1. Rotation and Orientation: Fundamentals
Jehee Lee
Seoul National University

2. What is Rotation ? Not intuitive Many different ways to describe
Formal definitions are also confusing
Many different ways to describe
Rotation (direction cosine) matrix
Euler angles
Axis-angle
Rotation vector
Helical angles
Unit quaternions

3. Orientation vs. Rotation
Circular movement
Orientation
The state of being oriented
Given a coordinate system, the orientation of an object can be represented as a rotation from a reference pose

4. Analogy
(point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement)
Reference coordinate system

5. Analogy
(point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement)
point : the 3d location of the bunny
vector : translational movement
Reference coordinate system

6. Analogy
(point : vector) is similar to (orientation : rotation) Both represent a sort of (state : movement)
point : the 3d location of the bunny
vector : translational movement
orientation : the 3d orientation of the bunny
rotation : circular movement
Reference coordinate system

7. 2D Orientation
Polar Coordinates

8. 2D Orientation Although the motion is continuous,
its representation could be discontinuous

9. 2D Orientation
Many-to-one correspondences between 2D orientations and their representations

10. Extra Parameter

11. Extra Parameter
2x2 Rotation matrix is yet another method of using extra parameters

12. Complex Number
Imaginary
Real

13. Complex Exponentiation
Imaginary
Real

14. 2D Rotation
Complex numbers are good for representing 2D orientations, but inadequate for 2D rotations
A complex number cannot distinguish different rotational movements that result in the same final orientation
Turn 120 degree counter-clockwise
Turn -240 degree clockwise
Turn 480 degree counter-clockwise
Imaginary
Real

15. 2D Rotation and Orientation
The consequence of any 2D rotational movement can be uniquely represented by a turning angle
2D Orientation
The non-singular parameterization of 2D orientations requires extra parameters
Eg) Complex numbers, 2x2 rotation matrices

16. Operations in 2D (orientation) : complex number
(rotation) : scalar value
exp(rotation) : complex number

17. 2D Rotation and Displacement
Imaginary
Real

18. 2D Rotation and Displacement
Imaginary
Real

19. 2D Orientation Composition
Imaginary
Real

20. 2D Rotation Composition
Imaginary
Real

21. Analogy

22. 3D Rotation
Given two arbitrary orientations of a rigid object,

23. 3D Rotation
We can always find a fixed axis of rotation and an angle about the axis

24. Euler’s Rotation Theorem
The general displacement of a rigid body with
one point fixed is a rotation about some axis
Leonhard Euler ( )
In other words,
Arbitrary 3D rotation equals to one rotation around an axis
Any 3D rotation leaves one vector unchanged

25. Euler angles Gimble Hardware implementation of Euler angles
Aircraft, Camera

26. Euler Angles Rotation about three orthogonal axes Gimble lock
12 combinations
XYZ, XYX, XZY, XZX
YZX, YZY, YXZ, YXY
ZXY, ZXZ, ZYX, ZYZ
Gimble lock
Coincidence of inner most and outmost gimbles’ rotation axes
Loss of degree of freedom

27. Euler Angles Euler angles are ambiguous
Two different Euler angles can represent the same orientation
This ambiguity brings unexpected results of animation where frames are generated by interpolation.

28. Ranges Euler angles in ZXZ Z-axis by a  ( -p , p ]
X’-axis by b  ( -p/2 , p/2 ]
Z’’-axis by g  ( -p , p ]

29. Dependence Compare Rotations about x-, y-, and z-axes are dependent
Rotation about z-axis by 180 degree
Rotation about y-axis by 180 degree, followed by another rotation about x-axis by 180 degree
Rotations about x-, y-, and z-axes are dependent

30. Rotation Vector Rotation vector (3 parameters)
Axis-Angle (2+1 parameters)

31. 3D Orientation Unhappy with three parameters Euler angles
Discontinuity (or many-to-one correspondences)
Gimble lock
Rotation vector (a.k.a Axis/Angle)

32. Using an Extra Parameter
Euler parameters

33. Quaternions William Rowan Hamilton (1805-1865)
Algebraic couples (complex number) 1833
where

34. Quaternions William Rowan Hamilton (1805-1865)
Algebraic couples (complex number) 1833
Quaternions 1843
where
where

35. Quaternions William Thomson Arthur Cayley
“… though beautifully ingenious, have been an unmixed evil to those who have touched them in any way.”
Arthur Cayley
“… which contained everything but had to be unfolded into another form before it could be understood.”

36. Unit Quaternions
Unit quaternions represent 3D rotations

37. Rotation about an Arbitrary Axis
Rotation about axis by angle
where
Purely Imaginary Quaternion

38. Unit Quaternion Algebra
Identity
Multiplication
Inverse
Opposite axis or negative angle
Unit quaternion space is
closed under multiplication and inverse,
but not closed under addition and subtraction

39. Unit Quaternion Algebra
Antipodal equivalence
q and –q represent the same rotation
ex) rotation by p-q about opposite direction
2-to-1 mapping between S and SO(3)
Twice as fast as in SO(3) 3

40. 3D Orientations and Rotations
Orientations and rotations are different in coordinate-invariant geometric programming
Use unit quaternions for orientation representation
3x3 orthogonal matrix is theoretically identical
Use 3-vectors for rotation representation

41. Tangent Vector (Infinitesimal Rotation)

42. Tangent Vector (Infinitesimal Rotation)

43. Tangent Vector (Infinitesimal Rotation)
Angular Velocity

44. Exp and Log
log
exp

45. Exp and Log
Euler
parameters
log
exp

46. Rotation Vector

47. Rotation Vector

48. Rotation Vector

49. Rotation Vector Finite rotation Infinitesimal rotation
Eg) Angular displacement
Be careful when you add two rotation vectors
Infinitesimal rotation
Eg) Instantaneous angular velocity
Addition of angular velocity vectors are meaningful

50. Spherical Linear Interpolation
SLERP [Shoemake 1985]
Linear interpolation of two orientations

51. Spherical Linear Interpolation

52. Coordinate-Invariant Operations

53. Adding Rotation Vectors

54. Affine Combination of Orientations

55. Analogy
(point : vector) is similar to (orientation : rotation)

56. Rotation Matrix vs. Unit Quaternion
Equivalent in many aspects
Redundant, No singularity
Exp & Log, Special tangent space
Why quaternions ?
Fewer parameters, Simpler algebra
Easy to fix numerical error
Cf) matrix orthogonalization (Gram-shmidt process, QR, SVD decomposition)
Why rotation matrices ?
One-to-one correspondence
Handle rotation and translation in a uniform way
Eg) 4x4 homogeneous matrices

57. Rotation Conversions
In theory, conversion between any representations is always possible
In practice, conversion is not straightward because of difference in convention
Quaternion to Matrix