v804-第四章opengl的2d圖形渲染課程4二維變換

1、二維變換學(xué)習(xí)如何使圖形運(yùn)動(dòng)平移變換、旋轉(zhuǎn)變換和放縮學(xué)會(huì)復(fù)雜變換的分解與合成學(xué)會(huì)使用OpenGL的幾何變換函數(shù)本章目標(biāo)數(shù)學(xué)基礎(chǔ)二維幾何變換齊次坐標(biāo)復(fù)合變換其它變換三維幾何變換圖形對(duì)象的幾何變換OpenGL的幾何變換函數(shù)主要內(nèi)容矢量（vector）連接兩個(gè)點(diǎn)的有向線段。又稱向量行向量和列向量?jī)煞N表示矢量和數(shù)學(xué)基礎(chǔ)矢量的數(shù)乘矢量的點(diǎn)積運(yùn)算性質(zhì)數(shù)學(xué)基礎(chǔ)矢量的長(zhǎng)度單位矢量 矢量間的夾角矢量的叉積右手法則數(shù)學(xué)基礎(chǔ)矩陣（Matrix)mn 階矩陣n階方陣（mn）單位矩陣 n階方陣，對(duì)角線元素為1， 其它元素為0數(shù)學(xué)基礎(chǔ)矩陣（續(xù)）行向量與列向量當(dāng)m1時(shí)，A退化為行向量a11, a12, , a1n當(dāng)n1時(shí)，

2、A退化為列向量a11, a21, , am1T矩陣的加法 A=(aij)mn，B(bij) mnA與B的和記為AB性質(zhì)：結(jié)合律和交換律數(shù)學(xué)基礎(chǔ)矩陣（續(xù)）矩陣的數(shù)乘 矩陣的乘法 性質(zhì)：結(jié)合律和分配律（不滿足交換律）數(shù)學(xué)基礎(chǔ)矩陣（續(xù)）矩陣的轉(zhuǎn)置 矩陣的逆n階方陣A是可逆的，若存在另一個(gè)n階方陣B，使得 ABBAIn，稱B是A的逆陣，記為BA1數(shù)學(xué)基礎(chǔ)平移變換 (translation transformation)將點(diǎn)P(x, y)在x軸方向、y軸方向分別平移距離tx，ty，得到點(diǎn)P(x, y)，則記為：T(tx , ty)矩陣表示：二維幾何變換旋轉(zhuǎn)變換(rotation transformati

3、on)如點(diǎn)P(x, y)的極坐標(biāo)表示（r為P 到原點(diǎn)的距離）繞坐標(biāo)原點(diǎn)（稱為參照點(diǎn)，基準(zhǔn)點(diǎn)）旋轉(zhuǎn)角度 （逆時(shí)針為正，順時(shí)針為負(fù)）二維幾何變換旋轉(zhuǎn)變換(續(xù)）記為：R()矩陣表示為：二維幾何變換放縮變換(scaling transformation)將點(diǎn)P(x, y)在x方向, y方向分別放縮 sx 和 sy 倍，得到點(diǎn)P(x, y)以坐標(biāo)原點(diǎn)為放縮參照（基準(zhǔn)）點(diǎn)不僅改變了物體的大小和形狀，也改變了它離原點(diǎn)的距離記為：S(sx, sy)二維幾何變換利用矩陣計(jì)算變換后的坐標(biāo)時(shí)，平移、旋轉(zhuǎn)和放縮變換分別為：運(yùn)算不統(tǒng)一，如何統(tǒng)一運(yùn)算？二維幾何變換為什么需要齊次坐標(biāo)?齊次坐標(biāo)就是將一個(gè)原本是n維的向量用一

4、個(gè)n+1維向量來(lái)表示。例如，二維點(diǎn)(x,y)的齊次坐標(biāo)表示為(hx,hy,h)。由此可以看出，一個(gè)向量的齊次表示是不唯一的，齊次坐標(biāo)的h取不同的值都表示的是同一個(gè)點(diǎn)，比如齊次坐標(biāo)(8,4,2)、(4,2,1)表示的都是二維點(diǎn)(4,2)齊次坐標(biāo)為什么需要齊次坐標(biāo)?對(duì)多個(gè)點(diǎn)計(jì)算多次不同的變換時(shí)，分別利用矩陣計(jì)算各變換導(dǎo)致計(jì)算量大運(yùn)算表示形式不統(tǒng)一平移為“”旋轉(zhuǎn)和放縮為“”統(tǒng)一運(yùn)算形式后，可以先合成變換運(yùn)算的矩陣，再作用于圖形對(duì)象齊次坐標(biāo)定義 Homogeneous Coordinate（x，y）點(diǎn)對(duì)應(yīng)的齊次坐標(biāo)定義為（x，y）點(diǎn)對(duì)應(yīng)的齊次坐標(biāo)為三維空間的一條直線 標(biāo)準(zhǔn)齊次坐標(biāo)（x，y，1）h0表示

