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 , 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  to its Lie group . 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." This proposal has received notable support, but some others have viewed the formula as just one of many variations of the Euler–Rodrigues formula, thereby crediting both. Statement
If  is a vector in  and  is a unit vector describing an axis of rotation about which  rotates by an angle  according to the right hand rule, the Rodrigues formula for the rotated vector  is

The intuition of the above formula is that the first term scales the vector down, while the second skews it (via vector addition) toward the new rotational position. The third term re-adds the height (relative to ) that was lost by the first term. An alternative statement is to write the axis vector as a cross product  of any two nonzero vectors  and  which define the plane of rotation, and the sense of the angle  is measured away from  and towards . Letting  denote the angle between these vectors, the two angles  and  are not necessarily equal, but they are measured in the same sense. Then the unit axis vector can be written

This form may be more useful when two vectors defining a plane are involved. An example in physics is the Thomas precession which includes the rotation given by Rodrigues' formula, in terms of two non-collinear boost velocities, and the axis of rotation is perpendicular to their plane. Derivation 

Let  be a unit vector defining a rotation axis, and let  be any vector to rotate about  by angle  (right hand rule, anticlockwise in the figure), producing the rotated vector . Using the dot and cross products, the vector  can be decomposed into components parallel and perpendicular to the axis ,

where the component parallel to  is called the vector projection of  on , 

,

and the component perpendicular to  is called the vector rejection of  from :

,

where the last equality follows from the vector triple product formula:  . Finally, the vector  is a copy of  rotated 90° around . Thus the three vectors  form a right-handed orthogonal basis of , with the last two vectors of equal length. Under the rotation, the component  parallel to the axis will not change magnitude nor direction:

while the perpendicular component will retain its magnitude but rotate its direction in the perpendicular plane spanned by  and , according to

in analogy with the planar polar coordinates  in the Cartesian basis , :

Now the full rotated vector is:

Substituting  or  in the last expression gives respectively:

Matrix notation 

The linear transformation on  defined by the cross product  is given in coordinates by representing  and  as column matrices:

That is, the matrix of this linear transformation (with respect to standard coordinates) is the cross-product matrix:
 
That is to say,
 
The last formula in the previous section can therefore be written as:

Collecting terms allows the compact expression

where

is the rotation matrix through an angle  counterclockwise about the axis , and  the  identity matrix. This matrix  is an element of the rotation group  of , and  is an element of the Lie algebra  generating that Lie group (note that  is skew-symmetric, which characterizes ). In terms of the matrix exponential,

To see that the last identity holds, one notes that

characteristic of a one-parameter subgroup, i.e. exponential, and that the formulas match for infinitesimal . For an alternative derivation based on this exponential relationship, see exponential map from  to . For the inverse mapping, see log map from  to . The Hodge dual of the rotation  is just  which  enables the extraction of both the axis of rotation and the sine of the angle of the rotation from the rotation matrix itself, with the usual ambiguity,

where . The above simple expression results from the fact that the Hodge duals of  and  are zero, and . When applying the Rodrigues' formula, however, the usual ambiguity could be removed with an extended form of the formula. See also 
 Axis angle
 Rotation (mathematics)
 SO(3) and SO(4)
 Euler–Rodrigues formula

References 

Leonhard Euler, "Problema algebraicum ob affectiones prorsus singulares memorabile", Commentatio 407 Indicis Enestoemiani, Novi Comm. Acad. Sci. Petropolitanae  15 (1770), 75–106. Olinde Rodrigues, "Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendants des causes qui peuvent les produire", Journal de Mathématiques Pures et Appliquées 5 (1840), 380–440. online. Friedberg, Richard (2022). "Rodrigues, Olinde: "Des lois géométriques qui régissent les déplacements d'un systéme solide...", translation and commentary". arXiv:2211.07787. Don Koks, (2006) Explorations in Mathematical Physics, Springer Science+Business Media,LLC. . Ch.4, pps 147 et seq. A Roundabout Route to Geometric Algebra

External links 
 Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007. For another descriptive example see: http://chrishecker.com/Rigid_Body_Dynamics#Physics_Articles, Chris Hecker, physics section, part 4. "The Third Dimension" – on page 3, section ``Axis and Angle'', http://chrishecker.com/images/b/bb/Gdmphys4.pdf

Rotation in three dimensions
Euclidean geometry
Orientation (geometry)

fr:Rotation vectorielle#Cas général