Orientations

1
Orientations

• Goal
• Convenient representation of orientation of
  objects characters in a scene
• Applications to
• inverse kinematics
• rigid body simulation
• rag doll physics
• etc.

2
Euler Angles

• so far, used one angle per axis, ie. Euler
  angles.
• Pros
• Simple!
• Cons
• Singularities (gimbal lock)
• Interpolation is tricky
• Composing rotations is tricky

3
Singularities

• Singularities may arise when axes are rotated to
  coincide
• Lose a degree of freedom
• Consider an airplanes roll-pitch-yaw
• eg. (0,90,0) vs. (90,90,90) identical position
• IK problems
• Infinitely many solutions if coincident
• Ill-posed when very close to coincident

4
Interpolation

• Vital for keyframe animation, blending mocap
  data, etc.
• Linear interpolating each angle independently
  gives odd behaviour near singularities
• Not coordinate independent
• For 2 orientations, rotate-interpolate gives a
  different result than interpolate-rotate

5
Composition

• Given two rotations applied sequentially, find a
  single orientation that gives the same result.
• Not at all obvious how to handle this

6
3x3 Orthogonal Matrices

• Eliminate many issues of Euler angles, but
• Hard to work with 9 components, 6 constraints
• Highly redundant
• Linear interpolation doesnt work

7
3x3 Orthogonal Matrices

• But
• Very effective for quickly transforming many
  points
• Composition is straightforward multiply
• Still useful, but not as our primary
  representation

8
Axis-angle

• Unit axis vector and an angle indicating how much
  to rotate.
• Very intuitive
• 4 numbers, 1 constraint less redundant
• Comparison, interpolation, composition still
  difficult.

9
2D Rotations

• Can use complex numbers to represent 2D rotations
• View a point (x,y) as x iy
• Rotation about origin is multiplication by
• cos(theta) isin(theta)
• Equivalently, e(itheta)
• ie. A unit magnitude complex number

10
2D Rotations

• And composition is again multiplication
• Isnt this overkill?
• In 2D, yes
• But this idea extends elegantly to 3D

11
Quaternions

• Generalizes complex numbers
• Two additional sqrt(-1) terms, j and k
• Form a bi cj dk
• Behave mostly like regular numbers
• But NOT commutative! ab ? ba

12
Quaternions

• Small set of simple multiplication rules
• ii -1, jj -1, kk -1
• ij k, jk i, ki j
• ji -k, kj-i, ik -j
• When j k components are 0, this reverts to
  basic complex numbers
• ie. 2D rotations.

13
Rotating points

• Given complex number
• Conjugate is
• Then
• If q is unit magnitude, just gives back 0 ix
• Real part remains 0, imaginary part retains its
  length, x

14
Rotating points

• Quaternion
• Conjugate
• Zero real-part quaternion
• Consider
• Real part stays 0, imaginary part keeps length
• For unit q, this is a rotation of the vector
  (x,y,z) in quaternion form

15
Quaternion Rotation

• So, to perform a rotation by unit quaternion q on
  vector v (x,y,z) to get new vector v
  (x,y,z) just do

16
Quaternion Rotation

• Unit-length quaternions can represent any
  orientation in 3D, without singularities
• Relation to axis-angle form
• Real part is cos(theta/2)
• Imaginary part is vector parallel to the rotation
  axis
• Redundancy q -q
• But since the two possible representations are
  maximally far apart, no harm done.

17
Quaternion Interpolation

• Interpolation is much more convenient
• linear interpolating, then renormalizing q to
  unit-length works fairly well
• For better results, ensure dot product of their
  imaginary parts is non-negative,by flipping sign,
  since q -q is the same orientation.
• We want the two quaternions axes to be
  pointing in the same direction, so that the
  interpolation path is as short as possible.
• For improved interpolation of unit vectors, look
  into slerp

18
Quaternion Composition

• Similar to matrix form, just multiply
• Consider 2 quaternion rotations applied to a
  point 0ixjykz
• or