# Three-dimensional rotation operator

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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. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (n − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.

## Contents

The three Euler rotations are one way to bring a rigid body to any desired orientation by sequentially making rotations about axis' fixed relative to the object. However, this can also be achieved with one single rotation (Euler's rotation theorem). Using the concepts of linear algebra it is shown how this single rotation can be performed.

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra.

In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it. A rigid body is usually considered as a continuous distribution of mass.

In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.

## Mathematical formulation

Let

${\displaystyle {\hat {e}}_{1}\ ,\ {\hat {e}}_{2}\ ,\ {\hat {e}}_{3}}$

be a coordinate system fixed in the body that through a change in orientation is brought to the new directions

In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

${\displaystyle \mathbf {A} {\hat {e}}_{1}\ ,\ \mathbf {A} {\hat {e}}_{2}\ ,\ \mathbf {A} {\hat {e}}_{3}.}$

Any vector

${\displaystyle {\bar {x}}\ =x_{1}{\hat {e}}_{1}+x_{2}{\hat {e}}_{2}+x_{3}{\hat {e}}_{3}}$

rotating with the body is then brought to the new direction

${\displaystyle \mathbf {A} {\bar {x}}\ =x_{1}\mathbf {A} {\hat {e}}_{1}+x_{2}\mathbf {A} {\hat {e}}_{2}+x_{3}\mathbf {A} {\hat {e}}_{3}}$

i.e. this is a linear operator

The matrix of this operator relative to the coordinate system

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, because there are two rows and three columns:

In mathematics, an operator is generally a mapping that acts on elements of a space to produce other elements of the same space. The most common operators are linear maps, which act on vector spaces. However, when using "linear operator" instead of "linear map", mathematicians often mean actions on vector spaces of functions, which also preserve other properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.

${\displaystyle {\hat {e}}_{1}\ ,\ {\hat {e}}_{2}\ ,\ {\hat {e}}_{3}}$

is

${\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}}={\begin{bmatrix}\langle {\hat {e}}_{1}|\mathbf {A} {\hat {e}}_{1}\rangle &\langle {\hat {e}}_{1}|\mathbf {A} {\hat {e}}_{2}\rangle &\langle {\hat {e}}_{1}|\mathbf {A} {\hat {e}}_{3}\rangle \\\langle {\hat {e}}_{2}|\mathbf {A} {\hat {e}}_{1}\rangle &\langle {\hat {e}}_{2}|\mathbf {A} {\hat {e}}_{2}\rangle &\langle {\hat {e}}_{2}|\mathbf {A} {\hat {e}}_{3}\rangle \\\langle {\hat {e}}_{3}|\mathbf {A} {\hat {e}}_{1}\rangle &\langle {\hat {e}}_{3}|\mathbf {A} {\hat {e}}_{2}\rangle &\langle {\hat {e}}_{3}|\mathbf {A} {\hat {e}}_{3}\rangle \end{bmatrix}}}$

As

${\displaystyle \sum _{k=1}^{3}A_{ki}A_{kj}=\langle \mathbf {A} {\hat {e}}_{i}|\mathbf {A} {\hat {e}}_{j}\rangle ={\begin{cases}0&i\neq j,\\1&i=j,\end{cases}}}$

or equivalently in matrix notation

${\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}}^{T}{\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}}$

the matrix is orthogonal and as a "right hand" base vector system is re-orientated into another "right hand" system the determinant of this matrix has the value 1.

An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors, i.e.

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the n-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of n-space.

### Rotation around an axis

Let

${\displaystyle {\hat {e}}_{1}\ ,\ {\hat {e}}_{2}\ ,\ {\hat {e}}_{3}}$

be an orthogonal positively oriented base vector system in ${\displaystyle R^{3}}$.

The linear operator

"Rotation with the angle ${\displaystyle \theta }$ around the axis defined by ${\displaystyle {\hat {e}}_{3}}$"

has the matrix representation

${\displaystyle {\begin{bmatrix}Y_{1}\\Y_{2}\\Y_{3}\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}}$

relative to this basevector system.

