cadcam实验贝齐尔曲线曲面的生成方法

1、1贝齐尔曲面设计1.1贝齐尔曲面定义设Pj(0,1,n； j = 0,1,m)为(n +1) 乂(m +1)个空间点列，则mXn次Bezier曲面定义为:P(u, v)=世 n P B (u)B (v)i=0 j=0u, V G 0,1(式 2-1)其中 B (u) = Ci ui (1 - u)m-i接点列中相邻两点所形成的空间网格 Bezier曲面的矩阵表示式是：气q,” ” 是Bernstein基函数。依次用线段连称之为特征网格。P(u,V) = B0 (u),B (u),B(u )-1P00P10P01P11P0 mP1m气(V)B (v)1,mPn1PnmBn m (V)(式 2-2

2、)在一般实际应用中，n、m(1)双线性Bezier曲面当m=n=1时不大于4。(u) B(w) pu, w 0,1(式 2-3)定义一张双线性Bezier曲面。已知四个角点之后，则0S (u, w) = (1 - w)(1 - u) + (1 - u) p0001+ (1 - w)up uwp 、1011(式 2-4)(2)双二次Bezier曲面当 m=n=2 时S(u, w) = y B.,2(u)Bj,2(w)p.i=0 j=0,W G0,l(式 2-5)由此式定义的曲面，其边界曲线及参数坐标曲线均为抛物线。(3)双三次Bezier曲面当m=n=3时S(u, w) = 22 B.,3(u)

3、B.,3(w)p,.i=0 j=0u, W G 0,1S (u,明=B 0,3(u) Bu) B 2,3(u) B 重(u)p 00p 01p 02p 03-B 0,3( w) _10111213B1,3(巧)p 20 p 21 p 22p 23B 2 3( w)2,3-p 30 p 31 p 32 p 33-_ B 3,3( w) _(式 2-6)(式 2-7)其矩阵表示为S(,仞=UMZBZMW-13 - 3 13 - 6 3 0U = u 3U 2U1,W = W3 W2W1, M =z - 3 30 0(式 2-8)100 01.2贝齐尔曲面性质1. Bezier曲面特征网格的四个角点

4、正好是Bezier曲面的四个角点，即P(0,0)=P00P(1,0)=PM0P(0,1)=P0NP(1,1)=PMN2. Bezier曲面特征网格最外一圈顶点定义Bezier曲面的四条边界；Bezier曲面边界的跨界切矢只能 与定义该边界的顶点及相邻一排顶点有关，且P00P10P01,P0nP1nP0,n-1,Pm0Pm-1,0Pm1,分别是四个角点的切平面；跨界二阶导矢只与定义该边界的及相邻两排顶点有关。几何不变性、凸 包性、对称性等性质可由Bezier曲线的相关性质容易推广得到。图2-1阴影三角形1.3贝齐尔曲面算法Bezier曲线的递推(de Casteljau)算法，可以推广到Bezi

5、er曲面的情形。若给定Bezier曲面特征网格的 控制顶点：PiJ(i = 0,1，m,j = 0,1,n)和一对参数值(u,v)，则：P (u, V) = m- n l PklB(u) B . (v) = . = Pm.nU, V G 0,1i=0 j=0式(一条曲线可以表示成两条低一次曲线的组合，一张曲面可以表示成低一次的四张曲面的线性组合。Pk ,l = v i,j其中：Pij(1 - U ) P k-1,0 + UP k-1,0iji+1, j(1 V) Pm ,l-1 + vPm,l-10, j0, j+1(k = l = 0)(k = 1,2,., m; l = 0)(k = m,

6、 l = 1,2,., n)式(2-10)或.PHPk,l = (1 一 v)P0,l-1 + VP0,l-1ijiji, j+1(1 U ) Pk -1,n + uPk-1,nIi 0i+1,0(k = l = 0)(k = 0; l = 1,2,., n)(k = 1,2,，m; l = n)式(2-11)上面给出了确定曲面上一点的两种方案。当按(1)式方案执行时，先以u参数值对控制网格u向的n+1 个多边形执行曲线de Casteljau算法，m级递推后，得到沿v向由n+1个顶点，0j )构成的中间多边形。再以v参数值对它执行曲线的de Casteljau算法，n级递推以后，得到一个P券

