1 Dr. Scott Schaefer Quaternions and Complex Numbers.

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Presentation transcript:

1 Dr. Scott Schaefer Quaternions and Complex Numbers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

17/75 Quaternions

18/75 Quaternions

19/75 Quaternions

20/75 Quaternions

21/75 Quaternion Multiplication

22/75 Quaternion Multiplication

23/75 Quaternion Multiplication

24/75 Quaternion Operations

25/75 Quaternion Operations

26/75 Quaternion Operations

27/75 Quaternion Operations

28/75 Quaternion Operations

29/75 Quaternion Operations

30/75 Quaternion Operations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

46/75 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

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

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

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

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

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

52/75 Quaternions and Interpolation Quaternion Interpolation

53/75 Quaternions and Interpolation Quaternion Interpolation

54/75 Quaternions and Interpolation Quaternion Interpolation

55/75 Quaternions and Interpolation Quaternion Interpolation

56/75 Quaternions and Interpolation Quaternion Interpolation

57/75 Quaternions and Interpolation Quaternion Interpolation

58/75 Quaternions and Interpolation Euler Angle Interpolation

59/75 Quaternions and Interpolation Euler Angle Interpolation

60/75 Quaternions and Interpolation Euler Angle Interpolation

61/75 Quaternions and Interpolation Euler Angle Interpolation

62/75 Quaternions and Interpolation Euler Angle Interpolation

63/75 Quaternions and Interpolation Euler Angle Interpolation

64/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

65/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

66/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

67/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

68/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

69/75 Quaternions and Interpolation Quaternion InterpolationEuler Angle Interpolation

70/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

71/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

72/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

73/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

74/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

75/75 Quaternions and Interpolation Euler Angle InterpolationQuaternion Interpolation

76/75

77/75 Exponential Forms Euler’s formula 

78/75 Quaternions in Exponential Form