openGL之glsl入门6--画三维图魔方、圆柱体

     这一章介绍坐标变换与矩阵相关内容，对应红宝书第5章内容，并通过两个简单的例子展示矩阵变换的效果。

1. 坐标变换

    变换的类型有多种，包括视图、模型、投影、视口变换等，概念可以参照红宝书5.1章节，概念虽不同，但最终作用到顶点坐标的方式是一致的，都是对顶点坐标进行运算（直接加减乘除或者使用矩阵运算），弄清楚这些概念，有利于清晰描述变换的效果，虽然不同的变化都可以得到相同的结果(显示器上输出)，但还是应该在合适的场景使用合适的变换达成目的，使程序更易理解。

  下面介绍最基本的坐标变换，这里用的是二维，三维的原理是一致的，只是复杂一些。 

1.1 平移(translate)

    平移很好理解，沿哪个轴平移，对应的坐标直接加对应长度就可以了

    如坐标(x，y)沿x轴和y轴分别平移Tx与Ty(正轴方向为正，负轴方向为负)，shader描述如下:

gl_Position = vec4(position.x +Tx， position.y + Ty，position.z，1.0); 

1.2 缩放(scale)

    缩放也很好理解，沿哪个轴方向缩放，乘上对应的值即可

    如坐标(x,y)沿x轴和y轴分别缩放Sx与Sy（缩写<1，放大>1），shader描述如下:

gl_Position = vec4(position.x * Sx，position.y * Sy，position.z ，1.0);

    使用这种方法进行平移与缩放前面的章节都有用到。 

1.3 旋转(rotate)

    旋转稍微复杂一些，2维情况下，绕原点旋转（即3维情况的Z轴），暗含的条件是该点到原点的距离R不变，旋转公式的推导用的是基本的三角公式，推导过程比较简短，这里写一下。

    坐标点( x ,y ) 可以表示为( R*cosα ，R * sinα )其中α为点到x轴的夹角（逆时针方向）。即：

x = R * cosα;y = R * sinα;

    逆时针 旋转β角度后得到( x',y' )，可以用以下公式表示与推导：

x' =R *cos(α+β) = R * (cosαcosβ - sinαsinβ) = xcosβ - ysinβ;y' = R *sin(α+β) = R * (sinαcosβ + cosαsinβ) = ycosβ + xsinβ;

    所以点( x ,y ) 旋转角度θ后，shader可以写成这样：

gl_Position = vec4(position. x * cosθ - position. y *sinθ ,position. y * cosθ + position. x * sinθ ,position.z ,1.0); 

    以上几种坐标变换时都需要指定坐标轴的（如没有指定，一定要搞清楚函数的默认值，如scale(0.5)，要弄清楚是x轴缩小一半还是整体缩小一半），后续可以看到坐标变换需要为每个轴指定变换值，如rotate(x,y,z)则表明绕x轴旋转x度，y轴旋转y度等。

    以上几种坐标变换是可以叠加的，可以同时平移、缩放、旋转，这几个操作，需要注意叠加顺序，先平移后缩放与先缩放后平移得出的坐标肯定不一样了，缩放的同时把平移的坐标也进行缩放了，旋转和平移一样的存在先后问题。 

2. 矩阵坐标变换

    使用上述的方式实现平移和缩放还比较简单，用来旋转就会复杂不少，如果多种操作组合的话，最终的公式就比较复杂了，很容易出错，实际编程过程中，坐标变换都是通过矩阵运算来完成的，而openGL显然对矩阵运算的支持非常好（glsl有很多向量vec*与矩阵mat*的定义及相关运算），前面的例子是供大家比照用的，如果一开始就使用矩阵运算来进行坐标变换，对没有计算机图形学基础的人来说要难上手一些，初步了解矩阵不难，内容也不多，这里介绍一下。

    矩阵基本的计算公式：

    矩阵相乘规则：m行n列的矩阵A与n行p列的矩阵B相乘，会得到m行p列的矩阵C，所以矩阵相乘要求矩阵A的列与矩阵B的行一致，得到的结果是，矩阵C的每个值为矩阵A的每一行的元素与矩阵B的每一列元素相乘后的和。上面的矩阵A的第一行位a b c d，矩阵B的第一列为x y z w，相乘的结果是对应位置相乘再相加，即ax+by+cz+dw。

    矩阵可以多级相乘，C = A*B

    矩阵相乘不满足交换律（A*B != B*A 如上面的矩阵,交换后A行与B列都不一致了,没法乘），但可以使用结合律,即 A *( B* C) = (A * B) * C。

    可以通过矩阵来实现平移、缩放、旋转等，看下面平移的例子

    可以看到矩阵A 对角线方向为1.0，除最后一列外，其他都是0.0，这个就是平移矩阵，矩阵A最后一列3个值对应x轴、y轴及z轴方向的平移。图中矩阵相乘结果是点（x，y，z）沿y轴平移0.3，公式y' = 0.0*x+1.0*y+0.0*z+0.3*1.0 = y+0.3。

    缩放矩阵改动的是对角线的值，旋转矩阵稍微复杂一些，但与平移矩阵的原理是一致的，这些都可以参考红宝书5.2章节。

 

