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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 linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \t ...
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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
to its
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
. This formula is variously credited to
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, 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 In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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 \textbf) that was lost by the first term. An alternative statement is to write the axis vector as a
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
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 :\mathbf = \frac = \frac\,. 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 In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
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 \mathbb_. Using the dot and
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
s, the vector can be decomposed into components parallel and perpendicular to the axis , : \mathbf = \mathbf_\parallel + \mathbf_\perp \,, where the component parallel to is called the vector projection of on , : \mathbf_\parallel = (\mathbf \cdot \mathbf) \mathbf , and the component perpendicular to is called the vector rejection of from : :\mathbf_ = \mathbf - \mathbf_ = \mathbf - (\mathbf \cdot \mathbf) \mathbf = - \mathbf\times(\mathbf\times\mathbf), where the last equality follows from the vector triple product formula: \mathbf\times (\mathbf \times \mathbf) = (\mathbf \cdot \mathbf)\mathbf - (\mathbf \cdot \mathbf)\mathbf. Finally, the vector \mathbf \times \mathbf_ = \mathbf \times \mathbf is a copy of \mathbf_ rotated 90° around \mathbf. Thus the three vectors \mathbf\,,\ \mathbf_\,,\, \mathbf \times \mathbf form a
right-handed In human biology, handedness is an individual's preferential use of one hand, known as the dominant hand, due to and causing it to be stronger, faster or more Fine motor skill, dextrous. The other hand, comparatively often the weaker, less dext ...
orthogonal basis of \mathbb^3, with the last two vectors of equal length. Under the rotation, the component \mathbf_ parallel to the axis will not change magnitude nor direction: :\mathbf_ = \mathbf_\parallel \,; while the perpendicular component will retain its magnitude but rotate its direction in the perpendicular plane spanned by \mathbf_ and \mathbf \times \mathbf, according to : \mathbf_ = \cos(\theta) \mathbf_\perp + \sin(\theta) \mathbf\times\mathbf_\perp = \cos(\theta) \mathbf_\perp + \sin(\theta) \mathbf\times\mathbf \,, in analogy with the planar
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
in the Cartesian basis , : :\mathbf = r\cos(\theta) \mathbf_x + r\sin(\theta) \mathbf_y \,. Now the full rotated vector is: : \mathbf_ = \mathbf_ + \mathbf_ = \mathbf_\parallel + \cos(\theta) \, \mathbf_\perp + \sin(\theta) \mathbf\times\mathbf . Substituting \mathbf_ = \mathbf - \mathbf_ or \mathbf_ = \mathbf - \mathbf_ in the last expression gives respectively: :\mathbf_ = \cos(\theta) \, \mathbf + \sin(\theta) \mathbf\times\mathbf + (1 - \cos\theta)(\mathbf \cdot \mathbf)\mathbf :\phantom = \mathbf + \sin(\theta) \mathbf\times\mathbf + (1-\cos\theta)\mathbf\times(\mathbf\times\mathbf).


Matrix notation

The linear transformation on \mathbf\isin\mathbb^3 defined by the cross product \mathbf \mapsto \mathbf \times \mathbf is given in coordinates by representing and as column matrices: :\begin (\mathbf\times\mathbf)_x \\ (\mathbf\times\mathbf)_y \\ (\mathbf\times\mathbf)_z \end = \begin k_y v_z - k_z v_y \\ k_z v_x - k_x v_z \\ k_x v_y - k_y v_x \end = \left begin 0\ \, & -k_z & k_y \\ k_z & 0\ \, & -k_x \\ -k_y & k_x & 0\ \, \end\right\begin v_x \\ v_y \\ v_z \end \,. That is, the matrix of this linear transformation (with respect to standard coordinates) is the cross-product matrix: : \mathbf= \left begin 0\ \, & -k_z & k_y \\ k_z & 0\ \, & -k_x \\ -k_y & k_x & 0\ \, \end\right,. That is to say, : \mathbf\times\mathbf=\mathbf\mathbf, \qquad\qquad \mathbf\times(\mathbf\times\mathbf)=\mathbf(\mathbf\mathbf) = \mathbf^2\mathbf \,. The last formula in the previous section can therefore be written as: :\mathbf_ = \mathbf + (\sin\theta) \mathbf\mathbf + (1 - \cos\theta)\mathbf^2\mathbf\,. Collecting terms allows the compact expression :\mathbf_\mathrm = \mathbf\mathbf where is the
rotation matrix In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation (mathematics), rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \t ...
through an angle counterclockwise about the axis , and the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
. This matrix is an element of the rotation group of , and is an element of the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
\mathfrak(3) generating that Lie group (note that is skew-symmetric, which characterizes \mathfrak(3)). In terms of the matrix exponential, :\mathbf = \exp (\theta\mathbf)\,. To see that the last identity holds, one notes that :\mathbf(\theta) \mathbf(\phi) = \mathbf (\theta+\phi), \quad \mathbf(0) = \mathbf\,, 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 \mathfrak(3) to . For the inverse mapping, see log map from to \mathfrak(3). The above result can be written in index notation as follows. The elements of the matrix for an active rotation by an angle \theta about an axis are given by : R_ = \cos\theta\, \delta_ + (1 - \cos\theta) n_i n_j - \sin\theta\, \epsilon_ n_k. Here, i, j, and k label the Cartesian components (x, y, z) or (1, 2, 3), \delta_ and \epsilon_ are the Kronecker and Levi-Civita symbols, and there is an implicit sum on repeated indices. The Hodge dual of the rotation \mathbf is just \mathbf^* = -\sin(\theta)\mathbf 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, :\begin \sin(\theta) &= \sigma \left, \mathbf^*\ \\ pt \mathbf &= -\frac \end where \sigma = \pm 1. The above simple expression results from the fact that the Hodge duals of \mathbf and \mathbf^2 are zero, and \mathbf^* = -\mathbf.


See also

* Axis angle *
Rotation (mathematics) Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a s ...
* SO(3) and
SO(4) In mathematics, the group (mathematics), group of rotations about a fixed point in four-dimensional space, four-dimensional Euclidean space is denoted SO(4). The name comes from the fact that it is the special orthogonal group of order 4. In this ...
* Euler–Rodrigues formula


References

*
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, "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
* Richard M. Friedberg (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 {{DEFAULTSORT:Rodrigues' Rotation Formula Rotation in three dimensions Euclidean geometry Orientation (geometry) fr:Rotation vectorielle#Cas général