H331: Computer Graphics

1. H331: Computer Graphics http://www.cs.kuleuven.ac.be/~graphics/H331/Philip Dutré Department of Computer Science Wednesday, March 3

2. Today • Transformations of Objects • Perspective Viewing • Book: Chapter 5, 7

3. Transformations • Last week: window-to-viewport transformation • But … we also want to transform objects in 2D and 3D.

4. Coordinate systems: 2D & 3D Y Z Y X Y X Left-handed Z X Right-handed

5. 3D Vectors • 3 scalars to describe vector • Add: a+b • Subtract: a-b b a b a

6. 3D Vectors • Linear combination (s and t scalars) c = s.a + t.b • Affine: s + t = 1 • Convex: s, t >= 0

7. 3D Vectors b c Convex combinations a

8. 3D Vectors • Length: |a| = sqrt(ax*ax+ay*ay+az*az) • Dot-product (inner product): a.b = ax*bx + ay*by + az*bz

9. 3D Vectors • Angle between 2 vectors • b.c = |b||c|cos(f) cos(f) = b.c / |b||c| b f a

10. 3D Vectors • Cross-product (outer product) a x b = (ay*bz - az*by, az*bx - ax*bz, ax*by - ay*bx ) Area = |a x b| b a x b a

11. Homogeneous coordinates • Points: (x y z 1) • Vectors: (x y z 0) • Why?Points and vectors can now be mixed in operations • Key concept in computer graphics

12. Homogeneous coordinates • E.g. subtraction:(* * * 1) – (* * * 1) = (* * * 0) • E.g. addition(* * * 1) + (* * * 0) = (* * * 1) • Affine linear combinations of points produce another point

13. Why are transformations useful? • Define objects only once, then transform to compose bigger objects • Translations, rotations, scaling, …

14. Why are transformations useful? • Define motif, then construct object from motif Use 8 times

15. Why are transformations useful? • Move camera around

16. Why are transformations useful? • Computer animation • Translate / rotate / warp object over time

17. 2D transformations • Each point can be expressed in a coordinate system: P = Pxi + Pyj + origin P (Px, Py,1) j Homogeneous coordinates origin i

18. 2D Transformations • Q = T(P) • Affine transformation: coordinates of Q are linear combination of coordinates of P

19. Affine 2D Transformations translation rotation scaling shear

20. 2D Translation Y Y (Qx, Qy,1) Tx Ty (Px, Py,1) X X

21. 2D Scaling Y Y (Qx, Qy,1) (Px, Py,1) X X

22. 2D Rotation Y (Qx, Qy,1) (Px, Py,1) R  R + f X

23. 2D Shear Y Y (Qx, Qy,1) (Px, Py,1) X X

24. Inverse transformations

25. Inverse transformations

26. Composing transformations • Rotate over 30 degrees • Translate Matrix multiply

27. Composite transformations • Order is important!!! Y Y Y rotate translate X X X Y Y Y rotate translate X X X

28. Composite transformations • E.g. Rotating about point • Standard rotation is about origin • Translate rotation centre to origin • Rotate around origin • Translate origin back to rotation centre

29. Composite transformations Y R X Y X Y Y X X R Tx -Ty Ty -Tx X

30. Composite transformations

31. Window to viewport: revisited • Window to viewport transformation can also be expressed as 2D transform!!! Object Coordinate System Screen Coordinate System viewport window

32. Properties of 2D transforms • Affine transformations preserve affine linear combinations

33. Properties of 2D transforms • Affine transformations preserve lines and planes • Line:

34. Properties of 2D transforms • Parallellism of lines and planes is preserved • Parallel lines:

35. Other properties • Relative ratios are preserved • Areas:

36. 3D affine transformations • Same idea, but 4x4 matrices

37. 3D Translation Z Z Y Y TZ TY X X TX

38. 3D Scaling Z Z Y Y SY SZ X X SX

39. 3D Rotation around X-axis Z Z qx Y Y X X

40. 3D Rotation around Y-axis Z Z qy Y Y X X

41. 3D Rotation around Z-axis Z Z Y Y qz X X

42. Composite 3D transformations • Same ideas as 2D • E.g. rotation around arbitrary axis u • 2 rotations such that u is aligned with x-axis • X-rotation over desired angle • Undo the two rotations to restore u to original direction

43. Properties of 3D transformations • Preservation of affine linear combinations • Preservation of lines and planes • Parallelism of lines and planes is preserved • Columns reveal transformed coordinate frame • Ratios are preserved • Volumes are scaled by |detM|

44. Changing coordinate systems • Different way of thinking about coordinate transforms • More natural: • Objects are modeled in their own coordinate system • What are the coordinates of the the transformed object in the world coordinate system?

45. Changing coordinate systems World coordinate system Object coordinate systems

46. Transform of coordinate system • origin = (0 0 1); i = (1 0 0); j = (0 1 0)

47. Transform of coordinate system m2 m1 m3 j origin i

48. Changing coordinate systems j’ i’ o’=M.origin i’ = M.i j’ = M.j o’ j origin i M transforms (origin, i, j) to (o’, i’, j’)

49. Changing coordinate systems j’ P b d i’ P = (c,d,1) in (o’, i’, j’) c o’ j origin a i What are the coordinates (a,b,1) of P in (origin, i, j)?

50. Changing coordinate systems j’ P b d i’ c o’ j origin a i