注意：GLSL里的矩阵，实际上都使用一维数组表示，且都是列优先(展开成一维数组时,先列后行,与c语言2维数组先行后列不一致)，如果自己构建矩阵，像上面的矩阵A，使用一维数组构建时，代码如下(注意0.3的位置)：

float MTA[] = {1.0,0.0,0.0,0.0,0.0,1.0,0.0,0.0,0.0,0.0,1.0,0.0,0.0,0.3.0.0,1.0}; 

3. 矩阵操作库

    自己构建各种矩阵当然是可行的，但我们一般不这么做，一般使用现成的库或者自己编写库来实现各种矩阵的构建，红宝书里用的vmath库，vmath库就一个头文件，你可以在红宝书源码里include目录中找到，即vmath.h,你也可以在http://bartipan.net/vmath/上找到最新版本。为保证源码的完整性，本文最后还是贴一下vmath的源码（为方便阅读，放到最后面，有点长）。

    vmath库的api如下：

/**@brief: 生成四维旋转 矩阵@param angle：为旋转的角度，单位为度。@param x,y,z：为对应xyz轴的布尔值变量，如x为1.0表示沿x轴旋转。@param return：返回四维旋转矩阵*/mat4 rotate(float angle,float x,float y, float z );

    这个函数用法可以参考openGL老api中的glRotatef，glRotatef是直接作用到3D模型上，而rotate返回一个四维旋转矩阵，需与顶点坐标相乘达成旋转目的。 

/**@brief: 生成四维平移矩阵@param x,y,z：为对应xyz轴平移的坐标，负数为负轴方向。@param return：返回四维平移 矩阵*/mat4  translate(float x,float y, float z );

    可参考openGL老API中的glTranslatef函数。 

/**@brief: 生成四维缩放矩阵@param x,y,z：为对应xyz轴缩放的比例，>1放大,<1缩小。@param return：返回四维缩放矩阵*/mat4  scale(float x,float y, float z );

    可参考openGL老API中的glscalef函数。

注意：原型为scale（float s）即只有一个参数的时候，是整体缩小模型，而不只是缩小x轴。

 

mat4 frustum(float left, float right, float bottom, float top, float near, float far)；mat4 perspective(float fovy， float aspect, float near , float far )；mat4 lookat(vec3 eye, vec3 center, vec3 up)；

    这3个函数都是做视图变换的，分别对应openGL的glFrustum，gluPerspective及gluLookAt函数，这几个函数的参数会比较复杂，网上相关的介绍很多，用法与rotate等函数是一致的。

    多矩阵相乘，最好放到客户端进行（用CPU去算），如放到shader中做，每个点都需算一遍，性能多少有些浪费。多矩阵相乘时，顺序需注意，避免顺序不对导致效果不符合要求的情况。 

