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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.04452 |
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| _version_ | 1866914141055221760 |
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| author | Díaz, Fernando Ricardo González Badenes, Vicent Martinez Montalvo, Teodoro Rivera García-Salcedo, Ricardo |
| author_facet | Díaz, Fernando Ricardo González Badenes, Vicent Martinez Montalvo, Teodoro Rivera García-Salcedo, Ricardo |
| contents | Quaternions provide a unified algebraic and geometric framework for representing three-dimensional rotations without the singularities that afflict Euler-angle parametrisations. This article develops a pedagogical and conceptual analysis of the \emph{Gimbal lock} phenomenon and demonstrates, step by step, how quaternion algebra resolves it. Beginning with the limitations of Euler representations, the work introduces the quaternionic rotation operator $v' = q\,v\,q^{*}$, derives the Rodrigues formula, and establishes the continuous, singularity-free mapping between unit quaternions and the rotation group $SO(3)$. The approach combines historical motivation, formal derivation, and illustrative examples designed for advanced undergraduate and graduate students. As an extension, Appendix~A presents the geometric and topological interpretations of quaternions, including their relation to the groups $\mathbb{Q}_8$ and $SU(2)$, and the Dirac belt trick, offering a visual analogy that reinforces the connection between algebra and spatial rotation. Overall, this work highlights the educational value of quaternions as a coherent and elegant framework for understanding rotational dynamics in physics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quaternionic approach to teaching 3D rotations and the resolution of gimbal lock Díaz, Fernando Ricardo González Badenes, Vicent Martinez Montalvo, Teodoro Rivera García-Salcedo, Ricardo Physics Education Quaternions provide a unified algebraic and geometric framework for representing three-dimensional rotations without the singularities that afflict Euler-angle parametrisations. This article develops a pedagogical and conceptual analysis of the \emph{Gimbal lock} phenomenon and demonstrates, step by step, how quaternion algebra resolves it. Beginning with the limitations of Euler representations, the work introduces the quaternionic rotation operator $v' = q\,v\,q^{*}$, derives the Rodrigues formula, and establishes the continuous, singularity-free mapping between unit quaternions and the rotation group $SO(3)$. The approach combines historical motivation, formal derivation, and illustrative examples designed for advanced undergraduate and graduate students. As an extension, Appendix~A presents the geometric and topological interpretations of quaternions, including their relation to the groups $\mathbb{Q}_8$ and $SU(2)$, and the Dirac belt trick, offering a visual analogy that reinforces the connection between algebra and spatial rotation. Overall, this work highlights the educational value of quaternions as a coherent and elegant framework for understanding rotational dynamics in physics. |
| title | A quaternionic approach to teaching 3D rotations and the resolution of gimbal lock |
| topic | Physics Education |
| url | https://arxiv.org/abs/2511.04452 |