1. 1 Dr. Scott Schaefer Quaternions and Complex Numbers

2. 2/75 Complex Numbers Defined by real and imaginary part  where

3. 3/75 Complex Numbers Defined by real and imaginary part  where

4. 4/75 Complex Numbers Defined by real and imaginary part  where

5. 5/75 Complex Numbers Defined by real and imaginary part  where

6. 6/75 Complex Numbers Defined by real and imaginary part  where

7. 7/75 Complex Numbers Defined by real and imaginary part  where

8. 8/75 Complex Numbers Defined by real and imaginary part  where

9. 9/75 Complex Numbers Defined by real and imaginary part  where

10. 10/75 Complex Numbers Defined by real and imaginary part  where

11. 11/75 Complex Numbers Defined by real and imaginary part  where

12. 12/75 Complex Numbers and Rotations Given a point (x,y), rotate that point about the origin by

13. 13/75 Complex Numbers and Rotations Given a point (x,y), rotate that point about the origin by

14. 14/75 Complex Numbers and Rotations Given a point (x,y), rotate that point about the origin by

15. 15/75 Complex Numbers and Rotations Given a point (x,y), rotate that point about the origin by Multiplication is rotation!!!

16. 16/75 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. 17/75 Quaternions

18. 18/75 Quaternions

19. 19/75 Quaternions

20. 20/75 Quaternions

21. 21/75 Quaternion Multiplication

22. 22/75 Quaternion Multiplication

23. 23/75 Quaternion Multiplication

24. 24/75 Quaternion Operations

25. 25/75 Quaternion Operations

26. 26/75 Quaternion Operations

27. 27/75 Quaternion Operations

28. 28/75 Quaternion Operations

29. 29/75 Quaternion Operations

30. 30/75 Quaternion Operations

31. 31/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

32. 32/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

33. 33/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

34. 34/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

35. 35/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

36. 36/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

37. 37/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

38. 38/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

39. 39/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

40. 40/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

41. 41/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

42. 42/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

43. 43/75 Quaternions and Rotations Claim: unit quaternions represent 3D rotation

44. 44/75 Quaternions and Rotations The quaternion representing rotation about the unit axis v by is

45. 45/75 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. 46/75 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 q 1 q 2 requires 16 multiples vs. 27 for matrices Quaternions are typically not hardware accelerated whereas matrices are

47. 47/75 Quaternions and Interpolation Given two orientations q 1 and q 2, find the orientation halfway between

48. 48/75 Quaternions and Interpolation Given two orientations q 1 and q 2, find the orientation halfway between

49. 49/75 Quaternions and Interpolation Unit quaternions represent points on a 4D hyper-sphere Interpolation on the sphere gives rotations that bend the least

50. 50/75 Quaternions and Interpolation Unit quaternions represent points on a 4D hyper-sphere Interpolation on the sphere gives rotations that bend the least

51. 51/75 Quaternions and Interpolation Unit quaternions represent points on a 4D hyper-sphere Interpolation on the sphere gives rotations that bend the least May need to interpolate between q 1 and q 2

52. 52/75 Quaternions and Interpolation Quaternion Interpolation

53. 53/75 Quaternions and Interpolation Quaternion Interpolation

54. 54/75 Quaternions and Interpolation Quaternion Interpolation

55. 55/75 Quaternions and Interpolation Quaternion Interpolation

56. 56/75 Quaternions and Interpolation Quaternion Interpolation

57. 57/75 Quaternions and Interpolation Quaternion Interpolation

58. 58/75 Quaternions and Interpolation Euler Angle Interpolation

59. 59/75 Quaternions and Interpolation Euler Angle Interpolation

60. 60/75 Quaternions and Interpolation Euler Angle Interpolation

61. 61/75 Quaternions and Interpolation Euler Angle Interpolation

62. 62/75 Quaternions and Interpolation Euler Angle Interpolation

63. 63/75 Quaternions and Interpolation Euler Angle Interpolation

64. 64/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

65. 65/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

66. 66/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

67. 67/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

68. 68/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

69. 69/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

70. 70/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

71. 71/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

72. 72/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

73. 73/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

74. 74/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

75. 75/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

76. 76/75

77. 77/75 Exponential Forms Euler’s formula 

78. 78/75 Quaternions in Exponential Form