Rotation: A review of useful theorems involving proper orthogonal matrices referenced to three-dimensional physical space

Abstract......Page 1
Introduction......Page 9
1. Orthogonal basis & coordinate transformations......Page 12
2. Rotation operations......Page 20
3. Axis and angle of rotation......Page 27
Euler-Rodrigues formula......Page 28
Computing the rotation tensor given axis and angle......Page 30
Corollary to the Euler-Rodrigues formula: Existence of a preferred basis......Page 39
Computing axis and angle given the rotation tensor.......Page 40
4. Rotations contrasted with reflections......Page 49
5. Quaternion representation of a rotation......Page 51
Relationship between quaternion and axis/angle forms......Page 52
6. Dyad form of an invertible linear operator......Page 53
Sequential rotations about fixed (laboratory) axes.......Page 55
EULER ANGLES: Sequential rotations about “follower” axes.......Page 57
8. Series expression for a rotation......Page 59
9. Spectrum of a rotation......Page 61
Difficult definition of the deformation gradient......Page 63
Intuitive definition of the deformation gradient......Page 67
The Jacobian of the deformation......Page 70
Sequential deformations......Page 71
Matrix analysis version of the polar decomposition theorem......Page 72
The polar decomposition theorem — a hindsight intuitive introduction......Page 73
A more rigorous (classical) presentation of the polar decomposition theorem......Page 76
The *FAST* way to do a polar decomposition in two dimensions......Page 81
proposal #1: Map and re-compute the polar decomposition......Page 83
proposal #2: Discard the “stretch” part of a mixed rotation.......Page 84
proposal #4: mix the quaternions......Page 86
The “spin” tensor......Page 87
The angular velocity vector......Page 88
Difference between vorticity and polar spin.......Page 89
The (commonly mis-stated) Gosiewski’s theorem......Page 92
Rates of sequential rotations......Page 94
Rates of simultaneous rotations......Page 95
Statistical notation......Page 97
Uniformly random unit vectors — the theory......Page 98
Uniformly random unit vectors — formalized implementation......Page 100
Uniformly random unit vectors — faster implementation......Page 102
Uniformly random unit vectors —The visualization......Page 103
Uniformly random rotations......Page 108
An easy algorithm for generating a uniformly random rotation.......Page 113
An alternative algorithm for generating a uniformly random rotation.......Page 116
Shoemake’s algorithm for uniformly random rotations.......Page 117
14. SCALARS and INVARIANTS......Page 125
What is a “superimposed rotation”?......Page 126
“Reference” and “Objective/spatial” tensors......Page 128
True or False: scalars are unaffected by a superimposed rigid rotation.......Page 129
Prelude to PMFI: A philosophical diversion......Page 130
PMFI: a sloppy introduction......Page 131
Translational frame invariance......Page 132
Rotational invariance.......Page 136
Here’s a model that satisfies the principle of material frame invariance.......Page 138
The principle of material frame indifference in general......Page 139
PMFI in rate forms of the constitutive equations......Page 147
Co-rotational rates (convected, Jaumann, Polar)......Page 149
Lie Derivatives and reference configurations......Page 151
Frame indifference is only an essential (not final) step......Page 155
16. Rigid Body Mechanics......Page 157
A relative description of rigid motion......Page 159
Velocity and angular velocity for rigid motion......Page 160
Time rate of a vector embedded in a rigid body......Page 161
Acceleration for rigid motion......Page 162
Important properties of a rigid body......Page 164
Linear momentum of a rigid body......Page 170
Kinetic energy......Page 171
EULER’S EQUATION (balance of angular momentum)......Page 173
REFERENCES......Page 175
Listing 1: Testing whether a matrix is a rotation......Page 177
Listing 2: Converting axis and angle to direction cosines......Page 178
Listing 3: Converting direction cosines to axis and angle......Page 179
Listing 4: Converting Euler angles to direction cosines.......Page 181
Listing 5: Converting direction cosines to Euler angles.......Page 182
Listing 6: Generating a uniformly random unit normal......Page 183
Listing 7: Generating a uniformly random rigid rotation.......Page 184
Tensor and vector notation......Page 187
Vectors......Page 188
Tensors......Page 189