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From Wikipedia, the free encyclopedia
Matrix representing a Euclidean rotation
A 180° rotation (middle) followed by a positive 90° rotation (left) is equivalent to a single negative 90° (positive 270°) rotation (right). Each of these figures depicts the result of a rotation relative to an upright starting position (bottom left) and includes the matrix representation of the permutation applied by the rotation (center right), as well as other related diagrams. See "Permutation notation" on Wikiversity for details.
A positive 90° rotation around the y -axis (left) after one around the z -axis (middle) gives a 120° rotation around the main diagonal (right).
In the top left corner are the rotation matrices, in the bottom right corner are the corresponding permutations of the cube with the origin in its center.

^ Note that if instead of rotating vectors, it is the reference frame that is being rotated, the signs on the sin θ terms will be reversed. If reference frame A is rotated anti-clockwise about the origin through an angle θ to create reference frame B, then R x (with the signs flipped) will transform a vector described in reference frame A coordinates to reference frame B coordinates. Coordinate frame transformations in aerospace, robotics, and other fields are often performed using this interpretation of the rotation matrix.

^ Note that

u

⊗

u

=

(

[

u

]

×

)

2

+

I

{\displaystyle \mathbf {u} \otimes \mathbf {u} ={\bigl (}[\mathbf {u} ]_{\times }{\bigr )}^{2}+{\mathbf {I} }}

so that, in Rodrigues' notation, equivalently,

R

=

I

+
(
sin
⁡
θ
)
[

u

]

×

+
(
1
−
cos
⁡
θ
)

(

[

u

]

×

)

2

.

{\displaystyle \mathbf {R} =\mathbf {I} +(\sin \theta )[\mathbf {u} ]_{\times }+(1-\cos \theta ){\bigl (}[\mathbf {u} ]_{\times }{\bigr )}^{2}.}

^ Note that this exponential map of skew-symmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to the third order,

e

2
A

−

I
+
A

I
−
A

=
−

2
3

A

3

+

O

(

A

4

)

.

{\displaystyle e^{2A}-{\frac {I+A}{I-A}}=-{\tfrac {2}{3}}A^{3}+\mathrm {O} \left(A^{4}\right).}

Conversely, a skew-symmetric matrix A specifying a rotation matrix through the Cayley map specifies the same rotation matrix through the map exp(2 artanh A ) .

^ For a detailed derivation, see Derivative of the exponential map . Issues of convergence of this series to the right element of the Lie algebra are here swept under the carpet. Convergence is guaranteed when || X || + || Y || < log 2 and || Z || < log 2 . If these conditions are not fulfilled, the series may still converge. A solution always exists since exp is onto [ clarification needed ] in the cases under consideration.

^ Swokowski, Earl (1979). Calculus with Analytic Geometry (Second ed.). Boston: Prindle, Weber, and Schmidt. ISBN 0-87150-268-2 .

^ W3C recommendation (2003). "Scalable Vector Graphics – the initial coordinate system" .

^ "Rotation Matrices" (PDF) . Retrieved 30 November 2021 .

^ Taylor, Camillo J.; Kriegman, David J. (1994). "Minimization on the Lie Group SO(3) and Related Manifolds" (PDF) . Technical Report No. 9405 . Yale University.

^ Cole, Ian R. (January 2015). Modelling CPV (thesis). Loughborough University. hdl : 2134/18050 .

^ Baker (2003) ; Fulton & Harris (1991)

^ ( Wedderburn 1934 , §8.02)

^ Hall 2004 , Ch. 3; Varadarajan 1984 , §2.15

^ ( Engø 2001 )

^ Curtright, T L ; Fairlie, D B ; Zachos, C K (2014). "A compact formula for rotations as spin matrix polynomials". SIGMA . 10 : 084. arXiv : 1402.3541 . Bibcode : 2014SIGMA..10..084C . doi : 10.3842/SIGMA.2014.084 . S2CID 18776942 .

^ Baker 2003 , Ch. 5; Fulton & Harris 1991 , pp. 299–315

^ ( Goldstein, Poole & Safko 2002 , §4.8)

^ Cid, Jose Ángel; Tojo, F. Adrián F. (2018). "A Lipschitz condition along a transversal foliation implies local uniqueness for ODEs" . Electronic Journal of Qualitative Theory of Differential Equations . 13 (13): 1–14. arXiv : 1801.01724 . doi : 10.14232/ejqtde.2018.1.13 .

