Rodrigues' rotation formula

In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in $SO(3)$ , the group of all rotation matrices, from an axis–angle representation. In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra $so (3)$ to its Lie group $SO(3)$ .

This formula is variously credited to Leonhard Euler, Olinde Rodrigues, or a combination of the two. A detailed historical analysis in 1989 concluded that the formula should be attributed to Euler, and recommended calling it "Euler's finite rotation formula."^[1] This proposal has received notable support,^[2] but some others have viewed the formula as just one of many variations of the Euler–Rodrigues formula, thereby crediting both.^[3]

^ Cheng, Hui; Gupta, K. C. (March 1989). "An Historical Note on Finite Rotations". Journal of Applied Mechanics. 56 (1). American Society of Mechanical Engineers: 139–145. doi:10.1115/1.3176034. Retrieved 2022-04-11.
^ Fraiture, Luc (2009). "A History of the Description of the Three-Dimensional Finite Rotation". The Journal of the Astronautical Sciences. 57 (1–2). Springer: 207–232. doi:10.1007/BF03321502. Retrieved 2022-04-15.
^ Dai, Jian S. (October 2015). "Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections". Mechanism and Machine Theory. 92. Elsevier: 144–152. doi:10.1016/j.mechmachtheory.2015.03.004. Retrieved 2022-04-14.

[1] Cheng, Hui; Gupta, K. C. (March 1989). "An Historical Note on Finite Rotations". Journal of Applied Mechanics. 56 (1). American Society of Mechanical Engineers: 139–145. doi:10.1115/1.3176034. Retrieved 2022-04-11.

[2] Fraiture, Luc (2009). "A History of the Description of the Three-Dimensional Finite Rotation". The Journal of the Astronautical Sciences. 57 (1–2). Springer: 207–232. doi:10.1007/BF03321502. Retrieved 2022-04-15.

[3] Dai, Jian S. (October 2015). "Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections". Mechanism and Machine Theory. 92. Elsevier: 144–152. doi:10.1016/j.mechmachtheory.2015.03.004. Retrieved 2022-04-14.

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