1. CSCE 689: Computer Animation Rotation Representation and Interpolation
Jinxiang Chai

2. Joints and Rotation
Rotational dofs are widely used in character animation
3 translation dofs
48 rotational dofs
1 dof: knee
2 dof: wrist
3 dof: shoulder

3. Orientation vs. Rotation
Orientation is described relative to some reference alignment
A rotation changes object from one orientation to another
Can represent orientation as a rotation from the reference alignment 

4. Ideal Orientation Format
Represent 3 degrees of freedom with minimum number of values
Allow concatenations of rotations
Math should be simple and efficient
concatenation
interpolation
rotation

5. Outline Rotation matrix Fixed angle and Euler angle Axis angle
Quaternion

6. 3D Transformation A 3D point (x,y,z) – x,y, and z coordinates
We will still use column vectors to represent points
Homogeneous coordinates of a 3D point
(x,y,z,1)T
Transformation will be performed using 4x4 matrix

7. Right-handed Coordinate System
Left hand coordinate system
Not used in this class and
Not in OpenGL

8. Homogenous coordinates
3D Transformation
Very similar to 2D transformation
Translation transformation
Homogenous coordinates

9. Homogenous coordinates
3D Transformation
Very similar to 2D transformation
Scaling transformation
Homogenous coordinates

10. 3D Rotation 3D rotation is done around a rotation axis
Fundamental rotations – rotate about x, y, or z axes
Counter-clockwise rotation is referred to as positive rotation (when you look down negative axis) y + x z

11. 3D Rotation
Rotation about z – similar to 2D rotation y + x z

12. 3D Rotation Rotation about y: z -> y, y -> x, x->z z y x y x

13. 3D Rotation Rotation about x (z -> x, y -> z, x->y) x z y y x
Transformation equations for rotations about the other two coordinate axes can be obtained with a cyclic permutation of the coordinate parameters, y x z

14. 3D Rotation
An arbitrary rotation can be defined by a matrix y x z

15. Matrices as Orientation
Matrices just fine, right?
No…
9 values to interpolate
don’t interpolate well

16. Representation of orientation
Homogeneous coordinates (review):
4X4 matrix used to represent translation, scaling, and rotation
a point in the space is represented as
Treat all transformations the same so that they can be easily combined

17. Rotation
New points
rotation matrix
old points

18. Interpolation
In order to “move things”, we need both translation and rotation
Interpolating the translation is easy, but what about rotations?

19. Interpolation of Orientation
How about interpolating each entry of the rotation matrix?
The interpolated matrix might no longer be orthonormal, leading to nonsense for the inbetween rotations

20. Interpolation of Orientation
Example: interpolate linearly from a positive 90 degree rotation about y axis to a negative 90 degree rotation about y
Linearly interpolate each component and halfway between, you get this...
Rotate about y-axis with 90
Rotate about y-axis with -90

21. Properties of Rotation Matrix
Easily composed?
Interpolation?
Compact representation?

22. Properties of Rotation Matrix
Easily composed? yes
Interpolation?
Compact representation?

23. Properties of Rotation Matrix
Easily composed? yes
Interpolation? not good
Compact representation?

24. Properties of Rotation Matrix
Easily composed? yes
Interpolation? not good
Compact representation?
- 9 parameters (only needs 3 parameters)

25. Outline Rotation matrix Fixed angle and Euler angle Axis angle
Quaternion

26. Fixed Angles Angles used to rotate about fixed axes
Orientations are specified by a set of 3 ordered parameters that represent 3 ordered rotations about fixed axes
Many possible orderings: x-y-z, x-y-x,y-x-z
- as long as axis does immediately follow itself such as x-x-y

27. Fixed Angles
Ordered triple of rotations about global axes, any triple can be used that doesn’t immediately repeat an axis, e.g., x-y-z, is fine, so is x-y-x. But x-x-z is not. X Z Y
E.g., (qz, qy, qx)
Q = Rx(qx). Ry(qy). Rz(qz). P

28. Euler Angles vs. Fixed Angles
One point of clarification
Euler angle
- rotates around local axes
Fixed angle
- rotates around world axes
Rotations are reversed
- x-y-z Euler angles == z-y-x fixed angles

29. Euler Angle Interpolation
Interpolating each components separately
Might have singularity problem
Halfway between (0, 90, 0) & (90, 45, 90)
Interpolate directly, get (45, 67.5, 45)
Desired result is (90, 22.5, 90) (verify this!)

30. Euler Angle Concatenation
Can't just add or multiply components
Best way:
Convert to matrices
Multiply matrices
Extract Euler angles from resulting matrix
Not cheap

31. Gimbal Lock Euler/fixed angles not well-formed
Different values can give same rotation
Example with z-y-x fixed angles:
( -90, 90, 90 ) = ( 0, 90, 0 )

32. Gimbal Lock Euler/fixed angles not well-formed
Different values can give same rotation
Example with z-y-x fixed angles:
( -90, 90, 90 ) = ( 0, 90, 0 ) z y x

33. Gimbal Lock Euler/fixed angles not well-formed
Different values can give same rotation
Example with z-y-x fixed angles:
( -90, 90, 90 ) = ( 0, 90, 0 ) z y x z
(-90,0,0) x y