5、無(wú)窮遠(yuǎn)點(diǎn)齊次坐標(biāo)二維變換的矩陣表示平移變換旋轉(zhuǎn)變換齊次坐標(biāo)放縮變換變換具有統(tǒng)一表示形式的優(yōu)點(diǎn)便于變換合成連續(xù)變換時(shí)，可以先得到變換的矩陣便于硬件實(shí)現(xiàn)齊次坐標(biāo)變換的性質(zhì)平移和旋轉(zhuǎn)變換具有可加性放縮變換具有可乘性齊次坐標(biāo)逆變換逆平移變換：正平移距離tx，ty逆旋轉(zhuǎn)變換：旋轉(zhuǎn)角度為齊次坐標(biāo)逆放縮變換：放縮系數(shù)為sx和sy齊次坐標(biāo)變換合成方法：連續(xù)變換時(shí)，先計(jì)算變換矩陣，再計(jì)算坐標(biāo)優(yōu)點(diǎn)：（1）提高了對(duì)圖形依次做多次變換的運(yùn)算效率如：圖形上有n個(gè)頂點(diǎn)Pi，如果依次施加的變換為T，R，那么頂點(diǎn)Pi 變換后的坐標(biāo)為每個(gè)頂點(diǎn)需要2次矩陣相乘只需要1次矩陣相乘復(fù)合變換變換合成（續(xù)）（2）提供構(gòu)造復(fù)雜變換的方法

6、對(duì)圖形作較復(fù)雜的變換時(shí)，不直接去計(jì)算這個(gè)變換，而是將其先分解成多個(gè)基本變換，再合成總的變換復(fù)合變換Composite transformation多個(gè)變換的組合可通過(guò)單個(gè)變換矩陣來(lái)計(jì)算矩陣乘積復(fù)合變換連續(xù)平移變換平移向量為(t1x，t1y)和(t2x，t2y)點(diǎn)P 經(jīng)變換為P，則有復(fù)合矩陣復(fù)合平移變換連續(xù)旋轉(zhuǎn)P 經(jīng)連續(xù)旋轉(zhuǎn)角度分別為 1 和 2 后連續(xù)旋轉(zhuǎn)具有相加性復(fù)合旋轉(zhuǎn)變換連續(xù)放縮連續(xù)放縮因子分別為：(s1x, s1y) 和 (s2x, s2y)復(fù)合放縮變換關(guān)于任意參照點(diǎn) 的旋轉(zhuǎn)變換步驟：（1）平移對(duì)象使參照（基準(zhǔn)）點(diǎn)移到原點(diǎn)（2）繞坐標(biāo)原點(diǎn)旋轉(zhuǎn)（3）平移對(duì)象使基準(zhǔn)點(diǎn)回到原始位置二維基準(zhǔn)點(diǎn)

7、旋轉(zhuǎn)關(guān)于任意參照點(diǎn) 的放縮變換 步驟：（1）平移對(duì)象使基準(zhǔn)點(diǎn)與坐標(biāo)原點(diǎn)重合（2）放縮變換（3）反向平移使得基準(zhǔn)點(diǎn)回到初始位置二維基準(zhǔn)點(diǎn)放縮變換合成時(shí)，矩陣相乘的順序單次變換：列向量表示點(diǎn)復(fù)合變換：先作用的放在連乘的右端，后作用的放在連乘的左端點(diǎn)表示成行向量呢？小結(jié)對(duì)稱變換（反射變換、鏡像變換：reflection）(1)關(guān)于 x 軸的對(duì)稱變換(2)關(guān)于 y 軸的對(duì)稱變換 其它變換(3)關(guān)于任意軸的對(duì)稱變換平移（tx, ty)使l 過(guò)坐標(biāo)原點(diǎn)，記為T1旋轉(zhuǎn)，記R1對(duì)稱, 記SYx旋轉(zhuǎn)-,記 R2平移(-tx, -ty)， 記T2總變換：T2R2 SYxR1 T1其它變換錯(cuò)切變換（shear）依賴

