Unlimited Release Printed May 9, 2002 ROTATION: A review of useful theorems involving proper orthogonal matrices referenced to three- dimensional physical space. Rebecca M. Brannon† and coauthors to be determined †Computational Physics and Mechanics T Sandia National Laboratories Albuquerque, NM 87185-0820 Abstract Useful and/or little-known theorems involving33× proper orthogonal matrices are reviewed. Orthogonal matrices appear in the transformation of tensor compo- nents from one orthogonal basis to another. The distinction between an orthogonal direction cosine matrix and a rotation operation is discussed. Among the theorems and techniques presented are (1) various ways to characterize a rotation including proper orthogonal tensors, dyadics, Euler angles, axis/angle representations, series expansions, and quaternions; (2) the Euler-Rodrigues formula for converting axis and angle to a rotation tensor; (3) the distinction between rotations and reflections, along with implications for “handedness” of coordinate systems; (4) non-commu- tivity of sequential rotations, (5) eigenvalues and eigenvectors of a rotation; (6) the polar decomposition theorem for expressing a general deformation as a se- quence of shape and volume changes in combination with pure rotations; (7) mix- ing rotations in Eulerian hydrocodes or interpolating rotations in discrete field approximations; (8) Rates of rotation and the difference between spin and vortici- ty, (9) Random rotations for simulating crystal distributions; (10) The principle of material frame indifference (PMFI); and (11) a tensor-analysis presentation of classical rigid body mechanics, including direct notation expressions for momen- tum and energy and the extremely compact direct notation formulation of Euler’s equations (i.e., Newton’s law for rigid bodies). Computer source code is provided for several rotation-related algorithms. A draft version of this document is available at http://www.me.unm.edu/~rmbrann/gobag.htmlDRAF Intentionally Left Blank DRAFTii Introduction ............................................................................................................ 1 Orthogonal basis & coordinate transformations.......................................... 4 Motivational discussion: Principal basis ...................................................... 4 Orthonormal basis transformations............................................................... 5 Alternative direction cosine matrix................................................................ 10 Coordinate transformations........................................................................... 10 Rotation operations.............................................................................................. 12 Example: rotation of an angle α about the X-axis ........................................ 14 Example: rotation of an angle α about the Y-axis. ....................................... 15 Example: rotation of an angle α about the Z-axis. ....................................... 16 Where does that negative sign go? ................................................................ 17 Faster way to write down the matrix of a rotation tensor ............................. 17 Specific example: A 90˚ rotation about the Z-axis......................................... 18 Axis and angle of rotation.................................................................................. 19 T Euler-Rodrigues formula ..................................................................................... 20 Computing the rotation tensor given axis and angle............................................ 22 Example.......................................................................................................... 23 Another example ............................................................................................ 24 Another similar example................................................................................ 24 Numerical example (for testing computer codes) .......................................... 26 Alternative way to construct the rotation tensor............................................ 27 Some properties of the axial tensor ............................................................... 28 Argyris’s form of the Euler-Rodrigues formula............................................. 30 Corollary to the Euler-Rodrigues formula: Existence of a preferred basis ........................................................................ 31 Computing axis and angle given the rotation tensor............................................ 32 Finding the principal rotation angle ............................................................. 32 Finding the principal rotation axis................................................................ 33 Method 1 algorithm for axis and angle of rotation ....................................... 36 Example.......................................................................................................... 37 Another example ............................................................................................ 38 Numerical Example (for testing codes).......................................................... 39 Method 2 algorithm for computing axis and angle........................................ 40 Rotations contrasted with reflections .............................................................. 41 Quaternion representation of a rotation ........................................................ 43 Shoemake’s form [3]...................................................................................... 43 A more structural direct form ........................................................................ 43 Relationship between quaternion and axis/angle forms....................................... 44 Dyad form of an invertible linear operator.................................................... 45 SPECIAL CASE: lab basis............................................................................. 45 SPECIAL CASE: rotation .............................................................................. 46 Sequential Rotations ............................................................................................ 47 Sequential rotations about fixed (laboratory) axes. ............................................. 47 EULER ANGLES: Sequential rotations about “follower” axes.......................... 49 Converting Euler angles to direction cosines................................................ 49 Example: ....................................................................................................... 49 DRAFiii Converting Euler angles to axis and angle.................................................... 50 Series expression for a rotation......................................................................... 51 Spectrum of a rotation......................................................................................... 53 Sanity check ................................................................................................... 54 Polar decomposition............................................................................................ 55 Difficult definition of the deformation gradient .................................................. 55 Intuitive definition of the deformation gradient................................................... 59 Converse problem: interpreting a deformation gradient matrix ................... 61 The Jacobian of the deformation.......................................................................... 62 Invertibility of a deformation............................................................................... 63 Sequential deformations....................................................................................... 63 Matrix analysis version of the polar decomposition theorem.............................. 64 The polar decomposition theorem — a hindsight intuitive introduction............. 65 A more rigorous (classical) presentation of the polar decomposition theorem ... 68 THEOREM:.................................................................................................... 69 PROOF: ......................................................................................................... 69 Working with the inverse gradient................................................................. 72 The *FAST* way to do a polar decomposition in two dimensions..................... 73 “Mixing” or interpolating rotations................................................................... 75 proposal #1: Map and re-compute the polar decomposition................................ 75 proposal #2: Discard the “stretch” part of a mixed rotation. ............................... 76 Proof: ............................................................................................................. 77 proposal #3: mix the pseudo-rotation-vectors...................................................... 78 proposal #4: mix the quaternions......................................................................... 78 Rates of rotation .................................................................................................... 79 The “spin” tensor ................................................................................................. 79 The angular velocity vector ................................................................................. 80 Angular velocity in terms of axis and angle of rotation......................................