Matrix-Rechnung I ' z =... Universität Frankfurt

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Noah Caldwell
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1 Matrix-Rechnung I x =Ax = = ' ' ' ' ' z y x a a a a w z y x

2 Matrix-Rechnung II Matrixmultiplikation ist assoziativ, abernicht kommutativ. OpenGL multipliziert Matrizen von rechts: ( A ( Ax) ) 1 x' = AAAx = A Merke: Die zuletzt definierte Transformation wird als erste auf das Objekt angewendet.

3 Transformation der Normalen I Normalenform der Tangential-Ebene: nx d = ( ) T nx ny nz x d = T ( a b c d) bezeichnet eine Ebene mit: T ( a b c d ) x = für alle Punkte der Ebene und T ( a b c ) Normalenvektor

4 Transformation der Normalen II n T x = n T Ex = n T A 1 Ax = ( ) ) T A 1 T n ( Ax) =

5 Modell-Transformation Translation = z y x t t t z y x x'

6 Modell-Transformation Rotation um die x-achse Θ Θ Θ Θ = 1 1 cos sin sin cos 1 z y x x'

7 Modell-Transformation Skalierung = 1 1 z y x s s s z y x x'

8 Transformationen I void glmatrixmode (GLenum mode) mode: GL_MODELVIEW, GL_PROJECTION,

9 Transformationen II void glloadidentity(void) void glloadmatrix{fd} (const TYPE *m) m[]={m 11,m 21,, m 12, m 22,, m 44 } void gltranslate{fd} (TYPE x, TYPE y, TYPE z) void glrotate{fd} (TYPE angle, TYPE x, TYPE y, TYPE z) void glscale{fd} (TYPE x, TYPE y, TYPE z)

10 Matrix-Stack I void glpushmatrix (void) void glpopmatrix (void)

11 Matrix-Stack II Beispiel

12 Matrix-Stack III glmatrixmode(gl_modelview); // Reset the modelview matrix glloadidentity(); // Translate the whole scene out and into view // This is the initial viewing transformation gltranslatef(.f,.f, -2.f); glcolor3ub(255,, ); // Red Nucleus glutsolidsphere(1.f, 15, 15); glcolor3ub(255,255,); // Yellow Electrons // First Electron Orbit glpushmatrix(); // Save viewing transformation glrotatef(felect1,.f, 1.f,.f); gltranslatef(9.f,.f,.f); glutwiresphere(6.f, 15, 15); // Rotate by angle of revolution // Translate out from origin to orbit distance // Draw the electron glpopmatrix(); // Restore the viewing transformation

13 // Second Electron Orbit glpushmatrix(); glrotatef(45.f,.f,.f, 1.f); glrotatef(felect1,.f, 1.f,.f); gltranslatef(-7.f,.f,.f); glutwiresphere(6.f, 15, 15); glpopmatrix(); Matrix-Stack IV // Third Electron Orbit glpushmatrix(); glrotatef(36.f-45.f,.f,.f, 1.f); glrotatef(felect1,.f, 1.f,.f); gltranslatef(.f,.f, 6.f); glutwiresphere(6.f, 15, 15); glpopmatrix(); // Increment the angle of revolution felect1 += 1.f; if(felect1 > 36.f) felect1 =.f;

14 Viewing Pipeline

15 Viewing Kamera-Analogie:

16 Viewing Transformation OpenGL: Augpunkt im Ursprung Blickrichtung entlang der negativen z-achse Viewing Transformation: void glulookat( GLdouble eye_x, GLdouble eye_y, GLdouble eye_z, GLdouble center_x, GLdouble center_y, GLdouble center_z, GLdouble up_x, GLdouble up_y, GLdouble up_z)

17 Projektions-Transformation I Parallel-Projektion

18 Parallel-Projektion I View/Clipping Volume

19 Parallel-Projektion II glmatrixmode(gl_projection); glloadidentity(); void glortho( GLdouble left, GLdouble right, GLdouble bottom, GLdouble top, Gldouble near, Gldouble far )