8、軸：坐標(biāo)保持不變的坐標(biāo)軸，又稱參考軸方向軸：余下的坐標(biāo)軸1、以y 軸為依賴軸的錯(cuò)切變換 （1）以 y = 0為參考軸（坐標(biāo)保持不變）Shx是對(duì)y=1上的點(diǎn)沿x軸移動(dòng)的距離其它變換（2）以 y = yref 為參考軸x = x + shx ( y yref)y = y其它變換2、以x軸為依賴軸的錯(cuò)切變換其它變換仿射變換affine transformation二維線性變換的一般形式平移，旋轉(zhuǎn)，放縮，對(duì)稱和錯(cuò)切是特例特點(diǎn)：保持平行線間的平行關(guān)系其它變換例：證明二維復(fù)合變換的矩陣總能表示為： 證明：其它變換圖形對(duì)象點(diǎn)，線段，多邊形，圓，字符方法先生成點(diǎn)集，再對(duì)其中的點(diǎn)進(jìn)行變換運(yùn)算量大對(duì)參數(shù)變換線段：

9、兩個(gè)端點(diǎn)多邊形：各頂點(diǎn)圓：圓心和半徑前提圖形對(duì)象的幾何表示不發(fā)生變化圖形對(duì)象的幾何變換說(shuō)明在核心庫(kù)中，每種幾何變換是一個(gè)獨(dú)立的函數(shù)所有變換都是在三維坐標(biāo)系中定義基本幾何變換函數(shù)平移函數(shù)：glTranslatefd(tx, ty, tz)對(duì)二維變換而言，取tz0旋轉(zhuǎn)函數(shù)：glRotatefd(theta, vx, vy, vz)向量(vx, vy, vz)為通過(guò)坐標(biāo)原點(diǎn)旋轉(zhuǎn)軸theta為旋轉(zhuǎn)角的度數(shù)放縮函數(shù)：glScalefd(sx, sy, sz)相對(duì)坐標(biāo)原點(diǎn)的縮放當(dāng)參數(shù)為負(fù)時(shí)，相對(duì)于平面進(jìn)行對(duì)稱變換OpenGL 幾何變換函數(shù)例程序 6-1void display (void) glColor

10、3f (0.0, 0.0, 1.0); glRecti (20, 50, 100, 80); glColor3f (1.0, 0.0, 0.0); glTranslatef (-120.0, -40.0, 0.0); glRecti (20, 50, 100, 80); glLoadIdentity ( ); glColor3f (1.0, 0.0, 1.0); glRotatef (135.0, 0.0, 0.0, 1.0);glRecti (20, 50, 100, 80); glLoadIdentity ( ); glColor3f ( 0.0, 1.0,0.0); glScalef (

11、0.5, -1.0, 1.0); glRecti (20, 50, 100, 80); glColor3f(0,0,0); glLoadIdentity ( ); glBegin(GL_LINES); glVertex2i(0,-Y/2+5); glVertex2i(0,Y/2-5); glVertex2i(X/2-5,0); glVertex2i(-X/2+5,0); glEnd(); glutSwapBuffers();注意：幾何變換矩陣只有1個(gè)OpenGL幾何變換函數(shù)例6-1（續(xù)）#include #include #include int X=0,Y=0;void Reshape(in

12、t width, int height) glViewport(0, 0, width, height); glMatrixMode(GL_PROJECTION); glLoadIdentity(); gluOrtho2D(-width/2, width/2, -height/2, height/2); glMatrixMode(GL_MODELVIEW);/定義模型觀察變換矩陣 glLoadIdentity(); glClear (GL_COLOR_BUFFER_BIT); X=width, Y=height;void init (void) glClearColor (1.0, 1.0,

13、1.0, 0.0); glMatrixMode(GL_PROJECTION); gluOrtho2D(0.0, 250.0, 0.0, 250.0);OpenGL幾何變換函數(shù)（續(xù)）int main(int argc, char* argv) glutInit(&argc, argv); glutInitDisplayMode (GLUT_DOUBLE | GLUT_RGB); glutInitWindowSize (250, 250); glutInitWindowPosition (300, 300); glutCreateWindow (Transformation); init ();

