1. CSCE 441: Computer Graphics Rotation Representation and Interpolation
Jinxiang Chai

2. Toy Example A 2D lamp with 6 degrees of freedom lower arm middle arm
Upper arm
base 2

3. Toy Example
A 2D lamp with 6 degrees of freedom
base

4. Toy Example
A 2D lamp with 6 degrees of freedom
Upper arm
base

5. Toy Example A 2D lamp with 6 degrees of freedom middle arm Upper arm
base

6. Toy Example A 2D lamp with 6 degrees of freedom lower arm middle arm
Upper arm
base

7. 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

8. 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 

9. 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

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

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

12. 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

13. Rotation
New points
rotation matrix
old points

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

15. 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

16. 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

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

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

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

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

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

22. Fixed Angles Angles are 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

23. 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

24. 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

25. Euler Angles vs. Fixed Angles
z-x-z Euler angles: (-60,30,45)
z-x-z fixed angles: (45,30,-60)

26. Euler Angle Interpolation
Interpolating each component 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!)

27. 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

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

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

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

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

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

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

34. 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

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

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

37. Axis Angle Rotate an 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.

38. 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

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

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

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

42. 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

43. 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

44. Axis-angle Rotation
Can interpolate rotation well

45. 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

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

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

48. 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

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

50. Quaternion Math Unit quaternion Multiplication Non-commutative
Associative

51. Quaternion Math
Conjugate
Inverse

52. Quaternion Example
let
then

53. How to Represent Rotation?
Axis-angle q r’ 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

54. Quaternion Rotation

55. Quaternion Rotation

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

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)
Point (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)
Quaternion rotation

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

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

62. Matrix Form
For a 3D point (x0,y0,z0)

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

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

65. 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

66. 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

67. Quaternion Interpolation
Direct linear interpolation does not work
Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle p0 p1

68. Quaternion Interpolation
Spherical linear interpolation (SLERP)
- Constant speed motion along a unit radius great circle
arc, given the ends and an interpolation parameter
between 0 and 1
- Normalize to regain unit quaternion p0 p1
p1(p1-1p0)1-t

69. Quaternion Interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion p0 p1
p1(p1-1p0)1-t

70. Quaternion Interpolation
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion p0 p1
p1(p1-1p0)1-t

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