1. UniS CS297 Graphics with Java and OpenGL Viewing, the model view matrix

2. UniS 2 coordinates to pixel positions A series computer operations convert an object's three-dimensional coordinates to pixel positions on the screen. Transformations, which are represented by matrix multiplication, include –modeling, –viewing, –projection operations. Such operations include –rotation, –translation, –scaling, –reflecting, –orthographic projection, –and perspective projection.

3. UniS 3 coordinates to pixel positions Generally, you use a combination of several transformations to draw a scene. Since the scene is rendered on a rectangular window, objects (or parts of objects) that lie outside the window must be clipped. In three-dimensional computer graphics, clipping occurs by throwing out objects on one side of a clipping plane. Finally, a correspondence must be established between the transformed coordinates and screen pixels. This is known as a viewport transformation.

4. UniS 4 Overview: The Camera Analogy Set up your tripod and pointing the camera at the scene (viewing transformation). Arrange the scene to be photographed into the desired composition (modeling transformation). Choose a camera lens or adjust the zoom (projection transformation). Determine how large you want the final photograph to be - for example, you might want it enlarged (viewport transformation). After these steps are performed, the picture can be snapped or the scene can be drawn.

5. UniS 5 Overview: The Camera Analogy Set up your tripod to look at where the model will be created

6. UniS 6 Overview: The Camera Analogy Position viewing model of world which will contain final model

7. UniS 7 Overview: The Camera Analogy Create geometric objects for adding to viewing model

8. UniS 8 Overview: The Camera Analogy Insert geometric model in viewing model, and decide on perspective

9. UniS 9 Perspective choices glu.gluPerspective(10.0, (float) w / (float) h, 1.0, 20.0); glu.gluPerspective(100.0, (float) w / (float) h, 1.0, 20.0); Difference of perspective given by choice of camera angle, and no change in any other parameters

10. UniS 10 Perspective choices glu.gluPerspective(50.0, 10.0f, 1.0, 20.0); Difference of perspective given by choice aspect, and no change in any other parameters glu.gluPerspective(50.0, 0.5f, 1.0, 20.0);

11. UniS 11 Viewport Choose which part of scene is shown on screen gl.glViewport(40, 50, w + 500, h +400); gl.glViewport(0, 0, w, h);

12. UniS 12 Computer Vertex Transformations Pipeline Note that these steps correspond to the order in which you specify the desired transformations in your program, not necessarily the order in which the relevant mathematical operations are performed on an object's vertices. The viewing transformations must precede the modeling transformations in your code. You can specify the projection and viewport transformations at any point before drawing occurs.

13. UniS 13 Vertex Transformations x0 y0 z0 w0 object coordinate model view matrix projection matrix perspective division viewport transformation eye coordinates clip coordinates normalised device coordinates window coordinates

14. UniS 14 modelview matrix The viewing and modeling transformations you specify are combined to form the modelview matrix, which is applied to the incoming object coordinates to yield eye coordinates. Next, if you've specified additional clipping planes to remove certain objects from the scene or to provide cutaway views of objects, these clipping planes are applied.

15. UniS 15 clip coordinates After that, OpenGL applies the projection matrix to yield clip coordinates. This transformation defines a viewing volume; objects outside this volume are clipped so that they're not drawn in the final scene. After this point, the perspective division is performed by dividing coordinate values by w (the forth coordinate in a vertex), to produce normalized device coordinates. (See Appendix F of the Redbook for more information.) Finally, the transformed coordinates are converted to window coordinates by applying the viewport transformation. You can manipulate the dimensions of the viewport to cause the final image to be enlarged, shrunk, or stretched.

16. UniS 16 Matrix transformations To specify viewing, modeling, and projection transformations, you construct a 4 × 4 matrix A, which is then multiplied by the coordinates of each vertex v in the scene to accomplish the transformation. Note that viewing and modeling transformations are automatically applied to surface normal vectors, in addition to vertices. a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), x0 y0 z0 w0 x1 y1 z1 w1 =

17. UniS 17 Matrix transformations, the formulae a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), v1 v2 v3 v4 u1 u2 u3 u4 = Where: ui = a(i,1)v1 + a(i,2)v2 +a(i,3)v3 + a(i,4)v4

18. UniS 18 Matrix transformations, the formulae a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), v1 v2 v3 v4 u1 u2 u3 u4 = Where: ui = a(i,1)v1 + a(i,2)v2 +a(i,3)v3 + a(i,4)v4

19. UniS 19 Matrix transformations, the formulae a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), v1 v2 v3 v4 u1 u2 u3 u4 = Where: ui = a(i,1)v1 + a(i,2)v2 +a(i,3)v3 + a(i,4)v4

20. UniS 20 Matrix transformations, the formulae a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), v1 v2 v3 v4 u1 u2 u3 u4 = Where: ui = a(i,1)v1 + a(i,2)v2 +a(i,3)v3 + a(i,4)v4

21. UniS 21 Matrix transformations, the formulae a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), v1 v2 v3 v4 u1 u2 u3 u4 = Where: ui = a(i,1)v1 + a(i,2)v2 +a(i,3)v3 + a(i,4)v4

