图形学之贝塞尔曲线实现贝塞尔曲线,c++,opengl秀玉轩晨的博客-

定义实现bezier曲线

定义

$B(t)=sum_{i=0}^n P_i complement_n^i t^i (1-t)^{n-i} ,其中[tepsilon (0,1]]$

B(t)=i=0∑n​Pi​∁ni​ti(1−t)n−i,其中[tϵ(0,1]]
不要被这么一坨公式给吓到,它求的其实是
$t$

t为定值时贝塞尔曲线上的点的坐标,而
$P_i$

Pi​则表示第
$i$

i个控制点的作用.不同的
$t$

t值可以画出不同的点,这些点就连成了贝塞尔曲线.

一阶贝塞尔曲线

$B(t) = (1-t)P_0 +tP_1$

B(t)=(1−t)P0​+tP1​

二阶贝塞尔曲线

$B(t)=(1-t)^2P_0 + 2t(1-t)P_1+t^2P_2$

B(t)=(1−t)2P0​+2t(1−t)P1​+t2P2​

三阶贝塞尔曲线

$B(t)=sum_{i=0}^3 P_i complement_3^i t^i (1-t)^{n-i}$

B(t)=i=0∑3​Pi​∁3i​ti(1−t)n−i

贝塞尔曲线升阶

升阶意思是增加控制点,但保持原曲线不变

$P_i^*= frac{i}{n+1}P_{i-1}+(1-frac{i}{n+1})P_i ,(i=0,1,2…,n+1)$

Pi∗​=n+1i​Pi−1​+(1−n+1i​)Pi​,(i=0,1,2...,n+1)

代码实现

#include <GL/glut.h> #include <stdlib.h> #include <math.h>  GLfloat xwcMin = -250.0, xwcMax = 250.0; GLfloat ywcMin = -250.0, ywcMax = 250.0;  class WcPt3d { public:  GLfloat x, y, z; };  // 画点 void plotPt(WcPt3d point) {  glBegin(GL_POINTS);  glVertex3f(point.x,point.y,point.z);  glEnd(); }  // 初始化背景色 void init(void) {  glClearColor(1.0, 1.0, 1.0, 0.0); }   // 计算二项式的系数 *** void BinomialCoeffi(GLint *C, GLint ctrlNums) {  GLint k, j;  for ( k = 0; k <= ctrlNums; k++)  {   C[k] = 1;   // n!/(k!(n-k)!)   for ( j = ctrlNums; j >= k+1 ; j--) // n!/k!   {    C[k] *= j;   }    for (j = ctrlNums - k; j >= 2; j--) // (n-k)!   {    C[k] /= j;   }  } }  void computeBezierPts(GLfloat t, WcPt3d * bezierCurvePt,GLint ctrlNums,GLint* C,WcPt3d* ctrlPts) {  GLint k, n = ctrlNums - 1;  GLfloat bezierCurveFcn;   bezierCurvePt->x = bezierCurvePt->y = bezierCurvePt->z = 0;  for ( k = 0; k < ctrlNums; k++)  {   bezierCurveFcn = C[k] * pow(t, k) * pow(1 - t, n - k);   bezierCurvePt->x += ctrlPts[k].x * bezierCurveFcn;   bezierCurvePt->y += ctrlPts[k].y * bezierCurveFcn;   bezierCurvePt->z += ctrlPts[k].z * bezierCurveFcn;  } } // 绘制连接线段 void plotLines(WcPt3d* points, GLint Ptnums) {  glBegin(GL_LINE_STRIP);  for (GLint i = 0; i < Ptnums; i++)  {   glVertex3f(points[i].x, points[i].y, points[i].z);  }  glEnd(); } // Bezier曲线的算法 void Bezier(WcPt3d *ctrlPts, GLint bezierCurveNums, GLint ctrlNums){  WcPt3d bezierCurvePt;  GLint k, *C; // 获取控制点的数目   GLfloat t;  C = new GLint[ctrlNums]; // 获取多项式的各项系数    BinomialCoeffi(C, ctrlNums - 1); // 计算各项的系数,并存储   for ( k = 0; k <= bezierCurveNums; k++)  {   t = (GLfloat)k / (GLfloat)bezierCurveNums;   computeBezierPts(t, &bezierCurvePt, ctrlNums,C,ctrlPts);   plotPt(bezierCurvePt);  }  // 绘制线段  plotLines(ctrlPts, ctrlNums);   delete[] C; } // 贝塞尔曲线的升阶 void ascendingBezier(WcPt3d* oldPts,GLint oldPtnums, GLint bezierCurveNums) {  GLint newPtNums = oldPtnums + 1;  WcPt3d* newPts = new WcPt3d[newPtNums];  GLint i;   // P[i]' = i/(n+1) P[i-1] + (n+1-i)/(n+1) Pi;  for (i = 0; i < newPtNums; i++)  {   GLfloat x = 0, y = 0, z= 0;   if (i==0)   {    x = oldPts[i].x;    y = oldPts[i].y;    z = oldPts[i].z;   }   else if(i==newPtNums - 1){    x = oldPts[i - 1].x;    y = oldPts[i - 1].y;    z = oldPts[i - 1].z;   }   else {    x = (GLfloat) oldPts[i - 1].x * i / oldPtnums + oldPts[i].x * (oldPtnums - i) / oldPtnums;    y = (GLfloat)oldPts[i - 1].y * i / oldPtnums + oldPts[i].y * (oldPtnums - i) / oldPtnums;    z = (GLfloat)oldPts[i - 1].z * i / oldPtnums + oldPts[i].z * (oldPtnums - i) / oldPtnums;   }   newPts[i] = { x,y,z };  }  Bezier(newPts, bezierCurveNums, newPtNums); }   // 初始化数据并进行绘制 void displayFcn1(void) {  WcPt3d ctrlPts[7] = { {-140,-40,0}, {-60,-80,0},{90,100,0},{120,200,0},{180,160,0},{200,130,0},{230,10,0}};    GLint bezierCurveNums = 2000 , ctrlNums = 7;  glClear(GL_COLOR_BUFFER_BIT);  glPointSize(2);  glColor3f(0, 0, 0);  // 将控制点,绘制精度代入  Bezier(ctrlPts,bezierCurveNums,ctrlNums);    glColor3f(1, 1, 0);  // 升阶  ascendingBezier(ctrlPts, ctrlNums,bezierCurveNums);   glFlush();  }  // 对绘图窗口进行缩放 void winReshapeFcn(GLint newWidth, GLint newHeight) {  glViewport(0, 0, newHeight, newHeight);   glMatrixMode(GL_PROJECTION);  glLoadIdentity();   gluOrtho2D(xwcMin, xwcMax, ywcMin, ywcMax);  glClear(GL_COLOR_BUFFER_BIT);  }  // 主函数 int main(int argc, char **argv){  glutInit(&argc, argv);   glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);  glutInitWindowPosition(50, 50);  glutInitWindowSize(600, 600);  glutCreateWindow("贝塞尔曲线");   init();   glutDisplayFunc(displayFcn1);  glutReshapeFunc(winReshapeFcn);  glutMainLoop(); } 