34. Gimbal Lock Euler/fixed angles not well-formed
Different values can give same rotation
Example with z-y-x fixed angles:
( -90, 90, 90 ) = ( 0, 90, 0 ) z y x z
(-90,0,0)
(-90,90,0) x x y z y

35. Gimbal Lock Euler/fixed angles not well-formed
Different values can give same rotation
Example with z-y-x fixed angles:
( -90, 90, 90 ) = ( 0, 90, 0 ) z y x z
(-90,0,0)
(-90,90,0)
(-90,90,90) y x x y z z y x

36. Gimbal Lock
A Gimbal is a hardware implementation of Euler angles used for mounting gyroscopes or expensive globes
Gimbal lock is a basic problem with representing 3D rotation using Euler angles or fixed angles

37. Gimbal Lock
When two rotational axis of an object pointing in the same direction, the rotation ends up losing one degree of freedom

38. Outline Rotation matrix Fixed angle and Euler angle Axis angle
Quaternion

39. Axis Angle Rotate object by q around A (Ax,Ay,Az,q) A Y q Z X
The axis is really only 2 DoFs - its length is irrelevant Z X
Euler’s rotation theorem: An arbitrary rotation may be described by only three parameters.

40. Axis-angle Rotation Given r – Vector in space to rotate
n – Unit-length axis in space about which to rotate
q – The amount about n to rotate
Solve
r’ – The rotated vector r’ r n

41. Axis-angle Rotation
Compute rpar: the projection of r along the n direction
rpar = (n·r)n r’
rpar r

42. Axis-angle Rotation
Compute rperp: rperp = r-rpar
rperp
rpar r’ r

43. Axis-angle Rotation
Compute v: a vector perpedicular to rpar and rperp: v= rparxrperp v
rperp
rpar r’ r

44. Axis-angle Rotation
Compute v: a vector perpedicular to rpar and rperp: v= rparxrperp v
rperp
rpar r’
Use rpar, rperp and v, θ to compute the new vector! r

45. Axis-angle Rotation rperp = r – (n·r) n q
V = n x (r – (n·r) n) = n x r r’
rpar = (n·r) n r n
r’ = r’par + r’perp
= r’par + (cos q) rperp + (sin q) V
=(n·r) n + cos q(r – (n·r)n) + (sin q) n x r
= (cos q)r + (1 – cos q) n (n·r) + (sin q) n x r

46. Axis-angle Rotation
Can interpolate rotation well

47. Axis-angle Interpolation
1. Interpolate axis from A1 to A2 Rotate axis about A1 x A2 to get A A1 q1 A Y q A2
A1 x A2
2. Interpolate angle from q1 to q2 to get q q2 Z X
3. Rotate the object by q around A

48. Axis-angle Rotation Can interpolate rotation well
Compact representation
Messy to concatenate
- might need to convert to matrix form

49. Outline Rotation matrix Fixed angle and Euler angle Axis angle
Quaternion

50. Quaternion Remember complex numbers: a+ib, where i2=-1
Quaternions are a non-commutative extension of complex numbers
Invented by Sir William Hamilton (1843)
Quaternion:
- Q = a + bi + cj + dk: where i2=j2=k2=ijk=-1,ij=k,jk=i,ki=j
- Represented as: q = (w, v) = w + xi + yj + zk

51. Quaternion 4 tuple of real numbers: w, x, y, z
Same information as axis angles but in a more computational-friendly form

52. Quaternion Math Unit quaternion Multiplication Non-commutative
Associative

53. Quaternion Math
Conjugate
Inverse

54. Quaternion Example
let
then

55. Quaternion Rotation

56. Quaternion Rotation

57. Quaternion Example
Rotate a point (1,0,0) about y-axis with -90 degrees z
(0,0,1) y x
(1,0,0)

58. Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degreesz
(0,0,1) y x
(1,0,0)

59. Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degreesz
(0,0,1) y x
(1,0,0)
Point (1,0,0)

60. Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degreesz
(0,0,1) y x
(1,0,0)
Point (1,0,0)
Quaternion rotation

61. Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degreesz
(0,0,1) y x
(1,0,0)

62. Quaternion Composition
Rotation by p then q is the same as rotation by qp

63. Matrix Form
For a 3D point (x,y,z)

64. Quaternion Interpolation
1-angle rotation can be
represented by a unit circle

65. Quaternion Interpolation
1-angle rotation can be
represented by a unit circle
2-angle rotation can be
represented by a unit sphere

66. Quaternion Interpolation
1-angle rotation can be
represented by a unit circle
2-angle rotation can be
represented by a unit sphere
Interpolation means moving on n-D sphere
Now imagine a 4-D sphere for 3-angle rotation

67. Quaternion Interpolation
Moving between two points on the 4D unit
Sphere:
• a unit quaternion at each step - another point on the 4D unit sphere
• move with constant angular velocity along the great circle between the two points on the 4D unit sphere

68. Quaternion Interpolation
Direct linear interpolation does not work
Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle

69. Quaternion Interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion

70. Quaternion Interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion

71. Quaternion Interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion

72. Quaternions Can interpolate rotation well Compact representation
Also easy to concatenate

73. Conversions
Maths - Conversion Quaternion to Matrix - Martin Baker