cadcam實驗貝齊爾曲線曲面的生成方法

1、1貝齊爾曲面設(shè)計1.1貝齊爾曲面定義設(shè)Pj(0,1,n； j = 0,1,m)為(n +1) 乂(m +1)個空間點列，則mXn次Bezier曲面定義為: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接點列中相鄰兩點所形成的空間網(wǎng)格 Bezier曲面的矩陣表示式是：氣q,” ” 是Bernstein基函數(shù)。依次用線段連稱之為特征網(wǎng)格。P(u,V) = B0 (u),B (u),B(u )-1P00P10P01P11P0 mP1m氣(V)B (v)1,mPn1PnmBn m (V)(式 2-2

2、)在一般實際應(yīng)用中，n、m(1)雙線性Bezier曲面當(dāng)m=n=1時不大于4。(u) B(w) pu, w 0,1(式 2-3)定義一張雙線性Bezier曲面。已知四個角點之后，則0S (u, w) = (1 - w)(1 - u) + (1 - u) p0001+ (1 - w)up uwp 、1011(式 2-4)(2)雙二次Bezier曲面當(dāng) m=n=2 時S(u, w) = y B.,2(u)Bj,2(w)p.i=0 j=0,W G0,l(式 2-5)由此式定義的曲面，其邊界曲線及參數(shù)坐標(biāo)曲線均為拋物線。(3)雙三次Bezier曲面當(dāng)m=n=3時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)其矩陣表示為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貝齊爾曲面性質(zhì)1. Bezier曲面特征網(wǎng)格的四個角點

4、正好是Bezier曲面的四個角點，即P(0,0)=P00P(1,0)=PM0P(0,1)=P0NP(1,1)=PMN2. Bezier曲面特征網(wǎng)格最外一圈頂點定義Bezier曲面的四條邊界；Bezier曲面邊界的跨界切矢只能 與定義該邊界的頂點及相鄰一排頂點有關(guān)，且P00P10P01,P0nP1nP0,n-1,Pm0Pm-1,0Pm1,分別是四個角點的切平面；跨界二階導(dǎo)矢只與定義該邊界的及相鄰兩排頂點有關(guān)。幾何不變性、凸 包性、對稱性等性質(zhì)可由Bezier曲線的相關(guān)性質(zhì)容易推廣得到。圖2-1陰影三角形1.3貝齊爾曲面算法Bezier曲線的遞推(de Casteljau)算法，可以推廣到Bezi

5、er曲面的情形。若給定Bezier曲面特征網(wǎng)格的 控制頂點：PiJ(i = 0,1，m,j = 0,1,n)和一對參數(shù)值(u,v)，則：P (u, V) = m- n l PklB(u) B . (v) = . = Pm.nU, V G 0,1i=0 j=0式(一條曲線可以表示成兩條低一次曲線的組合，一張曲面可以表示成低一次的四張曲面的線性組合。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)上面給出了確定曲面上一點的兩種方案。當(dāng)按(1)式方案執(zhí)行時，先以u參數(shù)值對控制網(wǎng)格u向的n+1 個多邊形執(zhí)行曲線de Casteljau算法，m級遞推后，得到沿v向由n+1個頂點，0j )構(gòu)成的中間多邊形。再以v參數(shù)值對它執(zhí)行曲線的de Casteljau算法，n級遞推以后，得到一個P券

7、,即所求曲面上的點。 也可以按(2)式方案執(zhí)行，先以v參數(shù)值對控制網(wǎng)格沿v向的m+1個多邊形執(zhí)行n級遞推，得沿u 向由m+1個頂點 =，m)構(gòu)成的中間多邊形。再以u參數(shù)值對它執(zhí)行n級遞推，得所求點P mn 00 。1.4程序設(shè)計步驟1.4.1 Bezier曲面的生成Bezier曲面是基于Beizer曲線而定義的，有特征網(wǎng)格決定其大致形狀，而特征網(wǎng)格是由控制點確定 的。因而，繪制Bezier曲面的步驟與繪制Bezie曲面的步驟是一致的：第一，定義控制點序列；第二，定義一個二維評價器；第三，激活二維評價器；最后，創(chuàng)建網(wǎng)格。定義一個二維評價器的函數(shù)為：glMap2d()或glMap2f()函數(shù)，gl

8、Map2d()函數(shù)的原型為：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參數(shù)表示評價器最后生成的坐標(biāo)類別，它可以取表2-1中的一個常量：常量含義GL_MAP2_VERTEX_3用(x,y,z)描述一個控制點GL_MAP2_VERTEX_4用(x,y,z,w)描述一個控制點GL_MAP2_INDEX控制點