14、glutDisplayFunc(display); glutReshapeFunc(Reshape); glutMainLoop(); return 0;OpenGL幾何變換函數(shù)矩陣操作模型觀察矩陣：glMatrixMode(GL_MODELVIEW)建立當(dāng)前觀察矩陣還有兩種方式：紋理和顏色模式，默認(rèn)為觀察模式設(shè)置當(dāng)前矩陣為單位矩陣：glLoadIdentity()矩陣棧4種：觀察、投影、紋理初始狀態(tài)下，每棧僅包含一單位棧壓棧：glPushMatrix()出棧：glPopMatrix()OpenGL幾何變換函數(shù)機(jī)器人手臂#include #define BASE_HEIGHT 2.0#defi

15、ne BASE_RADIUS 1.0#define LOWER_ARM_HEIGHT 5.0#define LOWER_ARM_WIDTH 0.5#define UPPER_ARM_HEIGHT 5.0#define UPPER_ARM_WIDTH 0.5typedef float point3;GLfloat theta = 0.0,0.0,0.0;GLint axis = 0;GLUquadricObj *p; /* pointer to quadric（二次曲面） object */void myinit() glClearColor(1.0, 1.0, 1.0, 1.0); glCol

16、or3f(1.0, 0.0, 0.0); p=gluNewQuadric(); /* allocate quadric object */ gluQuadricDrawStyle(p, GLU_LINE); /* render it as wireframe */OpenGL幾何變換函數(shù)例62（續(xù)）void base() glPushMatrix();/* rotate cylinder to align with y axis */ glRotatef(-90.0, 1.0, 0.0, 0.0);/* cyliner aligned with z axis*/ gluCylinder(p,

17、BASE_RADIUS, BASE_RADIUS, BASE_HEIGHT, 5, 5); glPopMatrix();void upper_arm() glPushMatrix(); glTranslatef(0.0, 0.5*UPPER_ARM_HEIGHT, 0.0); glScalef(UPPER_ARM_WIDTH, UPPER_ARM_HEIGHT, UPPER_ARM_WIDTH); glutWireCube(1.0); glPopMatrix();void lower_arm() glPushMatrix(); glTranslatef(0.0, 0.5*LOWER_ARM_H

18、EIGHT, 0.0); glScalef( LOWER_ARM_WIDTH, LOWER_ARM_HEIGHT, LOWER_ARM_WIDTH); glutWireCube(1.0); glPopMatrix();OpenGL幾何變換函數(shù)void display(void)/* Accumulate ModelView Matrix as we traverse tree */ glClear(GL_COLOR_BUFFER_BIT); glLoadIdentity(); glColor3f(1.0, 0.0, 0.0); glRotatef(theta0, 0.0, 1.0, 0.0);

19、 base(); glTranslatef(0.0, BASE_HEIGHT, 0.0); glRotatef(theta1, 0.0, 0.0, 1.0); lower_arm(); glTranslatef(0.0, LOWER_ARM_HEIGHT, 0.0); glRotatef(theta2, 0.0, 0.0, 1.0); upper_arm(); glutSwapBuffers();void mouse(int btn, int state, int x, int y)/* left button increase joint angle, right button decrea

20、ses it */if(btn=GLUT_LEFT_BUTTON & state = GLUT_DOWN)thetaaxis += 5.0; if( thetaaxis 360.0 ) thetaaxis -= 360.0;if(btn=GLUT_RIGHT_BUTTON & state = GLUT_DOWN) thetaaxis -= 5.0; if( thetaaxis 360.0 ) thetaaxis += 360.0; display();OpenGL幾何變換函數(shù)void menu(int id)/* menu selects which angle to change or wh

21、ether to quit */ if(id = 1 ) axis=0; if(id = 2) axis=1; if(id = 3 ) axis=2; if(id =4 ) exit(0);void myReshape(int w, int h) glViewport(0, 0, w, h); glMatrixMode(GL_PROJECTION); glLoadIdentity(); if (w = h) glOrtho(-10.0, 10.0, -5.0 * (GLfloat) h / (GLfloat) w, 15.0 * (GLfloat) h / (GLfloat) w, -10.0

22、, 10.0); else glOrtho(-10.0 * (GLfloat) w / (GLfloat) h, 10.0 * (GLfloat) w / (GLfloat) h, -5.0, 15.0, -10.0, 10.0); glMatrixMode(GL_MODELVIEW); glLoadIdentity();OpenGL幾何變換函數(shù)void main(int argc, char *argv) glutInit(&argc, argv); glutInitDisplayMode(GLUT_DOUBLE | GLUT_RGB | GLUT_DEPTH); glutInitWindo

