On representing rotations by Rodrigues parameters in non-orthonormal reference systems

Area of Science:

• Crystallography
• Materials Science
• Computational Materials Science

Background:

• Rodrigues vectors are commonly used for parameterizing rotations in 3D space.
• Their application in analyzing orientation maps of polycrystalline materials is frequent due to their properties.
• Conventional Rodrigues parameters are defined in orthonormal systems, posing challenges for non-orthogonal crystal lattices.

Purpose of the Study:

• To extend the definition of Rodrigues parameters for direct application to non-Cartesian crystal lattice bases.
• To develop a generalized formalism for crystallographic orientation analysis in oblique coordinate frames.
• To enhance the applicability of Rodrigues parameters in materials science.

Main Methods:

• Definition of generalized Rodrigues parameters as co- or contravariant vector components.
• Development of a formalism compatible with arbitrary metric tensors of crystal lattices.
• Adaptation of rotation composition and rotation matrix relationship formulas for generalized parameters.

Main Results:

• Generalized Rodrigues parameters directly link to non-Cartesian crystal bases.
• The formalism accommodates redundant crystallographic axes.
• Rotation composition and matrix relationships are adapted for wider applicability, including arbitrary metric tensors.
• The generalized parameters directly relate to rotation-invariant lattice directions and planes.

Conclusions:

• The extended Rodrigues parameter formalism simplifies crystallographic orientation analysis in non-orthogonal systems.
• This generalization enhances the utility of Rodrigues vectors for polycrystalline materials analysis.
• The approach offers a more versatile and direct method for handling crystallographic orientations.