4. 画魔方立方体

    魔方是规则的立方体，比较好绘制，代码如下：

#include <stdio.h>#include <stdlib.h>#include <string.h>#include <GL/glew.h>#include <GL/glut.h>/* 该头文件请到第6章下载*/#include "vmath.h"using namespace vmath;#define POINT_CNT       6       /* 每条边对应的点数*/typedef struct{    GLfloat x;    GLfloat y;    GLfloat z;} POINT_S;static const GLchar * vertex_source =    "#version 330 core\n"    "uniform mat4 pos_matrix;\n"    "uniform mat4 face_matrix;\n"    "uniform vec3 color_vec;\n"    "layout (location = 0) in vec3 in_position;\n"    "out vec3 color_frag;\n"    "void main(void)\n"    "{\n"        "       if( abs(in_position.z) == 0.49)\n"    "           color_frag = vec3(0.0,0.0,0.5);\n"        "       else\n"    "           color_frag = color_vec;\n"        "       gl_Position = pos_matrix * face_matrix * vec4(in_position,1.0);\n"    "}\n";static const GLchar * frag_source =    "#version 330 core\n"    "in vec3 color_frag;\n"    "out vec4 color;\n"    "void main(void)\n"    "{\n"    "    color = vec4(color_frag,1.0);\n"    "}\n";void loadShader(GLuint program, GLuint type, const GLchar * source){    GLint status = 0;    const GLchar * shaderSource[] = {source};    GLuint shader = glCreateShader(type);    glShaderSource(shader, 1, shaderSource, 0);    glCompileShader(shader);    glGetShaderiv(shader, GL_COMPILE_STATUS, &status);    glAttachShader(program, shader);}GLuint vao, vbo, ebo;mat4 pos_matrix,face_matrix;GLuint pos_matrix_idx,face_matrix_idx,color_vec_idx;void display(void){    glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);    glBindBuffer(GL_ARRAY_BUFFER, vao);    glUniformMatrix4fv(pos_matrix_idx, 1, GL_FALSE, pos_matrix);        static float pos[6][6] =        {                        {0.0f, 0.0f, 0.0f, 1.0f,1.0f,0.0f}, /* 顶面黄*/                        {90.0f, 1.0f, 0.0f, 0.0f,0.0f,1.0f}, /* 侧面绿*/                        {-90.0f, 1.0f, 0.0f, 0.0f,1.0f,0.0f},/* 侧面红*/                        {90.0f, 0.0f, 1.0f, 1.0f,0.0f,0.0f}, /* 侧面蓝*/                        {-90.0f, 0.0f, 1.0f, 1.0f,0.0f,1.0f},/* 侧面橙*/                        {180.0f, 1.0f, 0.0f, 1.0f,1.0f,1.0f}, /* 底面白*/        };        for( int i =0;i< 6;i++)        {                /* 告知各个面颜色*/                glUniform3fv(color_vec_idx,1,&pos[i][3]);                /* 坐标只定义了顶面,其他面通过旋转得到*/                face_matrix = vmath::rotate(pos[i][0], pos[i][1],pos[i][2],0.0f);            glUniformMatrix4fv(face_matrix_idx, 1, GL_FALSE, face_matrix);                /* 使用正方形绘制,每个正方形个顶点*/                glDrawElements(GL_QUADS, 10*4, GL_UNSIGNED_INT, (GLvoid *)(0));        }    glutSwapBuffers();}void init(void){    int i = 0, j = 0;    POINT_S * p = NULL;        /* 总宽度,一个格子,线宽*/        float quad[POINT_CNT] = {0.025f,0.325f,0.350f,0.650f,0.675f,0.975f};        POINT_S groud[4] =        {                        {-0.5f,0.5f,0.49f},                        {0.5f,0.5f,0.49f},                        {0.5f,-0.5f,0.49f},                        {-0.5f,-0.5f,0.49f},        };    /* 只分配一个面的顶点数据,其他面通过旋转绘制,额外的个点画面的底色*/    POINT_S vertex_list[POINT_CNT*POINT_CNT + 4];    for (i = 0; i < POINT_CNT ; i++)    {                for(j = 0;j< POINT_CNT;j++)                {                p = &vertex_list[i*POINT_CNT + j];                        p->x = quad[j] - 0.5f; /* quad从开始,挪中间*/                        p->y = quad[i] - 0.5f;                        p->z = 0.5;                }    }        memcpy(vertex_list+POINT_CNT*POINT_CNT,groud,sizeof(POINT_S)*4);    /* 绘制索引,一个面个正方形*/    GLuint index_list[10][4] =    {                {36,37,38,39},  /* 先画底色框,再画上面的格子*/                {0,1,7,6},{2,3,9,8},{4,5,11,10},                {12,13,19,18},{14,15,21,20},{16,17,23,22},                {24,25,31,30},{26,27,33,32},{28,29,35,34},   };    GLuint program = glCreateProgram();    loadShader(program, GL_VERTEX_SHADER, vertex_source);    loadShader(program, GL_FRAGMENT_SHADER, frag_source);    glLinkProgram(program);    glUseProgram(program);    glGenBuffers(1, &vao);    glBindBuffer(GL_ARRAY_BUFFER, vao);    glGenBuffers(1, &vbo);    glBindBuffer(GL_ARRAY_BUFFER, vbo);    glBufferData(GL_ARRAY_BUFFER, sizeof(vertex_list), vertex_list, GL_STATIC_DRAW);    glVertexAttribPointer(0, 3, GL_FLOAT, GL_FALSE, 0, (GLvoid *)0);    glEnableVertexAttribArray(0);    glGenBuffers(1, &ebo);    glBindBuffer(GL_ELEMENT_ARRAY_BUFFER, ebo);    glBufferData(GL_ELEMENT_ARRAY_BUFFER, sizeof(index_list), index_list, GL_STATIC_DRAW);    /* 定义旋转数组,主要存放当前旋转信息*/        face_matrix_idx = glGetUniformLocation(program, "face_matrix");        color_vec_idx = glGetUniformLocation(program, "color_vec");        pos_matrix_idx = glGetUniformLocation(program, "pos_matrix");    pos_matrix = vmath::rotate(30.0f, 1.0f, 1.0f,0.0f);    glClearColor(0.5f, 0.5f, 1.0f, 1.0f);    glClearDepth(1.0);    glEnable(GL_DEPTH_TEST);    glLineWidth(1.0);}void keyboard(unsigned char key, int x, int y){    switch (key)    {        case '-':            pos_matrix *= vmath::scale(0.95f);            break;        case '=':        case '+':            pos_matrix *= vmath::scale(1.05f);            break;        default:            break;    }    glutPostRedisplay();}void specialKey(GLint key, GLint x, GLint y){    float step = 2.0f;    switch (key)    {        case GLUT_KEY_UP:            pos_matrix *= vmath::rotate(step, 1.0f, 0.0f, 0.0f);            break;        case GLUT_KEY_DOWN:            pos_matrix *= vmath::rotate(-1.0f * step, 1.0f, 0.0f, 0.0f);            break;        case GLUT_KEY_LEFT:            pos_matrix *= vmath::rotate(step, 0.0f, 1.0f, 0.0f);            break;        case GLUT_KEY_RIGHT:            pos_matrix *= vmath::rotate(-1.0f * step, 0.0f, 1.0f, 0.0f);            break;        default:            break;    }    glutPostRedisplay();}int main(int argc, char * argv[]){    glutInit(&argc, argv);    glutInitDisplayMode(GLUT_RGBA | GLUT_DOUBLE | GLUT_DEPTH);    glutInitWindowPosition(200, 200);    glutInitWindowSize(400, 400);    glutCreateWindow("MagicCube");    glewInit();    init();    glutDisplayFunc(display);    glutKeyboardFunc(keyboard);    glutSpecialFunc(specialKey);    glutMainLoop();    return 0;}

效果如下：

 

    魔方的每个面由底色正方形和9个小的正方形组成，程序里只设置了一个面的顶点坐标，其他面都是通过旋转画出来的，画魔方各个面的时候，需先告知每个面的旋转角度和颜色，代码中的角度和颜色通过一个数组给出。相比二维绘图，绘制三维图都用的三维坐标，相关参数（glVertexAttribPointer 数据宽度，顶点着色器in变量的类型等）需匹配，查看时，旋转才能看到效果，其他的与二维绘制差不多。

