1. Quaternions and Complex Numbers Dr. Scott Schaefer

2. Complex Numbers • Defined by real and imaginary part • where

3. Complex Numbers • Defined by real and imaginary part • where

4. Complex Numbers • Defined by real and imaginary part • where

5. Complex Numbers • Defined by real and imaginary part • where

6. Complex Numbers • Defined by real and imaginary part • where

7. Complex Numbers • Defined by real and imaginary part • where

8. Complex Numbers • Defined by real and imaginary part • where

9. Complex Numbers • Defined by real and imaginary part • where

10. Complex Numbers • Defined by real and imaginary part • where

11. Complex Numbers • Defined by real and imaginary part • where

12. Complex Numbers and Rotations • Given a point (x,y), rotate that point about the origin by

13. Complex Numbers and Rotations • Given a point (x,y), rotate that point about the origin by

14. Complex Numbers and Rotations • Given a point (x,y), rotate that point about the origin by

15. Complex Numbers and Rotations • Given a point (x,y), rotate that point about the origin by Multiplication is rotation!!!

16. Quaternions – History • Hamilton attempted to extend complex numbers from 2D to 3D… impossible • 1843 Hamilton discovered a generalization to 4D and wrote it on the side of a bridge in Dublin • One real part, 3 complex parts

17. Quaternions

18. Quaternions

19. Quaternions

20. Quaternions

21. Quaternion Multiplication

22. Quaternion Multiplication

23. Quaternion Multiplication

24. Quaternion Operations

25. Quaternion Operations

26. Quaternion Operations

27. Quaternion Operations

28. Quaternion Operations

29. Quaternion Operations

30. Quaternion Operations

31. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

32. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

33. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

34. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

35. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

36. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

37. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

38. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

39. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

40. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

41. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

42. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

43. Quaternions and Rotations • Claim: unit quaternions represent 3D rotation

44. Quaternions and Rotations • The quaternion representing rotation about the unit axis v by is

45. Quaternions and Rotations • The quaternion representing rotation about the unit axis v by is • To convert to matrix, assume q=(s,v) and |q|=1

46. Quaternions vs. Matrices • Quaternions take less space (4 numbers vs. 9 for matrices) • Rotating a vector requires 28 multiplications using quaternions vs. 9 for matrices • Composing two rotations using quaternions q1q2 requires 16 multiples vs. 27 for matrices • Quaternions are typically not hardware accelerated whereas matrices are

47. Quaternions and Interpolation • Given two orientations q1 and q2, find the orientation halfway between

48. Quaternions and Interpolation • Given two orientations q1 and q2, find the orientation halfway between

49. Quaternions and Interpolation • Unit quaternions represent points on a 4D hyper-sphere • Interpolation on the sphere gives rotations that bend the least

50. Quaternions and Interpolation • Unit quaternions represent points on a 4D hyper-sphere • Interpolation on the sphere gives rotations that bend the least