ==