On the Composition of Rotations in $\mathbb{R}^3$

Year 2025, Volume: 8 Issue: 4, 183 - 188, 08.12.2025

François Dubeau

https://doi.org/10.33434/cams.1682530

https://izlik.org/JA35GK79SU

Abstract

Euler stated that the composition of two successive rotations is also a rotation, but did not solve the problem of finding the resultant (axis and angle of rotation) of the composition. It is Rodrigues who solved it. Based on the Rodrigues' formula for a rotation in $\mathbb{R}^3$, we present a unified new proof of both Euler's existence theorem and Rodrigues' identification. The proof relies only on basic algebraic properties of vectors in $\mathbb{R}^3$.

Keywords

Composition , Rotation formula , Vector analysis

References

• [1] L. Euler, Formulae generales pro translatione quacunque corporum rigidorum [E478], presented to the St. Petersburg Academy on October 9, 1775. Novi Comm. Acad. Sci. Petropol, 20 (1776), 189-207.
• [2] R. C. Alperin, The matrix of rotation, College Math. J., 20(3) (1989), 230. http://dx.doi.org/10.1080/07468342.1989.11973235
• [3] H. Cheng, K. C. Gupta, An historical note on finite rotations, J. Appl. Mech., 56(1) (1989), 139-145. http://dx.doi.org/10.1115/1.3176034
• [4] H. Cheng, K. C. Gupta, Erratam: An historical note on finite rotations, J. Appl. Mech., 56(1) (1989), 139-145, J. Appl. Mech., 56(2) (1989), 298. http://dx.doi.org/10.1115/1.3176082
• [5] H. W. Hickey, More on vector representation of rigid body, Math. Mag., 43(1) (1970), 38-39. http://dx.doi.org/10.1080/0025570X.1970.11975996
• [6] D. Kalman, The axis of rotation: Analysis, algebra, geometry, Math. Mag., 62(4) (1989), 248-252. http://dx.doi.org/10.1080/0025570X.1989.11977446
• [7] M. F. Maritz, Rotations in three dimensions, SIAM Rev., 63(2) (2021), 395-404. http://dx.doi.org/10.1137/19M128867X
• [8] O. Rodrigues, Des lois géométriques qui régissent les déplacements d’un objet solide dans l’espace, et de la variation des coordonnées provenant de ces déplacements considérées indépendamment des causes qui peuvent les produire, J. Math. Pures Appl., 5 (1840), 380-440.
• [9] J. S. Wong, On vector representation of rigid body, Math. Mag., 41(1) (1968), 28-29. http://dx.doi.org/10.1080/0025570X.1968.11975828
• [10] B. Palais, R. Palais, S. Radi, A disorienting look at Euler’s theorem on the axis of a rotation, Amer. Math. Monthly, 116(10) (2009), 892-909. http://dx.doi.org/10.4169/000298909X477014
• [11] T. G. Room, The composition of rotations in Euclidean three-space, Amer. Math. Monthly, 59(10) (1952), 688-692. http://dx.doi.org/10.2307/2307548
• [12] G. Trenkler, D. Trenkler, On the product of rotations, Int. J. Math. Educ. Sci. Technol., 39(1) (2008), 94-104. http: //dx.doi.org/10.1080/00207390601115054
• [13] B. Paul, On the composition of finite rotations, Amer. Math. Monthly, 70(8) (1963), 859-862. http://dx.doi.org/10.2307/2312674
• [14] H. P. Hsu, Vector Analysis, Simon & Schuster, New York, 1969.
• [15] M. Kinyon, Proof simplification and automated theorem proving, Phil. Trans. R. Soc. A, 377 (2019), Article ID 20180034. http://dx.doi.org/10.1098/rsta.2018.0034