    有一个细节需注意，客户端代码里并没有给出底色的颜色，而是顶点着色器代码里做判断来设置底色的颜色（配合背景的z坐标一起使用）。

 if( abs(in_position.z) == 0.49)\n"     color_frag = vec3(0.0,0.0,0.5);\n" else     color_frag = color_vec;

    为什么是0.49，并没有特殊的含义，就只是利用了深度测试特性，把底色与上面的小正方形错开，避免在同一平面绘制两种颜色，同一区域画颜色时（即把底色坐标的z坐标也设置成0.5时），有两种情况：

1. 开启深度测试（glEnable(GL_DEPTH_TEST);），深度一致的情况下，底色和魔方格子画在同一区域时，颜色是随机的，下左图就是这种情况。

2. 不开启深度测试，先绘制的会作为底色，后面绘制的会叠加在上面，下右图就是这种情况，无论如何转动，最后绘的白色总能显示出来。

    实际使用中，如果不是做颜色混合，同一区域（坐标平面与区域一致）不要重复绘制，浪费资源，且结果不好预期。 

5. 画圆柱体

代码如下：

#include <stdio.h>#include <stdlib.h>#include <string.h>#include <GL/glew.h>#include <GL/glut.h>#include "vmath.h"using namespace vmath;/* 绘制步长,单位为度*/#define STEP                    6#define SAMPLE_CNT              (360/STEP)static const GLchar * vertex_source =    "#version 330 core\n"    "uniform mat4 in_matrix;\n"    "layout (location = 0) in vec3 in_position;\n"    "layout (location = 1) in vec3 in_color;\n"    "flat out vec3 frag_color;\n"    "void main(void)\n"    "{\n"    "    gl_Position = in_matrix * vec4(in_position,1.0);\n"    "    frag_color = in_color;\n"    "}\n";static const GLchar * frag_source =    "#version 330 core\n"    "flat in vec3 frag_color;\n"    "out vec4 color;\n"    "void main(void)\n"    "{\n"    "    color = vec4(frag_color,1.0);\n"    "}\n";void loadShader(GLuint program, GLuint type, const GLchar * source){    GLint status = 0;    const GLchar * shaderSource[] = {source};    GLuint shader = glCreateShader(type);    glShaderSource(shader, 1, shaderSource, 0);    glCompileShader(shader);    glGetShaderiv(shader, GL_COMPILE_STATUS, &status);    glAttachShader(program, shader);}GLuint vao, vbo;GLuint matrix_idx;mat4 pos_matrix;GLuint ebo;void display(void){    glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);    glBindBuffer(GL_ARRAY_BUFFER, vao);    glUniformMatrix4fv(matrix_idx, 1, GL_FALSE, pos_matrix);                /* 画圆柱体侧面*/    glDrawArrays(GL_TRIANGLE_STRIP, 0, 2 * SAMPLE_CNT);        /* 画圆柱顶面和底面,底面使用索引偏移量的方式绘制*/        glDrawElements(GL_TRIANGLE_FAN, SAMPLE_CNT, GL_UNSIGNED_INT, (GLvoid *)(0));        glDrawElementsBaseVertex(GL_TRIANGLE_FAN, SAMPLE_CNT, GL_UNSIGNED_INT, (GLvoid *)(0),1);            glutSwapBuffers();}void init(void){    int i = 0;    float p = 0.0, r = 0.5;    GLfloat vertex_list[2 * 360 / STEP][3];    GLfloat color_list[2 * 360 / STEP][3];        GLuint index_list[360 / STEP];        /* 确定顶面和底面的坐标*/    for (i = 0; i < SAMPLE_CNT * 2; i += 2)    {        p = i * STEP * 3.14 / 180;        vertex_list[i][0] = cos(p) * r;        vertex_list[i][1] = sin(p) * r;        vertex_list[i][2] = 0.5f;        vertex_list[i + 1][0] = cos(p) * r;        vertex_list[i + 1][1] = sin(p) * r;        vertex_list[i + 1][2] = -0.5f;    }        /* 确定每个点的坐标*/    for (i = 0; i < SAMPLE_CNT * 2; i++)    {        if ((i / 2) % 2 == 0)        {            color_list[i][0] = 0.5f;            color_list[i][1] = 0.0f;            color_list[i][2] = 1.0f;        }        else        {            color_list[i][0] = 1.0f;            color_list[i][1] = 1.0f;            color_list[i][2] = 0.0f;        }    }        /* 确定顶面的索引*/    for (i = 0; i < SAMPLE_CNT; i++)    {                index_list[i] = i+2;    }       GLuint program = glCreateProgram();    loadShader(program, GL_VERTEX_SHADER, vertex_source);    loadShader(program, GL_FRAGMENT_SHADER, frag_source);    glLinkProgram(program);    glUseProgram(program);    glGenBuffers(1, &vao);    glBindBuffer(GL_ARRAY_BUFFER, vao);    glGenBuffers(1, &vbo);    glBindBuffer(GL_ARRAY_BUFFER, vbo);    glBufferData(GL_ARRAY_BUFFER, sizeof(vertex_list) + sizeof(color_list), NULL, GL_STATIC_DRAW);    glBufferSubData(GL_ARRAY_BUFFER, 0, sizeof(vertex_list), vertex_list);    glBufferSubData(GL_ARRAY_BUFFER, sizeof(vertex_list), sizeof(color_list), color_list);    glVertexAttribPointer(0, 3, GL_FLOAT, GL_FALSE, 0, (GLvoid *)0);    glVertexAttribPointer(1, 3, GL_FLOAT, GL_FALSE, 0, (GLvoid *)(sizeof(vertex_list)));    glEnableVertexAttribArray(0);    glEnableVertexAttribArray(1);    glGenBuffers(1, &ebo);    glBindBuffer(GL_ELEMENT_ARRAY_BUFFER, ebo);    glBufferData(GL_ELEMENT_ARRAY_BUFFER, sizeof(index_list), index_list, GL_STATIC_DRAW);    matrix_idx = glGetUniformLocation(program, "in_matrix");    pos_matrix = vmath::rotate(30.0f, 1.0f, 1.0f, 0.0f);        /* 使用非渐变的方式绘制颜色*/        glProvokingVertex(GL_FIRST_VERTEX_CONVENTION);     glClearColor(0.5f, 0.5f, 1.0f, 1.0f);    glClearDepth(1.0);    glEnable(GL_DEPTH_TEST);}void keyboard(unsigned char key, int x, int y){    switch (key)    {        case '-':            pos_matrix *= vmath::scale(0.9f);            break;        case '=':        case '+':            pos_matrix *= vmath::scale(1.1f);            break;        default:            break;    }    glutPostRedisplay();}void specialKey(GLint key, GLint x, GLint y){    float step = 2.0f;    switch (key)    {        case GLUT_KEY_UP:            pos_matrix *= vmath::rotate(step, 1.0f, 0.0f, 0.0f);            break;        case GLUT_KEY_DOWN:            pos_matrix *= vmath::rotate(-1.0f * step, 1.0f, 0.0f, 0.0f);            break;        case GLUT_KEY_LEFT:            pos_matrix *= vmath::rotate(step, 0.0f, 1.0f, 0.0f);            break;        case GLUT_KEY_RIGHT:            pos_matrix *= vmath::rotate(-1.0f * step, 0.0f, 1.0f, 0.0f);            break;        default:            break;    }    glutPostRedisplay();}int main(int argc, char * argv[]){    glutInit(&argc, argv);    glutInitDisplayMode(GLUT_RGBA | GLUT_DOUBLE | GLUT_DEPTH);    glutInitWindowPosition(200, 200);    glutInitWindowSize(500, 500);    glutCreateWindow("Cylinder");    glewInit();    init();    glutDisplayFunc(display);    glutKeyboardFunc(keyboard);    glutSpecialFunc(specialKey);    glutMainLoop();    return 0;}

