數學基礎
球面參數方程
球面的參數曲線可以用球座標表示,引入參數u,v,其中v是球面點與原點的連線與z軸正向的夾角,u表示連線在xy平面的投影與x軸正向的夾角,如下圖所示:
則球面參數方程可以表示爲:
球面法向量
已知球面的參數方程以後,很容易求得給定點的法向量,分別對u和v方向求偏導數,然後對兩個所得向量進行叉積即可:
實現細節
已知參數方程以後,需要進行離散,分別設定u和v的步長:ustep、vstep。然後通過不同的u和v,求得座標系中點的實際座標(x,y,z),在實現中有一點需要注意的是:
u=0與u=v這兩條線上點的是球體的兩個上下極點,所以進行渲染時需要區分,其中如圖中間段的離散點可以按照四邊形進行渲染,而上下兩段則需要按照三角形進行渲染。
代碼描述:
這裏只是繪製了球面,如果想繪製球體,只用在渲染時,加入點的法向量即可。
點的數據結構:
class Point
{
public:
Point(){};
Point(double a,double b,double c):x(a),y(b),z(c){};
public:
double x;
double y;
double z;
};
參數座標(u,v)轉換成時機座標(x,y,z)函數
Point getPoint(double u,double v)
{
double x = sin(PI*v)*cos(PI2*u);
double y = sin(PI*v)*sin(PI2*u);
double z = cos(PI*v);
return Point(x,y,z);
}
繪製球面
void drawWire()
{
double ustep = 1/(double)uStepsNum, vstep = 1/(double)vStepNum;
double u = 0,v = 0;
//繪製下端三角形組
for(int i = 0;i<uStepsNum;i++)
{
glBegin(GL_LINE_LOOP);
Point a = getPoint(0,0);
glVertex3d(a.x,a.y,a.z);
Point b = getPoint(u,vstep);
glVertex3d(b.x,b.y,b.z);
Point c = getPoint(u+ustep,vstep);
glVertex3d(c.x,c.y,c.z);
u += ustep;
glEnd();
}
//繪製中間四邊形組
u = 0, v = vstep;
for(int i=1;i<vStepNum-1;i++)
{
for(int j=0;j<uStepsNum;j++)
{
glBegin(GL_LINE_LOOP);
Point a = getPoint(u,v);
Point b = getPoint(u+ustep,v);
Point c = getPoint(u+ustep,v+vstep);
Point d = getPoint(u,v+vstep);
glVertex3d(a.x,a.y,a.z);
glVertex3d(b.x,b.y,b.z);
glVertex3d(c.x,c.y,c.z);
glVertex3d(d.x,d.y,d.z);
u += ustep;
glEnd();
}
v += vstep;
}
//繪製下端三角形組
u = 0;
for(int i=0;i<uStepsNum;i++)
{
glBegin(GL_LINE_LOOP);
Point a = getPoint(0,1);
Point b = getPoint(u,1-vstep);
Point c = getPoint(u+ustep,1-vstep);
glVertex3d(a.x,a.y,a.z);
glVertex3d(b.x,b.y,b.z);
glVertex3d(c.x,c.y,c.z);
glEnd();
}
}
初始化設置與繪圖函數
#define PI 3.14159265358979323846
#define PI2 6.28318530717958647692
GLsizei width = 600,height = 600;
int uStepsNum = 50,vStepNum = 50;
void init()
{
glClearColor(0,1,1,1);
glClearDepth(1.0);
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
glClearColor(0,1,1,1);
glClearDepth(1.0);
glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT);
GLfloat light_position [ ] = { 1.0f, 1.0f, 1.0f, 0.0f };
GLfloat light_ambient [ ] = { 0.2f, 0.2f, 0.2f, 0.2f };
GLfloat light_diffuse [ ] = { 0.5f, 0.5f, 0.5f, 0.2f };
GLfloat light_specular [ ] = { 0.5f, 0.5f, 0.5f, 0.2f };
glLightfv(GL_LIGHT0, GL_POSITION, light_position);
glLightfv(GL_LIGHT0, GL_AMBIENT, light_ambient);
glLightfv(GL_LIGHT0, GL_DIFFUSE, light_diffuse);
glLightfv(GL_LIGHT0, GL_SPECULAR, light_specular);
glEnable (GL_COLOR_MATERIAL);
glEnable(GL_LIGHTING);
glEnable(GL_LIGHT0);
glEnable(GL_AUTO_NORMAL);
glEnable (GL_NORMALIZE);
glEnable(GL_DEPTH_TEST);
glDepthFunc (GL_LESS);
}
void displayFunc()
{
glMatrixMode(GL_MODELVIEW);
glLoadIdentity();
glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT);
glColor3f(1.0,0.0,0.0);
glPointSize(1.0);
glRotated(30,1,0,0);
glRotated(60,0,1,0);
glRotated(90,0,0,1);
drawWire();
glutSwapBuffers();
}
主函數
int main(int argc,char* argv[])
{
glutInit(&argc,argv);
glutInitDisplayMode(GLUT_DOUBLE|GLUT_RGB);
glutInitWindowPosition(100,100);
glutInitWindowSize(width,height);
glutCreateWindow("Sphere");
init();
glutDisplayFunc(displayFunc);
glutMainLoop();
}