效果图

通过de Casteljau

de Casteljau递推法

由于通过贝塞尔方程计算太过复杂,所以提出了递推法,方程如下

$P_i^k= =begin{cases} P_i&text{k=0}\(1-t)P_i^{k-1}+tP_{i+1}^{k-1}&text{x!=0} end{cases}$

Pik​=={Pi​(1−t)Pik−1​+tPi+1k−1​​k=0x!=0​

代码实现

#include <GL/glut.h> #include <math.h> #include <stdlib.h>  class WcPt3D { public:  GLfloat x, y, z; };  GLfloat xwcMin = -250.0, xwcMax = 250.0; GLfloat ywcMin = -250.0, ywcMax = 250.0;  void plotPt(WcPt3D pt) {  glBegin(GL_POINTS);  glVertex3f(pt.x, pt.y, pt.z);  glEnd(); } // 当为1阶时 void oneCasteljau(WcPt3D* points, GLfloat t) {  WcPt3D point;  point.x = points[0].x * (1 - t) + points[1].x * t;  point.y = points[0].y * (1 - t) + points[1].y * t;  point.z = points[0].z * (1 - t) + points[1].z * t;  plotPt(point); } // 当为n阶时 void nCasteljau(WcPt3D* Points, GLint n, GLfloat t) { // 绘制的点 , 阶数 = 点数 -1, t 目前的比值   if (n > 1)   {   WcPt3D* newCtrls = new WcPt3D[n]; // 获取n-1阶的点    GLint k;   for ( k = 0; k < n; k++)   {    newCtrls[k].x = Points[k].x * (1 - t) + Points[k + 1].x * t;    newCtrls[k].y = Points[k].y * (1 - t) + Points[k + 1].y * t;    newCtrls[k].z = Points[k].z * (1 - t) + Points[k + 1].z * t;   }   nCasteljau(newCtrls, n - 1, t);  }  else {   oneCasteljau(Points,t);  } }  // 绘制连接线段 void plotLines(WcPt3D* points, GLint Ptnums) {  glBegin(GL_LINE_STRIP);  for (GLint i = 0; i < Ptnums; i++)  {   glVertex3f(points[i].x, points[i].y, points[i].z);  }  glEnd(); } // 使用算法绘制曲线 void deCasteljau(WcPt3D* ctrlPts, GLint ctrlNums, GLint bezierCurveNums) {   GLint k;   for ( k = 0; k <bezierCurveNums; k++)  {   GLfloat t = (GLfloat)k / (GLfloat)bezierCurveNums;   nCasteljau(ctrlPts, ctrlNums - 1, t);  }   // 绘制线段  plotLines(ctrlPts, ctrlNums); }  void displayFcn() {   WcPt3D ctrlPts[7] = { {-140,-40,0}, {-60,-80,0},{90,100,0},{120,200,0},{180,160,0},{200,130,0},{230,10,0} };   GLint bezierCurveNums = 2000, ctrlNums = 7;   glClear(GL_COLOR_BUFFER_BIT);  glPointSize(2);  glColor3f(0, 0, 0);    deCasteljau(ctrlPts, ctrlNums, bezierCurveNums);  glFlush();  }  // 对绘图窗口进行缩放 void winReshapeFcn(GLint newWidth, GLint newHeight) {  glViewport(0, 0, newHeight, newHeight);   glMatrixMode(GL_PROJECTION);  glLoadIdentity();   gluOrtho2D(xwcMin, xwcMax, ywcMin, ywcMax);  glClear(GL_COLOR_BUFFER_BIT);  }  void init() {  glClearColor(1.0, 1.0, 1.0, 0.0); } int main( int argc,  char **argv) {  glutInit(&argc, argv);   glutInitDisplayMode(GLUT_SINGLE | GLUT_RGB);  glutInitWindowPosition(50, 50);  glutInitWindowSize(600, 600);  glutCreateWindow("deCasteljau使用");  init();  glutDisplayFunc(displayFcn);  glutReshapeFunc(winReshapeFcn);  glutMainLoop(); } 

效果图

代码地址.