part 3 Martin Samuelčík Room I4

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1 art 3 Martin Sauelčík htt:// Roo I4

2 Vertex coordinates Fro inut coordinates to window coordinates Coordinates are always related to coordinates syste, sace, frae Transforing vertex = changing its coordinates = changing vertex coordinates fro one syste to another

3 Coordinate systes When working with vertex coordinates, it is always in soe sace Object sace this is inut sace for all vertices World sace ain coordinate syste of whole scene Eye (caera) sace syste where caera is always in oint (0, 0, 0) heading direction z

4 Coordinate systes Cli sace syste after rojection (3D to 2D) transforation is alied Noralized device coordinates interediate syste used for coutation of window coordinates, all coordinates of visible vertices are in the range <-1,1> Window (iage) sace syste inside window

5 Coordinate systes

6 Transforations Alying for vertices, norals, texture coordinates Changing coordinates of the sae vertex between coordinate systes Modeling transforation fro object to world syste Viewing transforation fro world to eye syste Projection transforation fro eye to cli syste Persective divide fro cli to noralized device syste Viewort transfor fro noralized devide to window syste

7 Transforations & systes

8 Transforations Transorfation 4x4 atrix Vertex 4x1 atrix Alying transforation to coordinates = atrices ultilication w z y x w z y x w z y x * ' ' ' '

9 Transforations Order of alying transforations is iortant Transforations are alied in reverse order as secified * * w z y x w z y x * * * *

10 OenGL transforations Model and view transforation in one atrix 3 tyes of transforations, state variables Modelview, Projection, Texture One active tye of transforation All oerations with atrix are alied to current tye of atrix void glmatrixmode(glenu ode) ode = GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE

11 Working with atrices void glloadidentity(void) set current atrix as identity atrix void glloadmatrix[fd](const TYPE *) set current atrix as atrix is array of floats or doubles with the count at least 16 void glmultmatrix[fd](const TYPE *) ultily current atrix with atrix and set current atrix to result is array of floats or doubles with the count at least 16

12 Exale glmatrixmode(gl_modelview); glloadidentity(); glmultmatrixf( 1 ); glmultmatrixf( 2 ); glmultmatrixf( 3 ); glbegin(gl_points); glvertex3fv(v); glend(); Vertex v is transfored using atrix 3, then 2, and then 1

13 Modelview atrix Modeling and view transforation together Model how object is ositioned in the world sace View how the caera is viewing world sace Modelview atrix = View * Model Each odelview atrix reresents soe local coordinate syste identity atrix is eye sace

14 Model atrix void gltranslate{fd}(type x, TYPE y, TYPE z) Coute translation atrix and utilies it with current atrix void glrotate{fd}(type angle, TYPE x, TYPE y, TYPE z) Coute rotation atrix and ultilies it with current atrix void glscale{fd}(type x, TYPE y, TYPE z) Coute rotation atrix and ultilies it with current atrix

15 Model atrix translation scaling rotation

16 Order atters gltranslate glrotate DrawFlowerot glrotate gltranslate DrawFlowerot

17 View atrix Transforation fro world to eye sace Secifies how caera views virtual sace In eye sace, caera has coordinates [0,0,0] and is heading in -z direction If view atrix is identity, then eye sace = world sace, vertices near [0,0,0] are not visible Basic view atrix gltranslatef(0,0,-5)

18 View atrix Function for setting view atrix fro GLU library Coutes view atrix and ultilies it with current atrix void glulookat(gldouble eyex, GLdouble eyey, GLdouble eyez, GLdouble centerx, GLdouble centery, GLdouble centerz, GLdouble ux, GLdouble uy, GLdouble uz) eye caera osition in world sace center interest osition in world sace u u vector in world sace

19 View atrix Let E be the 3d colun vector (eyex, eyey, eyez). Let C be the 3d colun vector (centerx, centery, centerz). Let U be the 3d colun vector (ux, uy, uz). Coute L = C - E. Noralize L. Coute S = L x U. Noralize S. Coute U' = S x L. M is the atrix whose coluns are, in order: (S, 0), (U', 0), (-L, 0), (-E, 1) (all colun vectors)

20 Modelview exale glmatrixmode(gl_modelview); glloadidentity(); // here is eye sace glulookat(5,4,3, 0,0,0, 0,1,0); // here is world sace gltranslatef(5,5,0); // here is first object sace DrawObject1(); gltranslatef(-5,0,2); glrotatef(45, 0,1,0); // here is second objects sace DrawObject2();

21 Projection atrix Transforing 3D to 2D Again 4x4 atrix result of rojection is 4-coonent vector Still reebering also z coordinate Defining viewing volue only vertices inside volue will be rendered

22 Persective rojection void glfrustu(gldouble left, GLdouble right, GLdouble botto, GLdouble to, GLdouble near, GLdouble far)

23 Persective rojection void glupersective(gldouble fovy, GLdouble asect, GLdouble znear, GLdouble zfar)

24 Orthograhic rojection void glortho(gldouble left, GLdouble right, GLdouble botto, GLdouble to, GLdouble near, GLdouble far) void gluortho2d(gldouble left, GLdouble right, GLdouble botto, GLdouble to)

25 Projection atrices glortho glfrustu

26 Cli & noralized sace Result of rojection transforation are coordinates of vertex in cli sace In cli sace, if x,y,z coordinates of vertex are in range <-w,w>, then are not clied Noralized sace x,y,z are divided by w In noralized sace, all reaining vertices have coordinates in range <-1,1> Noralized sace is used for better aing to window

27 Window sace Pixel coordinates of vertex together with its deth assed to rasterizer Viewort - secifies art of window where iage lane of rojection is aed void glviewort(glint x, GLint y, GLsizei width, GLsizei height) Asect ratio of viewing volue and viewort should be equal Deth range secifies interval for z coordinate gldethrange(n, f)

28 Projection & viewort void reshae(int w, int h) { glviewort(0, 0, w, h); gldethrange(0, 1); glmatrixmode(gl_projection); glloadidentity(); if (h == 0) h = 1; glupersective(80, (float)w/(float)h, 1.0, ); glmatrixmode(gl_modelview); glloadidentity(); }

29 Transforations conclusion

30 Retrieving transforations GLdouble odel[4*4]; glgetdoublev(gl_modelview_matrix, odel); GLdouble roj[4*4]; glgetdoublev(gl_projection_matrix, roj); GLint view[4]; glgetintegerv(gl_viewport, view);

31 Matrix stack Each tye of atrix (MODELVIEW, PROJECTON, TEXTURE) has its own stack Matrix on to of stack is current Working with current atrix void glpushmatrix(void) void glpomatrix(void) Pushing dulicate atrix on to Poing delete atrix fro to Use to reeber atrices for future use, for exale view atrix

32 Matrix stack exale glmatrixmode(gl_modelview); glloadidentity(); // here is eye sace glulookat(5,4,3, 0,0,0, 0,1,0); // here is world sace // reeber it for future use glpushmatrix(); gltranslatef(5,5,0); // here is first object sace DrawObject1(); // return to world sace glpomatrix(); // here is world sace again gltranslatef(-5,0,2); glrotatef(45, 0,1,0); // here is second objects sace DrawObject2();

33 The End! Questions?

OpenGL Transformations

OpenGL Transformations R. J. Renka Department of Computer Science & Engineering University of North Texas 02/18/2014 Introduction The most essential aspect of OpenGL is the vertex pipeline described in