In linear algebra , a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space . For example, using the convention below, the matrix

rotates points in the xy plane counterclockwise through an angle θ with respect to the positive x axis about the origin of a two-dimensional Cartesian coordinate system . To perform the rotation on a plane point with standard coordinates v = ( x , y ) , it should be written as a column vector , and multiplied by the matrix R :

If x and y are the endpoint coordinates of a vector, where x is cosine and y is sine, then the above equations become the trigonometric summation angle formulae . Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. One way to understand this is say we have a vector at an angle 30° from the x axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°.

The examples in this article apply to active rotations of vectors counterclockwise in a right-handed coordinate system ( y counterclockwise from x ) by pre-multiplication ( R on the left). If any one of these is changed (such as rotating axes instead of vectors, a passive transformation ), then the inverse of the example matrix should be used, which coincides with its transpose .

Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry , physics , and computer graphics . In some literature, the term rotation is generalized to include improper rotations , characterized by orthogonal matrices with a determinant of −1 (instead of +1). These combine proper rotations with reflections (which invert orientation ). In other cases, where reflections are not being considered, the label proper may be dropped. The latter convention is followed in this article.

Rotation matrices are square matrices , with real entries. More specifically, they can be characterized as orthogonal matrices with determinant 1; that is, a square matrix R is a rotation matrix if and only if R T = R −1 and det R = 1 . The set of all orthogonal matrices of size n with determinant +1 forms a group known as the special orthogonal group SO( n ) , one example of which is the rotation group SO(3) . The set of all orthogonal matrices of size n with determinant +1 or −1 forms the (general) orthogonal group O( n ) .

In two dimensions, the standard rotation matrix has the following form:

This rotates column vectors by means of the following matrix multiplication ,

Thus, the new coordinates ( x ′, y ′) of a point ( x , y ) after rotation are

is rotated by an angle θ , its new coordinates are

is rotated by an angle θ , its new coordinates are

The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. −90°). Thus the clockwise rotation matrix is found as

The two-dimensional case is the only non-trivial (i.e. not one-dimensional) case where the rotation matrices group is commutative, so that it does not matter in which order multiple rotations are performed. An alternative convention uses rotating axes, [1] and the above matrices also represent a rotation of the axes clockwise through an angle θ .

If a standard right-handed Cartesian coordinate system is used, with the x -axis to the right and the y -axis up, the rotation R ( θ ) is counterclockwise. If a left-handed Cartesian coordinate system is used, with x directed to the right but y directed down, R ( θ ) is clockwise. Such non-standard orientations are rarely used in mathematics but are common in 2D computer graphics , which often have the origin in the top left corner and the y -axis down the screen or page. [2]

See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix.

Particularly useful are the matrices

for 90°, 180°, and 270° counter-clockwise rotations.

form a ring isomorphic to the field of the complex numbers

C

{\displaystyle \mathbb {C} }

. Under this isomorphism, the rotation matrices correspond to circle of the unit complex numbers , the complex numbers of modulus 1 .

If one identify

R

2

{\displaystyle \mathbb {R} ^{2}}

with

C

{\displaystyle \mathbb {C} }

through the linear isomorphism

(
a
,
b
)
↦
a
+
i
b
,

{\displaystyle (a,b)\mapsto a+ib,}

the action of a matrix of the above form on vectors of

R

2

{\displaystyle \mathbb {R} ^{2}}

corresponds to the multiplication by the complex number x + iy , and rotations correspond to multiplication by complex numbers of modulus 1 .

As every rotation matrix can be written

the above correspondence associates such a matrix with the complex number

(this last equality is Euler's formula ).

A basic rotation (also called elemental rotation) is a rotation about one of the axes of a coordinate system. The following three basic rotation matrices rotate vectors by an angle θ about the x -, y -, or z -axis, in three dimensions, using the right-hand rule —which codifies their alternating signs. (The same matrices can also represent a clockwise rotation of the axes. [nb 1] )

For column vectors , each of these basic vector rotations appears counterclockwise when the axis about which they occur points toward the observer, the coordinate system is right-handed, and the angle θ is positive. R z , for instance, would rotate toward the y -axis a vector aligned with the x -axis , as can easily be checked by operating with R z on the vector (1,0,0) :

This is similar to the rotation produced by the above-mentioned two-dimensional rotation matrix. See below for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices.

Other rotation matrices can be obtained from these three using matrix multiplication . For example, the product

represents a rotation whose yaw, pitch, and roll angles are α , β and γ , respectively. More formally, it is an intrinsic rotation whose Tait–Bryan angles are α , β , γ , about axes z , y , x , respectively.
Similarly, the product

represents an extrinsic rotation whose (improper) Euler angles are α , β , γ , about axes x , y , z .

