给定点P0，P1，线性贝兹曲线是这两点之间连线的点，方程式如下：
B(t) = P0 + (P1 - P0)t t取[0,1];
假设P0（x0, y0),P1(x1, y1), Pn((1-t)x0+tx1),(1-t)y0+ty1)
1）通过点在直线上，斜率相等证明一阶贝塞罗曲线是一直线
K(PnP0) = K(PnP1)这种方法相对简单
2）通过P0P1两点根据点斜式求出直线的方程，然后将Pn点代入，
同样可以证明，相对繁琐
void myDisplay()
{
glClear(GL_COLOR_BUFFER_BIT);
glColor3f(1.0f,0.f,0.f);
int P1x = 10;
int P1y = 10;
int P2x = 200;
int P2y = 400;
glPointSize(3);
glLineWidth(1);
glBegin(GL_POINTS);
glVertex2d(P1x, P1y);
glVertex2d(P2x, P2y);
for (int i=0; i<10; i++)
{
double t = i*0.1;
double x = (1-t)*P1x + t* P2x;
double y = (1-t)*P1y + t* P2y;
glVertex2d(x, y);
}
glEnd();
glColor3f(0.f,1.f,0.f);
glBegin(GL_LINES);
glVertex2d(P1x, P1y);
glVertex2d(P2x, P2y);
glEnd();
glFlush();
}
二次方公式
二次方贝兹曲线的路径由给定点P0、P1、P2的函数B（t）：
B(t) = （1-t)*(1-t)P0 + 2*t*(1-t)P1 + t*tP2 t取[0,1];
#include <windows.h>
#include <gl/gl.h>
#include <gl/glu.h>
#include <gl/glut.h>
//////////////////////////////////
void myInit()
{
glClearColor(1.0,1.0,1.0,0.0);
glLineWidth(3.0);
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
gluOrtho2D(0.0,640,0.0,480);
}
/////////////////////////////////////
void myDisplay()
{
glClear(GL_COLOR_BUFFER_BIT);
glColor3f(1.0f,0.f,0.f);
int P1x = 10;
int P1y = 10;
int P2x = 200;
int P2y = 400;
int Pcx = 0;
int Pcy = 200;
glPointSize(3);
glLineWidth(1);
glBegin(GL_POINTS);
glVertex2d(P1x, P1y);
glVertex2d(P2x, P2y);
glVertex2d(Pcx, Pcy);
for (int i=0; i<100; i++)
{
double t = i*0.01;
double x = (1-t)*(1-t)*P1x + 2*(1-t)*Pcx + t*t*P2x;
double y = (1-t)*(1-t)*P1y + 2*(1-t)*Pcy + t*t*P2y;
glVertex2d(x, y);
}
glEnd();
glColor3f(0.f,1.f,0.f);
glBegin(GL_LINES);
glVertex2d(P1x, P1y);
glVertex2d(P2x, P2y);
glEnd();
glFlush();
}
///////////////////////////////////////
void main(int argc,char **argv)
{
glutInit(&argc,argv);
glutInitDisplayMode(GLUT_SINGLE|GLUT_RGB);
glutInitWindowSize(640,480);
glutInitWindowPosition(100,150);
glutCreateWindow("example");
glutDisplayFunc(myDisplay);
myInit();
glutMainLoop();
}
