OpenGL繪制簡單的參數曲線（二）——三次Bezier曲線

　　今天我們來介紹三次Bezier曲線，這曲線網上資料非常多，我這里只是簡單介紹下原理。

　　在二維空間中（三維也類似），給定n+1個點P₀、P₁、... 、P_n。參數t的n次的Bezier曲線是：

圖1

　　我們根據上面式子可以推出一次、二次、三次貝塞爾曲線，下面是一次貝塞爾曲線：

 

圖2

　　下面是二次貝塞爾曲線，表示的是從P₀P₁線段取Q₀，P₁P₂線段取Q₁，每一個Q₀Q₁都是曲線的切向量：

圖3

　　下面是三次貝塞爾曲線，表示的是從P₀P₁線段取Q₀，P₁P₂線段取Q₁，P₂P₃線段取Q₂，再從Q₀Q₁取R₀，Q₁Q₂取R₁，每一個R₀R₁都是曲線的切向量：

 

圖4

　　這樣就給出了公式，下面貼出三次Beizer曲線的代碼，同樣可以手動調節參數，大家參考一下。

 

#include <math.h>
#include <gl/glut.h>
#include <iostream>
using namespace std;
  
int xCoord[4], yCoord[4];
int num = 0;
 
bool finishBeizer = false;
bool mouseLeftDown = false;
bool mouseRightDown = false;
 
/*計算Bezier曲線*/
void Bezier(int n)
{
    float f1, f2, f3, f4;
    float deltaT = 1.0 / n;
    float T;
  
    glBegin(GL_LINE_STRIP);
    for (int i = 0; i <= n; i++) {
  
        T = i * deltaT;
  
        f1 = (1-T) *(1- T) * (1-T);
        f2 = 3 * T * (1-T) * (1- T);
        f3 = 3 * T * T * (1-T);
        f4 = T * T * T;
  
        glVertex2f( f1*xCoord[0] + f2*xCoord[1] + f3*xCoord[2] + f4*xCoord[3],
            f1*yCoord[0] + f2*yCoord[1] + f3*yCoord[2] + f4*yCoord[3]);
    }
    glEnd();
}
  
/*用鼠標進行繪制，完成后可改變控制點，拖動即可*/
void display(){
    glClear(GL_COLOR_BUFFER_BIT);
  
    glLineWidth(1.5);
    glColor3f (1.0, 0.0, 0.0);
    glBegin(GL_LINE_STRIP);
    for (int i = 0; i < num; i++)
        glVertex3f (xCoord[i], yCoord[i], 0.0);
    glEnd();
  
    glColor3f (0.0, 0.0, 1.0);
    if (num == 4)
        Bezier(20);
	
	glPointSize(10.0f);
	glBegin(GL_POINTS);
	glVertex2f(xCoord[0], yCoord[0]);
	glVertex2f(xCoord[1], yCoord[1]);
	glVertex2f(xCoord[2], yCoord[2]);
	glVertex2f(xCoord[3], yCoord[3]);
	glEnd(); 

    glFlush();
    glutSwapBuffers();
}
  
void init()
{
    glClearColor(1.0, 1.0, 1.0, 0.0);
    glShadeModel(GL_FLAT);
}
  
void myReshape(int w, int h)
{
    glViewport(0, 0, (GLsizei)w, (GLsizei)h);
    glMatrixMode(GL_PROJECTION);
    glLoadIdentity();
    gluOrtho2D(0.0, (GLsizei)w, (GLsizei)h, 0.0);
    glMatrixMode(GL_MODELVIEW);
    glLoadIdentity();
}
  
void mouse(int button, int state, int x, int y)
{
    if (!finishBeizer)
    {
        if (button == GLUT_LEFT_BUTTON && state == GLUT_DOWN)
        {
            xCoord[num] = x;
            yCoord[num] = y;
            num++;
 
            if (num == 4)
                finishBeizer = true;
 
            glutPostRedisplay();
        }
    }
    else
    {
        if (button == GLUT_LEFT_BUTTON && state == GLUT_DOWN)
        {
            mouseLeftDown = true;
        }
 
        if (button == GLUT_LEFT_BUTTON && state == GLUT_UP)
        {
            mouseLeftDown = false;
        }
 
        if (button == GLUT_RIGHT_BUTTON && state == GLUT_DOWN)
        {
            mouseRightDown = true;
        }
 
        if (button == GLUT_RIGHT_BUTTON && state == GLUT_UP)
        {
            mouseRightDown = false;
        }
    }
}
 
double distance(int x1, int y1, int x2, int y2)
{
	return sqrt((x1-x2) * (x1 -x2) + (y1-y2) * (y1-y2));
}

void motion(int x, int y)
{
    if (mouseLeftDown)
    {
		if (distance(xCoord[1], yCoord[1], x, y) < 20)
		{
			xCoord[1] = x;
			yCoord[1] = y;
		}

		if (distance(xCoord[2], yCoord[2], x, y) < 20)
		{
			xCoord[2] = x;
			yCoord[2] = y;
		}
    }
 
 
    glutPostRedisplay();
}
  
int  main(int argc, char** argv)
{
    glutInit(&argc, argv);
    glutInitDisplayMode (GLUT_DOUBLE | GLUT_RGB );
    glutInitWindowSize (450, 450);
    glutInitWindowPosition (200, 200);
    glutCreateWindow ("hello");
    init ();
  
    glutDisplayFunc(display);
    glutReshapeFunc(myReshape);
    glutMouseFunc(mouse);
    glutMotionFunc(motion);
  
    glutMainLoop();
    return 0;
}