7、,即所求曲面上的点。 也可以按(2)式方案执行，先以v参数值对控制网格沿v向的m+1个多边形执行n级递推，得沿u 向由m+1个顶点 =，m)构成的中间多边形。再以u参数值对它执行n级递推，得所求点P mn 00 。1.4程序设计步骤1.4.1 Bezier曲面的生成Bezier曲面是基于Beizer曲线而定义的，有特征网格决定其大致形状，而特征网格是由控制点确定 的。因而，绘制Bezier曲面的步骤与绘制Bezie曲面的步骤是一致的：第一，定义控制点序列；第二，定义一个二维评价器；第三，激活二维评价器；最后，创建网格。定义一个二维评价器的函数为：glMap2d()或glMap2f()函数，gl

8、Map2d()函数的原型为：Void glMap2d(GLenum target, GLdouble u1, GLdouble u2, Glint ustride,Glint uorder, GLdouble v1, GLdouble v2, Glint vorder, Glint uorder, const GLdouble * points);其中，target参数表示评价器最后生成的坐标类别，它可以取表2-1中的一个常量：常量含义GL_MAP2_VERTEX_3用(x,y,z)描述一个控制点GL_MAP2_VERTEX_4用(x,y,z,w)描述一个控制点GL_MAP2_INDEX控制点

9、代表一个颜色素引值GL_MAP2_COLOR_4控制点是RGBA颜色值GL_MAP2_NORMAL控制点是一个法线向量GL_MAP2_TEXTURE_COORD_1控制点是一个纹理坐标的s分量GL_MAP2_TEXTURE_COORD_2控制点是一个(s,t)纹理坐标GL_MAP2_TEXTURE_COORD_3控制点是一个(s,t,r)纹理坐标GL_MAP2_TEXTURE_COORD_4控制点是一个(s,t,r,q)纹理坐标表2-11.4.2绘制网格曲面(1)添加一个菜单添加一个名为Surface的菜单，在其属性对话框里的Caption栏中键入Surface0为其增加一个菜单项， 其名为

10、Bezier,在其属性对话框的 Caption 栏中键入 8。21。勾选 Pop-up 属性。按表 2-2 完成。CaptionID&MeahIDM_BZRSUFAC_MESH&FillIDM_BZRSUFAC_FILL&TextureIDM_BZRSUFAC_TEXTURE表2-2(2)添加保护成员函数添加一个保护成员函数：BzeSufacMesh(),编辑如下：Int I,j;GLfloat points443=-0.8f,-0.6f,0.8f,-0.2f,-0.6f,1.6f,0.2f,-0.6f,-0.4f,0.6f,-0.6f,0.8f,-0.6f,-0.2f,0.8f,-0.2f,

11、-0.2f,1.6f,0.2f,-0.2f,-0.4f,0.6f,-0.2f,0.8f,-0.6f,0.2f,0.8f,-0.2f,0.2f,0.4f,0.2f,0.2f,0.0f,0.3f,0.2f,-0.4f,-0.6f,0.6f,0.8f,-0.2f,0.6f,0.4f,-0.8f,0.6f,0.0f,0.8f,0.6f,-0.4f;/控制点glMap2f(GL_MAP2_VERTEX_3,0,1,3,4,0,1,12,4,(float*)controlPoints);/ 二维评价器glEnable(GL_MAP2_VERTEX_3);glColor3f(0,0,1);激活glPushM

12、atrix();glRotated(45,1,1,1);for (j=0;j8;j+)glBegin(GL_LINE_STRIP);for (i=0;i30;i+)glEvalCoord2f(float)i/30,(float)j/8);glEnd();glBegin(GL_LINE_STRIP);for (i=0;i30;i+)glEvalCoord2f(float)j/8,(float)i/30);glEnd();glPopMatrix();绘制glDisable(GL_MAP2_VERTEX_3);/挂起该函数将绘制一个网格曲面，因为glBegin()函数的参数为GL_LINE_STRI

13、Po编辑DrawScenc()函数在语句switch(m_typeControl).中添加语句：Case BezierSrufaceMesh:BzrSufacMesh(); break;(4)运行程序，得到如图2-2的结果图2-2 Bezier网格曲面1.4.3绘制一个填充曲面(1)增加两个保护函数Lighting()和 BzrSufacFill()函数，Lighting()函数光照处理，BzrSufacFill()函数则绘制曲面。Lighting();glEnable(GL_AUTO_NORMAL);glEnable(GL_NORMALIZE);/光照glMap2f(GL_MAP2_VERT

14、EX_3,0,1,3,4,0,1,12,4,(float*)controlPoints); 二维评价器glMapGrid2f(30,0,1,30,0,1); / 自动网格glEnable(GL_MAP2_VERTEX_3); 激活glPushMatrix();glRotated(45,1,1,1);glEvalMesh2(GL_FILL,0,30,0,30);glPopMatrix();绘制glDisable(GL_MAP2_VERTEX_3);glDisable(GL_AUTO_NORMAL);glDisable(GL_NORMALIZE);glDisable(GL_LIGHTING);/挂

