1. 3D orientation

2. Rotation matrix
Fixed angle and Euler angle
Axis angle
Quaternion
Exponential map

3. Joints and rotations
Rotational DOFs are widely used in character animation
3 translational DOFs
48 rotational DOFs
1.Rotation is the most important transformation in hierarchical modeling.
2.The rotations in a human model is represented by joints.
3.Different joint has different numbers of DOFs
4.The 3DOFs rotation is not organized in a simple 3 dimensional vector space
Each joint can have up to 3 DOFs
1 DOF: knee
2 DOF: wrist
3 DOF: arm

4. 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
since all the transformations are represented in the same way, we can combined those transformation together by simply multiplying the matrices

5. Translation
translation
matrix
new point
old point

6. Scaling
new point
scaling matrix
old point

7. Rotation
X axis
Y axis
The rows and columns are orthonormal
Z axis

8. Composite transformations
A series of transformations on an object can be applied as a series of matrix multiplications
: position in the global coordinate
: position in the local coordinate

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

10. Motivation
Finding the most natural and compact way to present rotation and orientations
Orientation interpolation which result in a natural motion
A closed mathematical form that deals with rotation and orientations (expansion for the complex numbers)

11. Rotation Matrix
A general rotation can be represented by a single 3x3 matrix
Length Preserving (Isometric)
Reflection Preserving
Orthonormal

12. Fixed Angle Representation
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, ie. first about x, then y, then z
Many possible orderings, don’t have to use all 3 axes, but can’t do the same axis back to back

13. Fixed Angle Representation
A rotation of 10,45, 90 would be written as
Rz(90) Ry(45), Rx(10) since we want to first rotate about x, y, z. It would be applied then to the point P…. RzRyRx P
Problem occurs when two of the axes of rotation line up on top of each other. This is called “Gimbal Lock”

14. Gimbal Lock
A 90 degree rotation about the y axis essentially makes the first axis of rotation align with the third.
Incremental changes in x,z produce the same results – you’ve lost a degree of freedom

15. Gimbal Lock
Phenomenon of two rotational axis of an object pointing in the same direction.
Simply put, it means your object won't rotate how you think it ought to rotate.

16. 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 in- between rotations

17. 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...

18. Properties of rotation matrix
Easily composed?
Interpolate?

19. Rotation matrix
Fixed angle and Euler angle
Axis angle
Quaternion
Exponential map

20. Fixed angle 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
We represent orientation using angles that rotate about fixed axes
x(90)y(90) is different from y(90)x(90)

21. Fixed angle A rotation of Rz(90)Ry(60)Rx(30) looks like

22. Euler angle
Same as fixed angles, except now the axes move with the object
An Euler angle is a rotation about a single Cartesian axis
Create multi-DOF rotations by concatenating Euler angles
evaluate each axis independently in a set order
book example

23. Euler angle vs. fixed angle
Rz(90)Ry(60)Rx(30) = Ex(30)Ey(60)Ez(90)
Euler angle rotations about moving axes written in reverse order are the same as the fixed axis rotations
What is the relation between Euler and fixed angles? Z Y X

24. Gimbal Lock (again!)
Rotation by 90o causes a loss of a degree of freedom x y z 1
/2 x’ 3

25. Gimbal Lock
Phenomenon of two rotational axis of an object pointing in the same direction.
Simply put, it means your object won't rotate how you think it ought to rotate.

26. 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

27. Gimbal lock
When two rotational axis of an object point in the same direction, the rotation ends up losing one degree of freedom

28. Properties of Euler angle
Easily composed?
Interpolate?
How about joint limit?
What seems to be the problem?
no, Eucleadian , yes, singularity
single rotation interpolates, but compound rotation doesn’t
spinning about each axis successively, the result of the compound orientation will look awful

29. Euler Angles
A general rotation is a combination of three elementary rotations: around the x-axis (x-roll) , around the y-axis (y-roll) and around the z-axis (z-roll).

30. Euler Angles

31. Euler Angles and Rotation Matrices

32. Euler angles interpolationz x y π
x-roll π
R(0,0,0),…,R(t,0,0),…,R(,0,0)
t[0,1] y z x
z-roll π π
y-roll π
R(0,0,0),…,R(0,t, t),…,R(0,, )

33. Euler Angles Interpolation Unnatural movement !

34. Rotation matrix
Fixed angle and Euler angle
Axis angle
Quaternion
Exponential map

35. Goal Find a parameterization in which
a simple steady rotation exists between two key orientations
moves are independent of the choice of the coordinate system

36. Axis angle Represent orientation as a vector and a scalar
vector is the axis to rotate about
scalar is the angle to rotate by

37. Angle and Axis Any orientation can be represented by a 4-tuple
angle, vector(x,y,z) where the angle is the amount to rotate by and the vector is the axis to rotate about
Can interpolate the angle and axis separately
No gimbal lock problems!
But, can’t efficiently compose rotations…must convert to matrices first!