23、wSize(500, 500); glutCreateWindow(robot); myinit(); glutReshapeFunc(myReshape); glutDisplayFunc(display); glutMouseFunc(mouse); glutCreateMenu(menu); glutAddMenuEntry(base, 1); glutAddMenuEntry(lower arm, 2); glutAddMenuEntry(upper arm, 3); glutAddMenuEntry(quit, 4); glutAttachMenu(GLUT_MIDDLE_BUTTO

24、N); glutMainLoop();OpenGL幾何變換函數(shù)使用雙緩沖以GLUT_DOUBLE為參數(shù)調(diào)用glutInitDisplayMode，設(shè)置雙緩存的窗口模式在顯示回調(diào)函數(shù)的最后調(diào)用glutSwapBuffers()函數(shù)空閑函數(shù)（回調(diào)函數(shù)）定義：如 void spinCube()功能：程序不受用戶干預(yù)時(shí)執(zhí)行spinCube()函數(shù)注冊(cè)：glutIdleFunc(spinCube) （在main函數(shù)中調(diào)用）重繪制函數(shù)glutPostRedisplay()OpenGL幾何變換函數(shù)旋轉(zhuǎn)的立方體8個(gè)頂點(diǎn)的顏色不同不停地旋轉(zhuǎn)鼠標(biāo)控制改變方向#include #include /定義頂點(diǎn)坐標(biāo)和顏色

25、GLfloat vertices3 = -1.0,-1.0,-1.0,1.0,-1.0,-1.0, 1.0,1.0,-1.0, -1.0,1.0,-1.0, -1.0,-1.0,1.0, 1.0,-1.0,1.0, 1.0,1.0,1.0, -1.0,1.0,1.0;GLfloat colors3 = 0.0,0.0,0.0,1.0,0.0,0.0, 1.0,1.0,0.0, 0.0,1.0,0.0, 0.0,0.0,1.0, 1.0,0.0,1.0, 1.0,1.0,1.0, 0.0,1.0,1.0; OpenGL幾何變換函數(shù)void polygon(int a, int b, int c

26、, int d)/* draw a polygon via list of vertices */ glBegin(GL_POLYGON); glColor3fv(colorsa); glVertex3fv(verticesa); glColor3fv(colorsb); glVertex3fv(verticesb); glColor3fv(colorsc); glVertex3fv(verticesc); glColor3fv(colorsd); glVertex3fv(verticesd); glEnd();void colorcube(void)/* map vertices to fa

27、ces */polygon(0,3,2,1);polygon(2,3,7,6);polygon(0,4,7,3);polygon(1,2,6,5);polygon(4,5,6,7);polygon(0,1,5,4);OpenGL幾何變換函數(shù)static GLfloat theta = 0.0,0.0,0.0;static GLint axis = 2;void display(void) /* display callback, clear frame buffer and z buffer, rotate cube and draw, swap buffers */ glClear(GL_C

28、OLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); glLoadIdentity(); glRotatef(theta0, 1.0, 0.0, 0.0); glRotatef(theta1, 0.0, 1.0, 0.0); glRotatef(theta2, 0.0, 0.0, 1.0); colorcube(); glutSwapBuffers();void spinCube()/* Idle callback, spin cube 2 degrees about selected axis */ thetaaxis += 2.0; if( thetaaxis 3

29、60.0 ) thetaaxis -= 360.0;/* display(); */ glutPostRedisplay();OpenGL幾何變換函數(shù)void mouse(int btn, int state, int x, int y)/* mouse callback, selects an axis about which to rotate */ if(btn=GLUT_LEFT_BUTTON & state = GLUT_DOWN) axis = 0; if(btn=GLUT_MIDDLE_BUTTON & state = GLUT_DOWN) axis = 1; if(btn=GL

30、UT_RIGHT_BUTTON & state = GLUT_DOWN) axis = 2;void myReshape(int w, int h) glViewport(0, 0, w, h); glMatrixMode(GL_PROJECTION); glLoadIdentity(); if (w = h) glOrtho(-2.0, 2.0, -2.0 * (GLfloat) h / (GLfloat) w, 2.0 * (GLfloat) h / (GLfloat) w, -10.0, 10.0); else glOrtho(-2.0 * (GLfloat) w / (GLfloat) h, 2.0 * (GLfloat) w / (GLfl