效果如下：

  

在绘制的过程中，顶面、底面和侧面的坐标的坐标是复用的，底面的坐标使用了glDrawElementsBaseVertex函数，采用相对索引方式绘制。

附vmath.h源码

(红宝书里用的vmath库，vmath库就一个头文件，你可以在红宝书源码里include目录中找到，即vmath.h,你也可以在http://bartipan.net/vmath/上找到最新版本。)

#ifndef __VMATH_H__#define __VMATH_H__#define _USE_MATH_DEFINES  1 // Include constants defined in math.h#include <math.h>namespace vmath{template <typename T> inline T radians(T angleInRadians){return angleInRadians * static_cast<T>(180.0/M_PI);}template <const bool cond>class ensure{public:    inline ensure() { switch (false) { case false: case cond: break; } }};template <typename T, const int len> class vecN;template <typename T, const int len>class vecN{public:    typedef class vecN<T,len> my_type;    // Default constructor does nothing, just like built-in types    inline vecN()    {        // Uninitialized variable    }    // Copy constructor    inline vecN(const vecN& that)    {        assign(that);    }    // Construction from scalar    inline vecN(T s)    {        int n;        for (n = 0; n < len; n++)        {            data[n] = s;        }    }    // Assignment operator    inline vecN& operator=(const vecN& that)    {        assign(that);        return *this;    }    inline vecN operator+(const vecN& that) const    {        my_type result;        int n;        for (n = 0; n < len; n++)            result.data[n] = data[n] + that.data[n];        return result;    }    inline vecN& operator+=(const vecN& that)    {        return (*this = *this + that);    }    inline vecN operator-() const    {        my_type result;        int n;        for (n = 0; n < len; n++)            result.data[n] = -data[n];        return result;    }    inline vecN operator-(const vecN& that) const    {        my_type result;        int n;        for (n = 0; n < len; n++)            result.data[n] = data[n] - that.data[n];        return result;    }    inline vecN& operator-=(const vecN& that)    {        return (*this = *this - that);    }    inline vecN operator*(const vecN& that) const    {        my_type result;        int n;        for (n = 0; n < len; n++)            result.data[n] = data[n] * that.data[n];        return result;    }    inline vecN& operator*=(const vecN& that)    {        return (*this = *this * that);    }    inline vecN operator*(const T& that) const    {        my_type result;        int n;        for (n = 0; n < len; n++)            result.data[n] = data[n] * that;        return result;    }    inline vecN& operator*=(const T& that)    {        assign(*this * that);        return *this;    }    inline vecN operator/(const vecN& that) const    {        my_type result;        int n;        for (n = 0; n < len; n++)            result.data[n] = data[n] * that.data[n];        return result;    }    inline vecN& operator/=(const vecN& that)    {        assign(*this * that);        return *this;    }    inline vecN operator/(const T& that) const    {        my_type result;        int n;        for (n = 0; n < len; n++)            result.data[n] = data[n] / that;        return result;    }    inline vecN& operator/(const T& that)    {        assign(*this / that);    }    inline T& operator[](int n) { return data[n]; }    inline const T& operator[](int n) const { return data[n]; }    inline static int size(void) { return len; }    inline operator const T* () const { return &data[0]; }protected:    T data[len];    inline void assign(const vecN& that)    {        int n;        for (n = 0; n < len; n++)            data[n] = that.data[n];    }};template <typename T>class Tvec2 : public vecN<T,2>{public:    typedef vecN<T,2> base;    // Uninitialized variable    inline Tvec2() {}    // Copy constructor    inline Tvec2(const base& v) : base(v) {}    // vec2(x, y);    inline Tvec2(T x, T y)    {        base::data[0] = x;        base::data[1] = y;    }};template <typename T>class Tvec3 : public vecN<T,3>{public:    typedef vecN<T,3> base;    // Uninitialized variable    inline Tvec3() {}    // Copy constructor    inline Tvec3(const base& v) : base(v) {}    // vec3(x, y, z);    inline Tvec3(T x, T y, T z)    {        base::data[0] = x;        base::data[1] = y;        base::data[2] = z;    }    // vec3(v, z);    inline Tvec3(const Tvec2<T>& v, T z)    {        base::data[0] = v[0];        base::data[1] = v[1];        base::data[2] = z;    }    // vec3(x, v)    inline Tvec3(T x, const Tvec2<T>& v)    {        base::data[0] = x;        base::data[1] = v[0];        base::data[2] = v[1];    }};template <typename T>class Tvec4 : public vecN<T,4>{public:    typedef vecN<T,4> base;    // Uninitialized variable    inline Tvec4() {}    // Copy constructor    inline Tvec4(const base& v) : base(v) {}    // vec4(x, y, z, w);    inline Tvec4(T x, T y, T z, T w)    {        base::data[0] = x;        base::data[1] = y;        base::data[2] = z;        base::data[3] = w;    }    // vec4(v, z, w);    inline Tvec4(const Tvec2<T>& v, T z, T w)    {        base::data[0] = v[0];        base::data[1] = v[1];        base::data[2] = z;        base::data[3] = w;    }    // vec4(x, v, w);    inline Tvec4(T x, const Tvec2<T>& v, T w)    {        base::data[0] = x;        base::data[1] = v[0];        base::data[2] = v[1];        base::data[3] = w;    }    // vec4(x, y, v);    inline Tvec4(T x, T y, const Tvec2<T>& v)    {        base::data[0] = x;        base::data[1] = y;        base::data[2] = v[0];        base::data[3] = v[1];    }    // vec4(v1, v2);    inline Tvec4(const Tvec2<T>& u, const Tvec2<T>& v)    {        base::data[0] = u[0];        base::data[1] = u[1];        base::data[2] = v[0];        base::data[3] = v[1];    }    // vec4(v, w);    inline Tvec4(const Tvec3<T>& v, T w)    {        base::data[0] = v[0];        base::data[1] = v[1];        