Editing Rodrigues' rotation formula (section)

{{Short description|Vector formula for a rotation in space, given its axis}}
{{Distinguish|text= the [[Euler–Rodrigues parameters]] and [[SO(4)#The Euler–Rodrigues formula for 3D rotations|The Euler–Rodrigues formula for 3D rotation]]}}

{{More footnotes needed|date=May 2021}}
In the theory of [[three-dimensional rotation]], '''Rodrigues' rotation formula''', named after [[Olinde Rodrigues]], is an efficient algorithm for rotating a [[vector (geometric)|vector]] in space, given an [[axis angle|axis]] and [[angle of rotation]]. By extension, this can be used to transform all three [[basis vector]]s to compute a [[rotation matrix]] in {{math|[[Rotation group SO(3)|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 (Riemannian geometry)|exponential map]] from the [[Lie algebra]] {{math|'''so'''(3)}} to its [[Lie group]] {{math|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."<ref>{{cite journal | last1 = Cheng | first1 = Hui | last2 = Gupta | first2 = K. C. | title = An Historical Note on Finite Rotations | journal = Journal of Applied Mechanics | volume = 56 | issue = 1 | pages = 139–145 | publisher = American Society of Mechanical Engineers | date = March 1989 | doi = 10.1115/1.3176034 | bibcode = 1989JAM....56..139C | url = https://iel.ucdavis.edu/publication/1989/j_ASMEAM.pdf | access-date = 2022-04-11}}</ref> This proposal has received notable support,<ref>{{cite journal | last = Fraiture | first = Luc | title = A History of the Description of the Three-Dimensional Finite Rotation | journal = The Journal of the Astronautical Sciences | volume = 57 | pages = 207–232 | publisher = Springer | date = 2009 | issue = 1–2 | doi = 10.1007/BF03321502 | bibcode = 2009JAnSc..57..207F | url = https://doi.org/10.1007/BF03321502 | access-date = 2022-04-15| url-access = subscription }}</ref> but some others have viewed the formula as just one of many variations of the [[Euler–Rodrigues formula]], thereby crediting both.<ref>{{cite journal | last = Dai | first = Jian S. | title = Euler–Rodrigues formula variations, quaternion conjugation and intrinsic connections | journal = Mechanism and Machine Theory | volume = 92 | pages = 144–152 | publisher = Elsevier | date = October 2015 | doi = 10.1016/j.mechmachtheory.2015.03.004 | doi-access = free }}</ref>