Transformations

1. Transformations 고려대학교 컴퓨터 그래픽스 연구실 kucg.korea.ac.kr

2. Contents • Affine transformations • rotation, translation, and scaling • Transformations in homogeneous coordinates • Concatenation of transformations • rotation about a fixed point • general rotation • instance transformation • rotation about an arbitrary axis • OpenGL transformation matrices • Smooth rotation with a virtual trackball kucg.korea.ac.kr

3. u v R P T Q Transformations • Take a point (or vector) and map that point (or vector) into another point (or vector) 4D column matrices homogeneous coordinate transformation function kucg.korea.ac.kr

4. Affine Transformations (1/2) • Linearity – linear function • Linear transformation • transform the representation of a point (or vector) into another representation of a point (or vector) point 44 matrix vector kucg.korea.ac.kr

5. Affine Transformations (2/2) • Linear transformation (cont’) • preserve lines – transform a line into another line  only transform the endpoints of a line segment • Most transformations in CG are affine • rotation, translation, scaling, and shear homogeneous coordinate affine transformation kucg.korea.ac.kr

6. Translation • Operation that displace points by a fixed distance in a given direction • displacement vector d (a) object in original position (b) object translated kucg.korea.ac.kr

7. Rotation (1/2) • Simple example of 2D rotation kucg.korea.ac.kr

8. Rotation (1/2) • Simple example of 2D rotation kucg.korea.ac.kr

9. Rotation (1/2) • Simple example of 2D rotation kucg.korea.ac.kr

10. Rotation (2/2) • Needs • fixed point – a point is unchanged by the rotation • rotation angle – positive rotation (counterclockwise in right hand system) • rotation axis in 3D – values on axis are unchanged by the rotation (a) rotation about a fixed point (b) 3D rotation kucg.korea.ac.kr

11. Rigid-Body Transformations • Rotation and translation • No combination of rotations and translations can alter the shape of object alter only the object’s location and orientation affine transformations, but non-rigid body transformations kucg.korea.ac.kr

12. Scaling (1/2) • Make an object bigger or smaller • uniform – scaling in all directions • Affine non-rigid body transformation • affine transformation: translation, rotation, scaling, shear nonuniform uniform kucg.korea.ac.kr

13. Scaling (2/2) • Needs • fixed point • direction to scale • scale factor • longer (α>1) or smaller (0≤α<1) • Reflection – negative scale factor effect of scale factor reflection kucg.korea.ac.kr

14. Transformations in Homogeneous Coordinates • Representations in homogeneous coordinates • Affine transformation – 44 matrix kucg.korea.ac.kr