base::data[2] = v[2];        base::data[3] = w;    }    // vec4(x, v);    inline Tvec4(T x, const Tvec3<T>& v)    {        base::data[0] = x;        base::data[1] = v[0];        base::data[2] = v[1];        base::data[3] = v[2];    }};typedef Tvec2<float> vec2;typedef Tvec2<int> ivec2;typedef Tvec2<unsigned int> uvec2;typedef Tvec2<double> dvec2;typedef Tvec3<float> vec3;typedef Tvec3<int> ivec3;typedef Tvec3<unsigned int> uvec3;typedef Tvec3<double> dvec3;typedef Tvec4<float> vec4;typedef Tvec4<int> ivec4;typedef Tvec4<unsigned int> uvec4;typedef Tvec4<double> dvec4;template <typename T, int n>static inline const vecN<T,n> operator * (T x, const vecN<T,n>& v){    return v * x;}template <typename T>static inline const Tvec2<T> operator / (T x, const Tvec2<T>& v){    return Tvec2<T>(x / v[0], x / v[1]);}template <typename T>static inline const Tvec3<T> operator / (T x, const Tvec3<T>& v){    return Tvec3<T>(x / v[0], x / v[1], x / v[2]);}template <typename T>static inline const Tvec4<T> operator / (T x, const Tvec4<T>& v){    return Tvec4<T>(x / v[0], x / v[1], x / v[2], x / v[3]);}template <typename T, int len>static inline T dot(const vecN<T,len>& a, const vecN<T,len>& b){    int n;    T total = T(0);    for (n = 0; n < len; n++)    {        total += a[n] * b[n];    }    return total;}template <typename T>static inline vecN<T,3> cross(const vecN<T,3>& a, const vecN<T,3>& b){    return Tvec3<T>(a[1] * b[2] - b[1] * a[2],                    a[2] * b[0] - b[2] * a[0],                    a[0] * b[1] - b[0] * a[1]);}template <typename T, int len>static inline T length(const vecN<T,len>& v){    T result(0);    for (int i = 0; i < v.size(); ++i)    {        result += v[i] * v[i];    }    return (T)sqrt(result);}template <typename T, int len>static inline vecN<T,len> normalize(const vecN<T,len>& v){    return v / length(v);}template <typename T, int len>static inline T distance(const vecN<T,len>& a, const vecN<T,len>& b){    return length(b - a);}template <typename T, const int w, const int h>class matNM{public:    typedef class matNM<T,w,h> my_type;    typedef class vecN<T,h> vector_type;    // Default constructor does nothing, just like built-in types    inline matNM()    {        // Uninitialized variable    }    // Copy constructor    inline matNM(const matNM& that)    {        assign(that);    }    // Construction from element type    // explicit to prevent assignment from T    explicit inline matNM(T f)    {        for (int n = 0; n < w; n++)        {            data[n] = f;        }    }    // Construction from vector    inline matNM(const vector_type& v)    {        for (int n = 0; n < w; n++)        {            data[n] = v;        }    }    // Assignment operator    inline matNM& operator=(const my_type& that)    {        assign(that);        return *this;    }    inline matNM operator+(const my_type& that) const    {        my_type result;        int n;        for (n = 0; n < w; n++)            result.data[n] = data[n] + that.data[n];        return result;    }    inline my_type& operator+=(const my_type& that)    {        return (*this = *this + that);    }    inline my_type operator-(const my_type& that) const    {        my_type result;        int n;        for (n = 0; n < w; n++)            result.data[n] = data[n] - that.data[n];        return result;    }    inline my_type& operator-=(const my_type& that)    {        return (*this = *this - that);    }    // Matrix multiply.    // TODO: This only works for square matrices. Need more template skill to make a non-square version.    inline my_type operator*(const my_type& that) const    {        ensure<w == h>();        my_type result(0);        for (int j = 0; j < w; j++)        {            for (int i = 0; i < h; i++)            {                T sum(0);                for (int n = 0; n < w; n++)                {                    sum += data[n][i] * that[j][n];                }                result[j][i] = sum;            }        }        return result;    }    inline my_type& operator*=(const my_type& that)    {        return (*this = *this * that);    }    inline vector_type& operator[](int n) { return data[n]; }    inline const vector_type& operator[](int n) const { return data[n]; }    inline operator T*() { return &data[0][0]; }    inline operator const T*() const { return &data[0][0]; }    inline matNM<T,h,w> transpose(void) const    {        matNM<T,h,w> result;        int x, y;        for (y = 0; y < w; y++)        {            for (x = 0; x < h; x++)            {                result[x][y] = data[y][x];            }        }        return result;    }    static inline my_type identity()    {        ensure<w == h>();        my_type result(0);        for (int i = 0; i < w; i++)        {            result[i][i] = 1;        }        return result;    }    static inline int width(void) { return w; }    static inline int height(void) { return h; }protected:    // Column primary data (essentially, array of vectors)    vecN<T,h> data[w];    // Assignment function - called from assignment operator and copy constructor.    