22. UniS 22 Rotation by matrix multiplication 0cos(θ),-sin(θ),0, sin(θ), 0, 1 0cos(θ),0, 0 1, x0 y0 z0 w0 x1 y1 z1 w1 =Rotation about the x-axis by θ degrees in a counter-clockwise direction

23. UniS 23 Rotation by matrix multiplication 0cos(θ), -sin(θ) 0, sin(θ), 0, 1 0 cos(θ), 0, 1, x0 y0 z0 w0 x1 y1 z1 w1 = Rotation about the y-axis by θ degrees in a counter-clockwise direction

24. UniS 24 Rotation by matrix multiplication Rotation about the z-axis by θ degrees in a counter-clockwise direction 01,0, 1 0cos(θ),-sin(θ),, 0sin(θ),cos(θ), x0 y0 z0 w0 x1 y1 z1 w1 =

25. UniS 25 Translation by matrix multiplication Translation along the X, Y and Z axes by x, y, and z respectively z1,0, 1 y1,0, x 1, v1 v2 v3 1 u1 u2 u3 1 = ASSUMES that final value in vertex is 1, if not then translation is scaled.

26. UniS 26 Matrix transformations, the formulae c(i,j) = a(i,1)b(1,j) + a(i,2)b(2,j) +a(i,3)b(3,j) + a(i,4)b(4,j) a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), b(3,4)b(3,3)b(3,2),b(3,1), b(4,3) b(2,3) b(1,3) b(4,4)b(4,2),b(4,1), b(2,4)b(2,2),b(2,1), b(1,4)b(1,2),b(1,1), X = c(3,4)c(3,3)c(3,2),c(3,1), c(4,3) c(2,3) c(1,3) c(4,4)c(4,2),c(4,1), c(2,4)c(2,2),c(2,1), c(1,4)c(1,2),c(1,1),

27. UniS 27 Matrix transformations, the formulae c(i,j) = a(i,1)b(1,j) + a(i,2)b(2,j) +a(i,3)b(3,j) + a(i,4)b(4,j) a(3,4)a(3,3)a(3,2),a(3,1), a(4,3) a(2,3) a(1,3) a(4,4)a(4,2),a(4,1), a(2,4)a(2,2),a(2,1), a(1,4)a(1,2),a(1,1), b(3,4)b(3,3)b(3,2),b(3,1), b(4,3) b(2,3) b(1,3) b(4,4)b(4,2),b(4,1), b(2,4)b(2,2),b(2,1), b(1,4)b(1,2),b(1,1), X = c(3,4)c(3,3)c(3,2),c(3,1), c(4,3) c(2,3) c(1,3) c(4,4)c(4,2),c(4,1), c(2,4)c(2,2),c(2,1), c(1,4)c(1,2),c(1,1),

28. UniS 28 Matrix transformations, the formulae To first apply a rotation (matrix R) and then a translation (matrix T) to a vertex v the matrix multiplication formula is new_v = T.R.v = (T.R) v = T (R.v) This is not the same as R.T.v. The order of matrix multiplication is important.

29. UniS 29 A Simple Example: Drawing a Cube

30. UniS 30 The code public void display(GLAutoDrawable drawable) { GL gl = drawable.getGL( ); GLU glu = new GLU(); GLUT glut = new GLUT(); gl.glClearColor(0.0f, 0.0f, 0.0f, 0.0f); gl.glClear(GL.GL_COLOR_BUFFER_BIT); gl.glColor3f(1.0f, 1.0f, 1.0f); gl.glLineWidth(3.0f); gl.glLoadIdentity (); gl.glTranslatef (0.0f, 0.0f, -3.0f); gl.glRotatef (30.0f,0.0f,1.0f,0.0f); glut.glutWireCube(0.9f); } public void reshape( GLAutoDrawable drawable, int x, int y, int width, int height) { GL gl = drawable.getGL( ); GLU glu = new GLU(); gl.glViewport(0, 0, width, height); gl.glMatrixMode(GL.GL_PROJECTION); gl.glLoadIdentity (); glu.gluPerspective(50.0, 1.0, 1.0, -5.0); gl.glMatrixMode (GL.GL_MODELVIEW); }

31. UniS 31 The matrix transformations gl.glLoadIdentity (); gl.glTranslatef(0.0f, 0.0f, -3.0f); gl.glRotatef(30.0f,0.0f,1.0f,0.0f);

32. UniS 32 glLoadIdentity In the particular context of the display method, this command will set the current model view matrix to the identity matrix Id: 01,0, 1 01,0, 0 1,

33. UniS 33 glTranslatef(x,y,z); this command multiplies the current model view matrix M = Id, by this: z1,0, 1 y1,0, x 1, T = So that the model matrix is then M1 = M.T = Id.T

34. UniS 34 glRotatef( θ, 0,1,0) this command multiplies the current model view matrix M, by this: R = So that the model view matrix is then M1.R = M.T.R = Id.T.R 0cos(θ), -sin(θ) 0, sin(θ), 0, 1 0 cos(θ), 0, 1,

35. UniS 35 glut.glutWireCube(0.9f) When this line is now executed: each vertex v in the cube is created then each vertex is multiplied by the current model view matrix M1 the new vertices M1.v are then used to create the geometric model.