20 Projektions-Transformation II Perspektivische Projektion

21 Perspektivische Projektion I View/Clipping Volume

22 Perspektivische Projektion II void glfrustum (GLdouble left, GLdouble right, GLdouble bottom, GLdouble top, GLdouble near, GLdouble far)

23 Perspektivische Projektion III void gluperspective (GLdouble fovy, GLdouble aspect_ratio, GLdouble near, GLdouble far)

24 Viewport Transformation void glviewport(glint x, Glint y, GLsizei width, GLsizei height)

25 void reshape (int w, int h) { if (h==) h=1; WindowWidth = w; WindowHeight = h; Viewing Putting it all together I glmatrixmode (GL_PROJECTION); glloadidentity (); gluperspective(3., (GLfloat)w / (GLfloat)h, 1.5, 2); glmatrixmode (GL_MODELVIEW); }

26 void display(void) { glclear (GL_COLOR_BUFFER_BIT); Viewing Putting it all together II glviewport (,, (GLsizei) WindowWidth/2, (GLsizei) WindowHeight/2); //Set Viewport 1 glcolor3f (1., 1., 1.); glloadidentity (); /* clear the modeling matrix */ // viewing transformation glulookat (vx, 3., vz,.,.,.,., 1.,.); glscalef (1., 2., 1.); /* modeling transformation */ glutwirecube (1.); //Set Viewport 2 glviewport (, WindowHeight/2, (GLsizei) WindowWidth/2, (GLsizei) WindowHeight/2); glcolor3f (1., 1.,.); glloadidentity (); /* clear the modeling matrix */ //viewing transformation glulookat (vx,., vz,.,.,.,., 1.,.); glscalef (1., 2., 1.); /* modeling transformation */ glutwirecube (1.);

27 Viewing Putting it all together III //Set Viewport 3 glviewport (WindowWidth/2, WindowHeight/4, (GLsizei) WindowWidth/2, (GLsizei) WindowHeight/2); glcolor3f (., 1., 1.); glloadidentity (); /* clear the modeling matrix */ /* viewing transformation */ glulookat (vx, 5., vz,.,.,.,., 1.,.); glscalef (1., 3., 1.); /* modeling transformation */ glutwirecube (1.); } glutswapbuffers();

28 Viewing Reverse Transformation int gluunproject (GLdouble winx, GLdouble winy, GLdouble winz, const GLdouble modelmatrix[16], const GLdouble projmatrix[16], const Glint viewport[4], GLdouble *worldx, GLdouble *worldy, GLdouble *worldz )

29 Reverse Transformation Beispiel I void reshape(int w, int h) { glviewport (,, (GLsizei) w, (GLsizei) h); glmatrixmode(gl_projection); glloadidentity(); gluperspective (45., (GLfloat) w/(glfloat) h, 1., 1.); glmatrixmode(gl_modelview); glloadidentity(); }

30 Reverse Transformation Beispiel II void mouse(int button, int state, int x, int y) { GLint viewport[4]; GLdouble mvmatrix[16], projmatrix[16]; GLint realy; /* OpenGL y coordinate position is different from MS Windows coordinate */ GLdouble wx, wy, wz; /* returned world x, y, z coords */ switch (button) { case GLUT_LEFT_BUTTON: if (state == GLUT_DOWN) { glgetintegerv (GL_VIEWPORT, viewport); glgetdoublev (GL_MODELVIEW_MATRIX, mvmatrix); glgetdoublev (GL_PROJECTION_MATRIX, projmatrix); /* note viewport[3] is height of window in pixels */ realy = viewport[3] - (GLint) y - 1; printf ("Coordinates at cursor are (%4d, %4d)\n", x, realy); gluunproject ((GLdouble) x, (GLdouble) realy,., mvmatrix, projmatrix, viewport, &wx, &wy, &wz); printf ("World coords at z=. are (%f, %f, %f)\n", wx, wy, wz); gluunproject ((GLdouble) x, (GLdouble) realy, 1., mvmatrix, projmatrix, viewport, &wx, &wy, &wz); printf ("World coords at z=1. are (%f, %f, %f)\n", wx, wy, wz); } break;...

Computer Graphics. Bing-Yu Chen National Taiwan University

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