38. Angular displacement
(,n) defines an angular displacement of  about an axis n

39. Properties of axis angle
Can avoid Gimbal lock. Why?
Can interpolate the vector and the scalar separately. How?
euler angle representation separates the orientation into three steps, which introduce the chances that the last rotational axis overlap with the first axis
axis angle representation does everything in one rotation and there is no ordering issue

40. Axis angle interpolation
if you sweep A1 on this plan to A2, you are essentially doing the interpolation between two vectors

41. Rotation matrix
Fixed angle and Euler angle
Axis angle
Quaternion
Exponential map

42. Quaternion 4 tuple of real numbers:
scalar
vector
Quaternion is originally represented as an extended complex number. Hamilton first defined quaternion using four real number. Geometrically, quaternion can not be easily organized in Euaclidean space. We need to define different algebraic rules for quaternions from the regular vector algebra
v and the rotating axis r are related by sin(theta/2)
Same information as axis angles but in a different form

43. Quaternions ? Extend the concept of rotation in 3D to 4D.
Avoids the problem of "gimbal-lock" and allows for the implementation of smooth and continuous rotation.
In effect, they may be considered to add a additional rotation angle to spherical coordinates ie. Longitude, Latitude and Rotation angles
A Quaternion is defined using four floating point values |x y z w|. These are calculated from the combination of the three coordinates of the rotation axis and the rotation angle.

44. How do quaternions relate to 3D animation?
Solution to "Gimbal lock"
Instead of rotating an object through a series of successive rotations, a quaternion allows the programmer to rotate an object through a single arbitary rotation axis.
Because the rotation axis is specifed as a unit direction vector, it may be calculated through vector mathematics or from spherical coordinates ie (longitude/latitude).
Quaternions interpolation : smooth and predictable rotation effects.

45. Quaternion to Rotation Matrix

46. Quaternions Definition
Extension of complex numbers
4-tuple of real numbers
s,x,y,z or [s,v]
s is a scalar
v is a vector
Same information as axis/angle but in a different form
Can be viewed as an original orientation or a rotation to apply to an object

47. Quaternions Math

48. Quaternions properties
The conjugate and magnitude are similar to complex numbers
Quaternions are non commutative
q1 = (s1,v1) q2 = (s2,v2)
q1*q2 = (s1s2 – v1.v2 , s1v2 + s2v1 + v1 x v2)
inverse:
unit quaternion:

49. Quaternion Rotation To rotate a vector, v using quaternion math
represent the vector as [0,v]
represent the rotation as a quaternion, q

50. Quaternions as Rotations
Rotation of P=(0,r) about the unit vector n by an angle θ using the unit quaternion q=(s,v)
but q=(cos½θ, sin½θ•n) where |n|=1

51. Quaternions as Rotations
Concatenating rotations – rotate using q1 and then using q2 is like rotation using q2*q1

52. Quaternion math
Unit quaternion
Multiplication

53. Quaternion math
Conjugate
Inverse
the unit length quaternion

54. Quaternion Rotation If is a unit quaternion and
then results in rotating about by
proof: see Quaternions by Shoemaker

55. Quaternion Rotation

56. Quaternion composition
If and are unit quaternion
the combined rotation of first rotating by and then by is equivalent to

57. Matrix form

58. Quaternion interpolation
2-angle rotation can be represented by a unit sphere
1-angle rotation can be represented by a unit circle
there is a way that helps us to visualize the interpolation of quaternion
all the possible 1D orientation can be located on a circle
Interpolation means moving on n-D sphere
Now imagine a 4-D sphere for 3-angle rotation

59. 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

60. Quaternion interpolation
Direct linear interpolation does not work
Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle
Spherical linear interpolation (SLERP)
Normalize to regain unit quaternion

61. Rotations in Reality
It’s easiest to express rotations in Euler angles or Axis/angle
We can convert to/from any of these representations
Choose the best representation for the task
input:Euler angles
interpolation: quaternions
composing rotations: quaternions, orientation matrix

62. Rotation matrix
Fixed angle and Euler angle
Axis angle
Quaternion
Exponential map

63. Exponential map Represent orientation as a vector
direction of the vector is the axis to rotate about
magnitude of the vector is the angle to rotate by
Zero vector represents the identity rotation

64. Properties of exponential map
No need to re-normalize the parameters
Fewer DOFs
Good interpolation behavior
Singularities exist but can be avoided

65. Choose a representation
Choose the best representation for the task
input:
joint limits:
interpolation:
compositing:
rendering:
Euler angles
Euler angles, quaternion (harder)
quaternion or exponential map
quaternions or orientation matrix
orientation matrix ( quaternion can be represented as matrix as well)