1. 1 Lecture 7 Geometrical Transformations: 2D transformations 3D transformations Matrix representation OpenGL functions

2. 2 Rendering Pipeline

3. 3

4. 4

5. 5

6. 6 Basic 2D Transformations

7. 7

8. 8

9. 9 Matrix representation

10. 10 Combination of 2D transformations

11. 11 2x2 matrices

12. 12 2x2 matrices

13. 13 2x2 matrices

14. 14 2x2 matrices

15. 15 2x2 matrices Only rotation, scaling, and shear can be represented in 2x2 matrix form. These are linear transformations.

16. 16 Linear transformations

17. 17 2D Translation: 3x3 matrix

18. 18 Homogeneous coordinates

19. 19 Basic 2D transformations in homogeneous coordinates

20. 20 Affine transformations

21. 21 Projections

22. 22 Matrix compositions

23. 23 Matrix compositions

24. 24 Matrix compositions

25. 25 3D transformations

26. 26 Basic 3D transformations

27. 27 Basic 3D transformations

28. 28 OpenGL Matrix Stacks OpenGL works with stacks of 4x4 matrices: glMatrixMode(enum mode); mode: GL_MODELVIEW GL_PROJECTION GL_TEXTURE Specifies whether the modelview, projection or texture matrix will be modified.

29. 29 OpenGL Matrix Stacks glLoadIdentity(void); Clears the currently modifiable matrix for future transformation instructions. Typically we call this instruction before specifying modeling, viewing or projection transformations.

30. 30 OpenGL Matrix Stacks Manipulating stacks: the OpenGL calls to push, pop, or multiply top of stack glLoadMatrix(void); glMultMatrix(void); glPushMatrix(void); the topmost matrix is copied glPopMatrix(void); the topmost matrix is destroyed All vertices of the object are multiplied by the matrix.

31. 31 OpenGL Matrix Stacks Manipulating stacks: the OpenGL calls to push, pop, or multiply top of stack glLoadMatrix(const type *m); glMultMatrix(const type *m); m1 m5 m9 m13 m2 m6 m10 m14 m3 m7 m11 m15 m4 m8 m12 m16

32. 32 OpenGL Matrix Stacks Manipulating stacks: the OpenGL calls to push, pop, or multiply top of stack glPushMatrix(void); the topmost matrix is copied glPopMatrix(void); the topmost matrix is destroyed All vertices of the object are multiplied by the matrix.

33. 33 OpenGL Transformations glTranslate(TYPE x, TYPE y, TYPE z); Multiplies the current matrix by a translation matrix. glRotate(TYPE angle, TYPE x, TYPE y, TYPE z); Multiplies the current matrix by a rotation matrix. glScale(TYPE x, TYPE y, TYPE z); Multiplies the current matrix by a scaling matrix.

34. 34 Summary Types of transformations: –linear –affine –projective Representation of transformations: –3x3 and 4x4 matrices –homogeneous coordinates Compositions of transformations OpenGL Matrix Instructions