Version 1
: Received: 14 September 2024 / Approved: 14 September 2024 / Online: 16 September 2024 (12:19:40 CEST)
How to cite:
Katrusiak, A.; Le, H. Q. Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-cartesian Reference Systems. Preprints2024, 2024091146. https://doi.org/10.20944/preprints202409.1146.v1
Katrusiak, A.; Le, H. Q. Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-cartesian Reference Systems. Preprints 2024, 2024091146. https://doi.org/10.20944/preprints202409.1146.v1
Katrusiak, A.; Le, H. Q. Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-cartesian Reference Systems. Preprints2024, 2024091146. https://doi.org/10.20944/preprints202409.1146.v1
APA Style
Katrusiak, A., & Le, H. Q. (2024). Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-cartesian Reference Systems. Preprints. https://doi.org/10.20944/preprints202409.1146.v1
Chicago/Turabian Style
Katrusiak, A. and Hien Quy Le. 2024 "Quaternion and Biquaternion Representations of Proper and Improper Transformations in Non-cartesian Reference Systems" Preprints. https://doi.org/10.20944/preprints202409.1146.v1
Abstract
Quaternion and biquaternion symmetry transformations have been applied to non-Cartesian reference systems of direct and reciprocal crystal lattices. The transformations performed directly in the sets of crystal reference axes simplify the calculations, eliminate the need for orthogonalization, permit the use of crystallographic vectors for defining the directions of rotations and perform the computations directly in the crystal coordinates. The applications of the general quaternion transformations are envisioned for physical, chemical, crystallographic and engineering applications. The general quaternion multiplication rules for any symmetry-unrestricted lattices have been derived for the triclinic crystallographic system and have been applied to the biquaternion representations of all point-group symmetry elements, including the crystallographic hexagonal system. Cayley multiplication matrices for point groups, based on the biquaternion symbols of proper and improper symmetry elements, have been exemplified.
Keywords
rotations; proper and improper symmetry; general transformation; non-Cartesian system; crystal lattice; quaternion; biquaternion; computations optimization; Caylay table
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