These matrices produce the desired effect only if they are used to premultiply column vectors , and (since in general matrix multiplication is not commutative ) only if they are applied in the specified order (see Ambiguities for more details). The order of rotation operations is from right to left; the matrix adjacent to the column vector is the first to be applied, and then the one to the left. [3]

Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis ( Euler rotation theorem ).

There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation ). Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.

Given a 3 × 3 rotation matrix R , a vector u parallel to the rotation axis must satisfy

since the rotation of u around the rotation axis must result in u . The equation above may be solved for u which is unique up to a scalar factor unless R = I .

Further, the equation may be rewritten

which shows that u lies in the null space of R − I .

Viewed in another way, u is an eigenvector of R corresponding to the eigenvalue λ = 1 . Every rotation matrix must have this eigenvalue, the other two eigenvalues being complex conjugates of each other. It follows that a general rotation matrix in three dimensions has, up to a multiplicative constant, only one real eigenvector.

One way to determine the rotation axis is by showing that:

Since ( R − R T ) is a skew-symmetric matrix , we can choose u such that

The matrix–vector product becomes a cross product of a vector with itself, ensuring that the result is zero:

The magnitude of u computed this way is || u || = 2 sin θ , where θ is the angle of rotation.

This does not work if R is symmetric. Above, if R − R T is zero, then all subsequent steps are invalid. In this case, it is necessary to diagonalize R and find the eigenvector corresponding to an eigenvalue of 1.

To find the angle of a rotation, once the axis of the rotation is known, select a vector v perpendicular to the axis. Then the angle of the rotation is the angle between v and R v .

A more direct method, however, is to simply calculate the trace : the sum of the diagonal elements of the rotation matrix. Care should be taken to select the right sign for the angle θ to match the chosen axis:

from which follows that the angle's absolute value is

The matrix of a proper rotation R by angle θ around the axis u = ( u x , u y , u z ) , a unit vector with u 2 x + u 2 y + u 2 z = 1 , is given by: [4]

A derivation of this matrix from first principles can be found in section 9.2 here. [5] The basic idea to derive this matrix is dividing the problem into few known simple steps.

This can be written more concisely as

where [ u ] × is the cross product matrix of u ; the expression u ⊗ u is the outer product , and I is the identity matrix . Alternatively, the matrix entries are:

where ε jkl is the Levi-Civita symbol with ε 123 = 1 . This is a matrix form of Rodrigues' rotation formula , (or the equivalent, differently parametrized Euler–Rodrigues formula ) with [nb 2]

In

R

3

{\displaystyle \mathbb {R} ^{3}}

the rotation of a vector x around the axis u by an angle θ can be written as:

If the 3D space is right-handed and θ > 0 , this rotation will be counterclockwise when u points towards the observer ( Right-hand rule ). Explicitly, with

(

α

,

β

,

u

)

{\displaystyle ({\boldsymbol {\alpha }},{\boldsymbol {\beta }},\mathbf {u} )}

a right-handed orthonormal basis,

Note the striking merely apparent differences to the equivalent Lie-algebraic formulation below .

For any n -dimensional rotation matrix R acting on

R

n

,

{\displaystyle \mathbb {R} ^{n},}

A rotation is termed proper if det R = 1 , and improper (or a roto-reflection) if det R = –1 . For even dimensions n = 2 k , the n eigenvalues λ of a proper rotation occur as pairs of complex conjugates which are roots of unity: λ = e ± iθ j for j = 1, ..., k , which is real only for λ = ±1 . Therefore, there may be no vectors fixed by the rotation ( λ = 1 ), and thus no axis of rotation. Any fixed eigenvectors occur in pairs, and the axis of rotation is an even-dimensional subspace.

For odd dimensions n = 2 k + 1 , a proper rotation R will have an odd number of eigenvalues, with at least one λ = 1 and the axis of rotation will be an odd dimensional subspace. Proof:

Here I is the identity matrix, and we use det( R T ) = det( R ) = 1 , as well as (−1) n = −1 since n is odd. Therefore, det( R – I ) = 0 , meaning there is a null vector v with ( R – I ) v = 0 , that is R v = v , a fixed eigenvector. There may also be pairs of fixed eigenvectors in the even-dimensional subspace orthogonal to v , so the total dimension of fixed eigenvectors is odd.

For example, in 2-space n = 2 , a rotation by an
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