15、起编辑 DrawScene()函数在语句switch(m_typeControl).中添加语句；Case BezierSurfaceFill:BzeSufacFill(); break;运行程序，得到如图2-3的结果这是Bezier曲面的一个特性。说明：程序输出的是斜二测投影图，X轴正方向向右，Y轴正方向向下，X、Y、Z三个轴正方成右手坐标系.曲面显示在Z 轴负坐标的位置上。图2-4中所示为轮船船艏，首柱线与设计水线交于曲面的右上角；最大横剖线 与设计水线交于曲面左上角：曲面右下角为首柱线与龙骨线的切点或交点：曲面左下角为最大横剖 线与底部龙骨起跑线的交点。#include StdAfx.h#

16、include OpenGL.h/添加头文件名 /#include #include bitmap.h #define PI (double)3.1415926) #define RADIANS(fAngle) (double)(fAngle)*PI/180.0) COpenGL:COpenGL(void):type(MESH)/类型网格/GLfloat points443=/ GLfloat points4*4*3= /-0.8f,-0.6f,0.8f,-0.2f,-0.6f,1.6f,0.2f,-0.6f,-0.4f,0.6f,-0.6f,0.8f, /-0.6f,-0.2f,0.8f,-

17、0.2f,-0.2f,1.6f,0.2f,-0.2f,-0.4f,0.6f,-0.2f,0.8f, / -0.6f,0.2f,0.8f,-0.2f,0.2f,0.4f,0.2f,0.2f,0.0f,0.3f,0.2f,-0.4f, / -0.6f,0.6f,0.8f,-0.2f,0.6f,0.4f,-0.8f,0.6f,0.0f,0.8f,0.6f,-0.4f /;/输入各点坐标/ for(int i=0;i4;i+) / for (int j=0;j4;j+) /for (int m=0;m3;m+)/controlPointsijm=pointsi*4*3+j*3+m; TOC o 1-5

18、 h z /COpenGL:COpenGL(void)/添加函数类型为 void/ wglMakeCurrent(hDC, NULL); wglDeleteContext(hRC); void COpenGL:Init(void) glClearColor(1.0,1.0,1.0,1.0);glClearDepth(1.0);glEnable(GL_DEPTH_TEST);glShadeModel(GL_SMOOTH);glDepthFunc(GL_LEQUAL);glHint(GL_PERSPECTIVE_CORRECTION_HINT, GL_NICEST); bool COpenGL:S

19、etupPixelFormat(HDC hDC0) int nPixelFormat;hDC=hDC0;PIXELFORMATDESCRIPTOR pfd = sizeof(PIXELFORMATDESCRIPTOR), 1, PFD_DRAW_TO_WINDOW | PFD_SUPPORT_OPENGL | PFD_DOUBLEBUFFER, PFD_TYPE_RGBA, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, PFD_MAIN_PLANE, 0, 0, 0, 0 ; if (!(nPixelFormat = ChoosePi

20、xelFormat(hDC, &pfd) MessageBox(NULL,can not find proper mode,Error,MB_OK|MB_ICONEXCLAMATION); /若错误则显示 can not find proper modeerror/return FALSE;/返回 FLASH/ SetPixelFormat(hDC,nPixelFormat,&pfd); hRC = wglCreateContext(hDC); wglMakeCurrent(hDC, hRC); return TRUE;/返回 TURE/ void COpenGL:Reshape(int wi

21、dth,int height) / glViewport(0,0,width,height); / glMatrixMode(GL_PROJECTION); / glLoadIdentity(); / gluPerspective(60,(GLfloat)width/(GLfloat)height,0.1,3000.0);/ glMatrixMode(GL_MODELVIEW);glViewport(0,0,width,height);glMatrixMode(GL_PROJECTION);glLoadIdentity();gluPerspective(60,(GLfloat)width/(G

22、Lfloat)height,0.1,3000.0);glMatrixMode(GL_MODELVIEW);/glMatrixMode(GL_MODELVIEW);enum AspectFRONT,BACK,LEFT,RIGHT,UP,DOWN;enum ColorBLUE,GREEN,RED,ORANGE,WHITE,YELLOW;/代表颜色索引值/const GLubyte colorTable63=0,0,255,0,255,0,255,0,0,255,102,0,255,255,255,255,255,0;void COpenGL:Render()glClear(GL_COLOR_BUF

