Computer Graphics transformations

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1. Computer Graphics 2D transformations

2. Overview Why transformations? Basic transformations: translation, rotation, scaling Combining transformations homogenous coordinates, transform. Matrices

3. Why transformation? Model of objects world coordinates: km, mm, etc. Hierarchical models:: human = torso + arm + arm + head + leg + leg arm = upperarm + lowerarm + hand … Viewing zoom in, move drawing, etc. Animation

4. Translation Translate over vector ( t x , t y ) x’=x+ t x , y’=y+ t y or x y P P+T T

5. Translation polygon Translate polygon: Apply the same operation on all points. Works always, for all transformations of objects defined as a set of points. x y T

6. Rotation x y P x y P’ a

7. Rotation around a point Q x y P x y P’ a Q P  Q

8. Scale with factor s x and s y : x’= s x x , y’= s y y or Scaling x y P x P’ Q’ Q

9. Scaling with respect to a point F Scale with factors s x and s y : P x ’= s x P x , P y ’= s y P y With respect to F : P x ’  F x = s x ( P x  F x ), P y ’  F y = s y ( P y  F y ) or P x ’= F x + s x ( P x  F x ), P y ’= F y + s y ( P y  F y ) x y P x P’ Q’ Q F P  F

10. Transformations Translate with V : T = P + V Scale with factor s x = s y = s : S = s P Rotate over angle a: R’ x = cos a P x  sin a P y R’ y = sin a P x + cos a P y x y P T S R

11. Transformations… Messy! Transformations with respect to points: even more messy! How to combine transformations?

12. Homogeneous coordinates 1 Uniform representation of translation, rotation, scaling Uniforme representation of points and vectors Compact representation of sequence of transformations

13. Homogeneous coordinates 2 Add extra coordinate: P = ( p x , p y , p h ) or x = ( x, y, h ) Cartesian coordinates: divide by h x = ( x/h , y/h ) Points: h = 1 (for the time being…), vectors: h = 0

14. Translation matrix

15. Rotation matrix

16. Scaling matrix

17. Inverse transformations

18. Combining transformations 1

19. Combining transformations 2

20. Combining transformations 3

21. Rotation around a point 1 R 1) 2) 3)

22. Rotation around a point 2 R 1) 2) 3)

23. Rotation around point 3 R 1) 2) 3)

24. Rotation around point 4 R 1) 2) 3)

25. Scaling w.r.t. point 1 F 1) 2) 3)

26. Scaling w.r.t.point 2 F 1) 2) 3)

27. Scaling w.r.t.point 3 F 1) 2) 3)

28. Order of transformations 1 x y x’ y’ x’’ y’’ x y x’ y’ x’’ y’’ Matrix multiplication does not commute. The order of transformations makes a difference! Rotation, translation… Translation, rotation…

29. Order of transformations 2 Pre-multiplication: P’ = M n M n-1 … M 2 M 1 P Transformation M n in global co o rdinates Post-multiplication: P’ = M 1 M 2 … M n-1 M n P Transformation M n in local co o rdinates: the coordinate system after application of M 1 M 2 … M n-1

30. Order of transformations 3 OpenGL: glRotate, glScale , etc.: Post-multiplication of current transformation matrix Always transformation in local coordinates Global coordinate version: read in reverse order

31. Order of transformations 4 glTranslate(…); glRotate(…); x y x’’ y’’ x’ y’ Local trafo interpretation Local transformations : x y x’’ y’’ Global transformations: x’ y’ Global trafo interpretation

32. Matrices in general rotation and scaling translation

33. Direct construction of matrix x y v A B T u If you know the target frame: Construct matrix directly. Define shape in nice local u,v coordinates, use matrix transformation to put it in x,y space.

34. Direct construction of matrix x y v A B T u If you know the target frame: Construct matrix directly.

35. Other 2D transformations Reflection Shear Can also be combined

36. Reflection over axis Reflext over x- axis: x’= x , y’=  y or x y

37. Reflect over origin Reflect over origin: x’=  x , y’=  y or x y

38. Shear Shear the y -as: x’= x+f , y’=y or x y 

39. 2D transformations summarized Transformations: modelling, viewing, animation; Several kinds of transformations; Homogeneous co ordinates; Combine transformations using matrix multiplication.