15. Translation • Point p to p’ by displacing by a distance d ? translation matrix kucg.korea.ac.kr

16. Translation • Point p to p’ by displacing by a distance d kucg.korea.ac.kr

17. Translation • Point p to p’ by displacing by a distance d • Inverse of a translation matrix ? kucg.korea.ac.kr

18. Translation • Point p to p’ by displacing by a distance d • Inverse of a translation matrix kucg.korea.ac.kr

19. Scaling • Scaling matrix with a fixed point of the origin ? scaling matrix kucg.korea.ac.kr

20. Scaling • Scaling matrix with a fixed point of the origin kucg.korea.ac.kr

21. Scaling • Scaling matrix with a fixed point of the origin • Inverse of a scaling matrix ? kucg.korea.ac.kr

22. Scaling • Scaling matrix with a fixed point of the origin • Inverse of a scaling matrix kucg.korea.ac.kr

23. Rotation (1/2) • Rotation with a fixed point at the origin ? rotation matrix kucg.korea.ac.kr

24. Rotation (1/2) • Rotation with a fixed point at the origin ? kucg.korea.ac.kr

25. Rotation (1/2) • Rotation with a fixed point at the origin ? kucg.korea.ac.kr

26. Rotation (1/2) • Rotation with a fixed point at the origin kucg.korea.ac.kr

27. Rotation (2/2) • Inverse of a rotation matrix ? kucg.korea.ac.kr

28. Rotation (2/2) • Inverse of a rotation matrix : orthogonal matrix kucg.korea.ac.kr

29. Shear (1/2) • One more affine transformation shear the object in the x direction ? kucg.korea.ac.kr

30. Shear (1/2) • One more affine transformation shear the object in the x direction kucg.korea.ac.kr

31. Shear (2/2) • Shear in the x direction ? shearing matrix kucg.korea.ac.kr

32. Shear (2/2) • Shear in the x direction kucg.korea.ac.kr

33. Shear (2/2) • Shear in the x direction • Inverse of a shearing matrix ? kucg.korea.ac.kr

34. Shear (2/2) • Shear in the x direction • Inverse of a shearing matrix kucg.korea.ac.kr

35. CBA p M q Concatenation of Transformations • Concatenating • affine transformations by multiplying together • sequences of the basic transformations  define an arbitrary transformation directly • ex) three successive transformations p A B C q kucg.korea.ac.kr

36. Rotation about a Fixed Point (1/3) • Fixed point: pf • apply Rz() to rotation about a fixed point rotation of a cube about its center kucg.korea.ac.kr

37. Rotation about a Fixed Point (2/3) sequence of transformations kucg.korea.ac.kr

38. Rotation about a Fixed Point (3/3) kucg.korea.ac.kr

39. General Rotation (1/2) • Three successive rotations about the three axes rotation of a cube about the z axis rotation of a cube about the y axis ? rotation of a cube about the x axis kucg.korea.ac.kr

40. General Rotation (2/2) kucg.korea.ac.kr

41. ? Instance Transformation (1/2) • Instance of an object’s prototype • occurrence of that object in the scene • Instance transformation • applying an affine transformation to the prototype to obtain desired size, orientation, and location instance transformation kucg.korea.ac.kr

42. Instance Transformation (2/2) kucg.korea.ac.kr

43. Rotation about an Arbitrary Axis (1/6) • Needs • fixed point: p0 • rotation angle: θ • rotation axis: vector p2-p1 rotation of a cube about an arbitrary axis kucg.korea.ac.kr

44. Rotation about an Arbitrary Axis (2/6) • First transformation is translation T(-p0) and the final one is T(p0) • Rotation problem!!! • we can get an arbitrary rotation from three rotations about individual axes • carry out two rotations to align the axis of rotation with the z axis • rotate by θ about the the z axis movement of the fixed point to the origin kucg.korea.ac.kr

45. Rotation about an Arbitrary Axis (3/6) • Determine x and y • direction angles and cosines sequence of rotations direction angles kucg.korea.ac.kr

46. Rotation about an Arbitrary Axis (4/6) • Determine x and y (cont’) • projection line segment into plane y=0 • look at the projection of line segment (before rotation) on the plane x=0 computation of the x rotation kucg.korea.ac.kr

47. Rotation about an Arbitrary Axis (5/6) • Determine x and y (cont’) • projection line segment into z axis • rotation about y axis • caution!!! – clockwise angle computation of the y rotation kucg.korea.ac.kr

48. Rotation about an Arbitrary Axis (6/6) • Finally concatenate all the matrices • Ex) rotate an object by 45 degrees about the line passing through the origin and the point (1,2,3), fixed point is the origin • solution – textbook p.195~196 kucg.korea.ac.kr

49. OpenGL Transformation Matrices (1/2) • CTM (Current Transformation Matrix) • matrix that is applied to any vertex • part of the pipeline • : replacement • : initialization operation: • : postmultiplication vertices vertices CTM kucg.korea.ac.kr

50. OpenGL Transformation Matrices (2/2) • Matrix modes • model-view and projection matrices • Three functions: rotation, translation, scaling vertices vertices model-view projection CTM glRotatef(angle, vx, vy, vz); glTranslatef(dx, dy, dz); glScalef(sx, sy, sz); kucg.korea.ac.kr