inline void assign(const matNM& that)    {        int n;        for (n = 0; n < w; n++)            data[n] = that.data[n];    }};/*template <typename T, const int N>class TmatN : public matNM<T,N,N>{public:    typedef matNM<T,N,N> base;    typedef TmatN<T,N> my_type;    inline TmatN() {}    inline TmatN(const my_type& that) : base(that) {}    inline TmatN(float f) : base(f) {}    inline TmatN(const vecN<T,4>& v) : base(v) {}    inline my_type transpose(void)    {        my_type result;        int x, y;        for (y = 0; y < h; y++)        {            for (x = 0; x < h; x++)            {                result[x][y] = data[y][x];            }        }        return result;    }};*/template <typename T>class Tmat4 : public matNM<T,4,4>{public:    typedef matNM<T,4,4> base;    typedef Tmat4<T> my_type;    inline Tmat4() {}    inline Tmat4(const my_type& that) : base(that) {}    inline Tmat4(const base& that) : base(that) {}    inline Tmat4(const vecN<T,4>& v) : base(v) {}    inline Tmat4(const vecN<T,4>& v0,                 const vecN<T,4>& v1,                 const vecN<T,4>& v2,                 const vecN<T,4>& v3)    {        base::data[0] = v0;        base::data[1] = v1;        base::data[2] = v2;        base::data[3] = v3;    }};typedef Tmat4<float> mat4;typedef Tmat4<int> imat4;typedef Tmat4<unsigned int> umat4;typedef Tmat4<double> dmat4;static inline mat4 frustum(float left, float right, float bottom, float top, float n, float f){    mat4 result(mat4::identity());    if ((right == left) ||        (top == bottom) ||        (n == f) ||        (n < 0.0) ||        (f < 0.0))       return result;    result[0][0] = (2.0f * n) / (right - left);    result[1][1] = (2.0f * n) / (top - bottom);    result[2][0] = (right + left) / (right - left);    result[2][1] = (top + bottom) / (top - bottom);    result[2][2] = -(f + n) / (f - n);    result[2][3]= -1.0f;    result[3][2] = -(2.0f * f * n) / (f - n);    result[3][3] =  0.0f;    return result;}static inline mat4 perspective(float fovy /* in degrees */, float aspect, float n, float f){float  top = n * tan(radians(0.5f*fovy)); // bottom = -topfloat  right = top * aspect; // left = -rightreturn frustum(-right, right, -top, top, n, f);}template <typename T>static inline Tmat4<T> lookat(vecN<T,3> eye, vecN<T,3> center, vecN<T,3> up){    const Tvec3<T> f = normalize(center - eye);    const Tvec3<T> upN = normalize(up);    const Tvec3<T> s = cross(f, upN);    const Tvec3<T> u = cross(s, f);    const Tmat4<T> M = Tmat4<T>(Tvec4<T>(s[0], u[0], -f[0], T(0)),                                Tvec4<T>(s[1], u[1], -f[1], T(0)),                                Tvec4<T>(s[2], u[2], -f[2], T(0)),                                Tvec4<T>(T(0), T(0), T(0), T(1)));    return M * translate<T>(-eye);}template <typename T>static inline Tmat4<T> translate(T x, T y, T z){    return Tmat4<T>(Tvec4<T>(1.0f, 0.0f, 0.0f, 0.0f),                    Tvec4<T>(0.0f, 1.0f, 0.0f, 0.0f),                    Tvec4<T>(0.0f, 0.0f, 1.0f, 0.0f),                    Tvec4<T>(x, y, z, 1.0f));}template <typename T>static inline Tmat4<T> translate(const vecN<T,3>& v){    return translate(v[0], v[1], v[2]);}template <typename T>static inline Tmat4<T> scale(T x, T y, T z){    return Tmat4<T>(Tvec4<T>(x, 0.0f, 0.0f, 0.0f),                    Tvec4<T>(0.0f, y, 0.0f, 0.0f),                    Tvec4<T>(0.0f, 0.0f, z, 0.0f),                    Tvec4<T>(0.0f, 0.0f, 0.0f, 1.0f));}template <typename T>static inline Tmat4<T> scale(const Tvec4<T>& v){    return scale(v[0], v[1], v[2]);}template <typename T>static inline Tmat4<T> scale(T x){    return Tmat4<T>(Tvec4<T>(x, 0.0f, 0.0f, 0.0f),                    Tvec4<T>(0.0f, x, 0.0f, 0.0f),                    Tvec4<T>(0.0f, 0.0f, x, 0.0f),                    Tvec4<T>(0.0f, 0.0f, 0.0f, 1.0f));}template <typename T>static inline Tmat4<T> rotate(T angle, T x, T y, T z){    Tmat4<T> result;    const T x2 = x * x;    const T y2 = y * y;    const T z2 = z * z;    float rads = float(angle) * 0.0174532925f;    const float c = cosf(rads);    const float s = sinf(rads);    const float omc = 1.0f - c;    result[0] = Tvec4<T>(T(x2 * omc + c), T(y * x * omc + z * s), T(x * z * omc - y * s), T(0));    result[1] = Tvec4<T>(T(x * y * omc - z * s), T(y2 * omc + c), T(y * z * omc + x * s), T(0));    result[2] = Tvec4<T>(T(x * z * omc + y * s), T(y * z * omc - x * s), T(z2 * omc + c), T(0));    result[3] = Tvec4<T>(T(0), T(0), T(0), T(1));    return result;}template <typename T>static inline Tmat4<T> rotate(T angle, const vecN<T,3>& v){    return rotate<T>(angle, v[0], v[1], v[2]);}#ifdef min#undef min#endiftemplate <typename T>static inline T min(T a, T b){    return a < b ? a : b;}#ifdef max#undef max#endiftemplate <typename T>static inline T max(T a, T b){    return a >= b ? a : b;}template <typename T, const int N>static inline vecN<T,N> min(const vecN<T,N>& x, const vecN<T,N>& y){    vecN<T,N> t;    int n;    for (n = 0; n < N; n++)    {        t[n] = min(x[n], y[n]);    }    return t;}template <typename T, const int N>static inline vecN<T,N> max(const vecN<T,N>& x, const vecN<T,N>& y){    vecN<T,N> t;    int n;    for (n = 0; n < N; n++)    {        t[n] = max<T>(x[n], y[n]);    }    return t;}template <typename T, const int N>static inline vecN<T,N> clamp(const vecN<T,N>& x, const vecN<T,N>& minVal, const vecN<T,N>& maxVal){    return min<T>(max<T>(x, minVal), maxVal);}template <typename T, const int N>static inline vecN<T,N> smoothstep(const vecN<T,N>& edge0, const vecN<T,N>& edge1, const vecN<T,N>& x){    vecN<T,N> t;    t = clamp((x - edge0) / (edge1 - edge0), vecN<T,N>(T(0)), vecN<T,N>(T(1)));    return t * t * (vecN<T,N>(T(3)) - vecN<T,N>(T(2)) * t);}template <typename T, const int N, const int M>static inline matNM<T,N,M> matrixCompMult(const matNM<T,N,M>& x, const matNM<T,N,M>& y){    matNM<T,N,M> result;    int i, j;    for (j = 0; j < M; ++j)    {        for (i = 0; i < N; ++i)        {            result[i][j] = x[i][j] * y[i][j];        }    }    return result;}template <typename T, const int N, const int M>static inline vecN<T,N> operator*(const vecN<T,M>& vec, const matNM<T,N,M>& mat){    int n, m;    vecN<T,N> result(T(0));    for (m = 0; m < M; m++)    {        for (n = 0; n < N; n++)        {            result[n] += vec[m] * mat[n][m];        }    }    return result;}};#endif /* __VMATH_H__ */