9、代表一個顏色素引值GL_MAP2_COLOR_4控制點是RGBA顏色值GL_MAP2_NORMAL控制點是一個法線向量GL_MAP2_TEXTURE_COORD_1控制點是一個紋理坐標(biāo)的s分量GL_MAP2_TEXTURE_COORD_2控制點是一個(s,t)紋理坐標(biāo)GL_MAP2_TEXTURE_COORD_3控制點是一個(s,t,r)紋理坐標(biāo)GL_MAP2_TEXTURE_COORD_4控制點是一個(s,t,r,q)紋理坐標(biāo)表2-11.4.2繪制網(wǎng)格曲面(1)添加一個菜單添加一個名為Surface的菜單，在其屬性對話框里的Caption欄中鍵入Surface0為其增加一個菜單項， 其名為

10、Bezier,在其屬性對話框的 Caption 欄中鍵入 8。21。勾選 Pop-up 屬性。按表 2-2 完成。CaptionID&MeahIDM_BZRSUFAC_MESH&FillIDM_BZRSUFAC_FILL&TextureIDM_BZRSUFAC_TEXTURE表2-2(2)添加保護(hù)成員函數(shù)添加一個保護(hù)成員函數(shù)：BzeSufacMesh(),編輯如下：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;/控制點glMap2f(GL_MAP2_VERTEX_3,0,1,3,4,0,1,12,4,(float*)controlPoints);/ 二維評價器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();繪制glDisable(GL_MAP2_VERTEX_3);/掛起該函數(shù)將繪制一個網(wǎng)格曲面，因為glBegin()函數(shù)的參數(shù)為GL_LINE_STRI

13、Po編輯DrawScenc()函數(shù)在語句switch(m_typeControl).中添加語句：Case BezierSrufaceMesh:BzrSufacMesh(); break;(4)運(yùn)行程序，得到如圖2-2的結(jié)果圖2-2 Bezier網(wǎng)格曲面1.4.3繪制一個填充曲面(1)增加兩個保護(hù)函數(shù)Lighting()和 BzrSufacFill()函數(shù)，Lighting()函數(shù)光照處理，BzrSufacFill()函數(shù)則繪制曲面。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); 二維評價器glMapGrid2f(30,0,1,30,0,1); / 自動網(wǎng)格glEnable(GL_MAP2_VERTEX_3); 激活glPushMatrix();glRotated(45,1,1,1);glEvalMesh2(GL_FILL,0,30,0,30);glPopMatrix();繪制glDisable(GL_MAP2_VERTEX_3);glDisable(GL_AUTO_NORMAL);glDisable(GL_NORMALIZE);glDisable(GL_LIGHTING);/掛

15、起編輯 DrawScene()函數(shù)在語句switch(m_typeControl).中添加語句；Case BezierSurfaceFill:BzeSufacFill(); break;運(yùn)行程序，得到如圖2-3的結(jié)果這是Bezier曲面的一個特性。說明：程序輸出的是斜二測投影圖，X軸正方向向右，Y軸正方向向下，X、Y、Z三個軸正方成右手坐標(biāo)系.曲面顯示在Z 軸負(fù)坐標(biāo)的位置上。圖2-4中所示為輪船船艏，首柱線與設(shè)計水線交于曲面的右上角；最大橫剖線 與設(shè)計水線交于曲面左上角：曲面右下角為首柱線與龍骨線的切點或交點：曲面左下角為最大橫剖 線與底部龍骨起跑線的交點。#include StdAfx.h#

16、include OpenGL.h/添加頭文件名 /#include #include bitmap.h #define PI (double)3.1415926) #define RADIANS(fAngle) (double)(fAngle)*PI/180.0) COpenGL:COpenGL(void):type(MESH)/類型網(wǎng)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 /;/輸入各點坐標(biāo)/ 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)/添加函數(shù)類型為 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); /若錯誤則顯示 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;/代表顏色索引值/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();/繪制網(wǎng)格/ break;case FILL:DrawFill();break;/繪制填充曲面/繪制填充曲面/DrawTexture();break;/繪制貼圖曲面/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;/定義控制點/in

26、t i,j;glMap2f(GL_MAP2_VERTEX_3,0,1,3,4,0,1,12,4,(float*)controlPoints);/二維評價器/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); / 構(gòu)造紋 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);/二維評價器：生成幾何坐標(biāo)和紋理坐標(biāo)/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();/繪制 /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);/掛起/else(MessageBox(NULL,failed to load image, ,MB_OK);void COpenGL:Lighting()/編輯光照函數(shù)/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