From Rotation Matrices to Quaternions: Derivations, Efficient Algorithms, and Applications in Computing and Physics

Chongjing Li

2025

Abstract

This paper explores the mathematical foundations and applications of quaternions in representing three-dimensional rotations. Quaternions, discovered by William Rowan Hamilton in 1843, provide a powerful and efficient alternative to traditional rotation matrices. The paper begins by deriving the rotational quaternion from rotation matrices, highlighting the mathematical relationship between the two. It then compares the advantages and disadvantages of using quaternions versus rotation matrices for 3D rotations, emphasizing the efficiency and numerical stability of quaternions. The paper also introduces new algorithms for reducing the computational complexity of quaternion operations while maintaining accuracy. Finally, it discusses the extensive applications of quaternions in computer animation and modern physics, demonstrating their versatility and importance in enhancing computational efficiency and realism. By addressing both theoretical and practical aspects, this paper aims to provide a comprehensive overview of quaternions and their significance in various fields.

Paper Citation

in Harvard Style

Li C. (2025). From Rotation Matrices to Quaternions: Derivations, Efficient Algorithms, and Applications in Computing and Physics. In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA; ISBN 978-989-758-774-0, SciTePress, pages 342-347. DOI: 10.5220/0013825500004708

in Bibtex Style

@conference{iampa25,
author={Chongjing Li},
title={From Rotation Matrices to Quaternions: Derivations, Efficient Algorithms, and Applications in Computing and Physics},
booktitle={Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA},
year={2025},
pages={342-347},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0013825500004708},
isbn={978-989-758-774-0},
}

in EndNote Style

TY - CONF

JO - Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy - Volume 1: IAMPA
TI - From Rotation Matrices to Quaternions: Derivations, Efficient Algorithms, and Applications in Computing and Physics
SN - 978-989-758-774-0
AU - Li C.
PY - 2025
SP - 342
EP - 347
DO - 10.5220/0013825500004708
PB - SciTePress