23、FER_BIT|GL_DEPTH_BUFFER_BIT);glLoadIdentity();gluLookAt(0,0,3,0,0,0,0,1,0);switch(type)case MESH:DrawMesh();/绘制网格/ break;case FILL:DrawFill();break;/绘制填充曲面/绘制填充曲面/DrawTexture();break;/绘制贴图曲面/case TEXTURE:/GLint vertices83=-1,-1,-1,1,-1,-1,-1,-1,1,1,-1,1,-1,1,-1,1,1,-1,-1,1,1,1,1,1default:break;/draw

24、 stuff/ draw cube;/in order:front ,back ,left ,right,top,bottomGLubyte vertIndex64=6,2,3,7,5,1,0,4,4,0,2,6,7.3.1.5, 2,0,1,3;/ for (int i=0;i6;i+)(/glColor3ubv(colorTablei);/glBegin(GL_QUADS);/for (int j=0;j4;j+)/glVertex3iv(verticesvertIndexij);/glEnd();/glFlush();SwapBuffers(hDC);void COpenGL:DrawM

25、esh()GLfloat controlPoints443=-0.8f,-0.6f,0.8f,-0.2f,-0.6f,1.6f,0.2f,-0.6f,-0.4f,0.6f,-0.6f,0.8f,-0.6f,-0.2f,0.8f,-0.2f,-0.2f,1.6f,0.2f,-0.2f,-0.4f,0.6f,-0.2f,0.8f,-0.6f,0.2f,0.8f,-0.2f,0.2f,0.4f,0.2f,0.2f,0.0f,0.3f,0.2f,-0.4f,-0.6f,0.6f,0.8f,-0.2f,0.6f,0.4f,-0.8f,0.6f,0.0f,0.8f,0.6f,-0.4f;/定义控制点/in

26、t i,j;glMap2f(GL_MAP2_VERTEX_3,0,1,3,4,0,1,12,4,(float*)controlPoints);/二维评价器/glEnable(GL_MAP2_VERTEX_3);glColor3f(0,0,1);/激活/glPushMatrix();glRotated(45,1,1,1);for (j=0;j8;j+)glBegin(GL_LINE_STRIP);for (i=0;i30;i+)glEvalCoord2f(float)i/30,(float)j/8);glEnd();glBegin(GL_LINE_STRIP);for (i=0;isizeX,t

27、extureImage-sizeY0,GL_BGR_EXT, GL_UNSIGNED_BYTE,textureImage-data); glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_WRAP_S,GL_CLAMP);glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_WRAP_T,GL_CLAMP); glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_MAG_FILTER,GL_NEAREST); glTexParameterf(GL_TEXTURE_2D,GL_TEXTURE_MIN_FILTE

28、R,GL_NEAREST);glTexEnvf(GL_TEXTURE_ENVGL_TEXTURE_ENV_MODE,GL_DECAL); / 构造纹 理/ glMap2f(GL_MAP2_VERTEX_3,0,1,3,4,0,1,12,4,(float*)controlPoints);glMap2f(GL_MAP2_TEXTURE_COORD_2,0,1,2,2,0,1,4,2,(float*)controlPoints);/二维评价器：生成几何坐标和纹理坐标/Lighting();glEnable(GL_TEXTURE_2D);glEnable(GL_MAP2_TEXTURE_COORD_2

29、);glEnable(GL_MAP2_VERTEX_3);glEnable(GL_AUTO_NORMAL);glEnable(GL_NORMALIZE);/光照/glMapGrid2f(30,0,1,30,0,1);glPushMatrix();glScaled(1.2,1.2,1.2);glRotated(45,0,0,1);glRotated(45,1,1,1);glEvalMesh2(GL_FILL,0,30,0,30);glPopMatrix();/绘制 /glDisable(GL_MAP2_VERTEX_3);glDisable(GL_MAP2_TEXTURE_COORD_2);gl

30、Disable(GL_TEXTURE_2D);glDisable(GL_AUTO_NORMAL);glDisable(GL_NORMALIZE);glDisable(GL_LIGHTING);/挂起/else(MessageBox(NULL,failed to load image, ,MB_OK);void COpenGL:Lighting()/编辑光照函数/GLfloat lightAmb=0.2,0.2,0.2,1;GLfloat lightPos=1,1,2,1;GLfloat matDiffuse=0,0.7,0.7,1;GLfloat matSpecular=1,1,1,1;GLfloat matShininess=60;glEnable(GL_LIGHTING);glEnable(GL_LIGHT0);glLightfv(GL_LIGHT0,GL_AMBIENT,lightAmb);glLightfv(GL_LIGHT0,GL_POSITION,lightPos);glMat