Transformation of objects 2D transformations 3D transformations Matrix representation

1. Transformation of objects 2D transformations 3D transformations Matrix representation OpenGL functions

2. Rendering Pipeline

3. Rendering Pipeline

4. Basic 2D Transformations

5. Basic 2D Transformations

6. Matrix representation

7. Combination of 2D transformations

8. 2x2 matrices

9. 2x2 matrices

10. 2x2 matrices

11. 2x2 matrices • Only rotation, scaling, and shear can be represented in 2x2 matrix form. • These are linear transformations.

12. Linear transformations

13. 2D Translation: 3x3 matrix

14. Homogeneous coordinates

15. Basic 2D transformations in homogeneous coordinates

16. Affine transformations

17. Matrix compositions

18. Matrix compositions

19. 3D transformations

20. Basic 3D transformations

21. Basic 3D transformations

22. Rotation about an arbitrary point Alternative Form for a Rotation About a Point. From Example 5.2.5, we have Qx= cos() Px - sin() Py + dx = cos() Px - sin() Py - cos() Vx + sin() Vy + Vx = cos()( Px – Vx) - sin() (Py – Vy) + Vx Qy is same as Qx.

25. P=(4,7) 45 degrees about V(5,4)

26. Reflection about the axis at angle 

28. (-3, 3), (0,3), (0,5) to (0,0) (2,0) (1, sqrt(3))

30. Rotate P(1,2,-1) after 45 degrees

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

33. 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.

34. OpenGL Matrix Stacks Manipulating stacks: the OpenGL calls to push, pop, or multiply top of stack glLoadMatrix {fd}(const TYPE *m); sets 16 value of the current matrix glMultMatrix {fd}(const TYPE *m)); multiply by m glPushMatrix(void); the topmost matrix is copied glPopMatrix(void); the topmost matrix is destroyed All vertices of the object are multiplied by the matrix.

35. 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

36. 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.

38. //additional methods for Canvas class //components of 2D transformation //<<<<<<<<<<<<<<< initCT >>>>>>>>>>>>>>>>> void Canvas:: initCT(void) { glMatrixMode(GL_MODELVIEW); glLoadIdentity(); // set CT to the identity matrix } //<<<<<<<<<<<<<< scale2D >>>>>>>>>>>>>>>>>>>> void Canvas:: scale2D(double sx, double sy) { glMatrixMode(GL_MODELVIEW); glScaled(sx, sy, 1.0); // set CT to CT * (2D scaling) } //<<<<<<<<<<<<<<< translate2D >>>>>>>>>>>>>>>>> void Canvas:: translate2D(double dx, double dy) { glMatrixMode(GL_MODELVIEW); glTranslated(dx, dy, 1.0); // set CT to CT * (2D translation) } //<<<<<<<<<<<<<<<< rotate2D >>>>>>>>>>>>>>>>>>>> void Canvas:: rotate2D(double angle) { glMatrixMode(GL_MODELVIEW); glRotated(angle, 0.0, 0.0, 1.0); // set CT to CT * (2D rotation) }

39. void Canvas:: pushCT(void) { glMatrixMode(GL_MODELVIEW); glPushMatrix(); // push a copy of the top matrix } void Canvas:: popCT(void) { glMatrixMode(GL_MODELVIEW); glPopMatrix(); // pop the top matrix from the stack }

41. gluLookAt()