This then means that a vector

${\displaystyle {\bar {x}}={\begin{bmatrix}{\hat {e}}_{1}&{\hat {e}}_{2}&{\hat {e}}_{3}\end{bmatrix}}{\begin{bmatrix}X_{1}\\X_{2}\\X_{3}\end{bmatrix}}}$

is rotated to the vector

${\displaystyle {\bar {y}}={\begin{bmatrix}{\hat {e}}_{1}&{\hat {e}}_{2}&{\hat {e}}_{3}\end{bmatrix}}{\begin{bmatrix}Y_{1}\\Y_{2}\\Y_{3}\end{bmatrix}}}$

by the linear operator.

The determinant of this matrix is

${\displaystyle \det {\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}=1}$

and the characteristic polynomial is

{\displaystyle {\begin{aligned}\det {\begin{bmatrix}\cos \theta -\lambda &-\sin \theta &0\\\sin \theta &\cos \theta -\lambda &0\\0&0&1-\lambda \end{bmatrix}}&={\big (}{(\cos \theta -\lambda )}^{2}+{\sin \theta }^{2}{\big )}(1-\lambda )\\&=-\lambda ^{3}+(2\ \cos \theta \ +\ 1)\ \lambda ^{2}-(2\ \cos \theta \ +\ 1)\ \lambda +1\\\end{aligned}}}

The matrix is symmetric if and only if ${\displaystyle \sin \theta =0}$, i.e. for ${\displaystyle \theta =0}$ and for ${\displaystyle \theta =\pi }$.

The case ${\displaystyle \theta =0}$ is the trivial case of an identity operator.

For the case ${\displaystyle \theta =\pi }$ the characteristic polynomial is

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients. The characteristic polynomial of an endomorphism of vector spaces of finite dimension is the characteristic polynomial of the matrix of the endomorphism over any base; it does not depend on the choice of a basis. The characteristic equation is the equation obtained by equating to zero the characteristic polynomial.

${\displaystyle -(\lambda -1){(\lambda +1)}^{2}}$

i.e. the rotation operator has the eigenvalues

${\displaystyle \lambda =1\quad \lambda =-1}$

The eigenspace corresponding to ${\displaystyle \lambda =1}$ is all vectors on the rotation axis, i.e. all vectors

${\displaystyle {\bar {x}}=\alpha \ {\hat {e}}_{3}\quad -\infty <\alpha <\infty }$

The eigenspace corresponding to ${\displaystyle \lambda =-1}$ consists of all vectors orthogonal to the rotation axis, i.e. all vectors

${\displaystyle {\bar {x}}=\alpha \ {\hat {e}}_{1}+\beta \ {\hat {e}}_{2}\quad -\infty <\alpha <\infty \quad -\infty <\beta <\infty }$

For all other values of ${\displaystyle \theta }$ the matrix is un-symmetric and as ${\displaystyle {\sin \theta }^{2}>0}$ there is only the eigenvalue ${\displaystyle \lambda =1}$ with the one-dimensional eigenspace of the vectors on the rotation axis:

${\displaystyle {\bar {x}}=\alpha \ {\hat {e}}_{3}\quad -\infty <\alpha <\infty }$

The rotation matrix by angle ${\displaystyle \theta }$ around a general axis of rotation ${\displaystyle \mathbf {k} =\left[{\begin{array}{ccc}k_{1}\\k_{2}\\k_{3}\end{array}}\right]}$ is given by Rodrigues' rotation formula.

${\displaystyle R=I\cos \theta +[\mathbf {k} ]_{\times }\sin \theta +(1-\cos \theta )\mathbf {k} \mathbf {k} ^{\mathsf {T}}}$,

where ${\displaystyle I}$ is the identity matrix and ${\displaystyle [\mathbf {k} ]_{\times }}$ is the dual 2-form of ${\displaystyle \mathbf {k} }$ or cross product matrix,

${\displaystyle [\mathbf {k} ]_{\times }=\left[{\begin{array}{ccc}0&-k_{3}&k_{2}\\k_{3}&0&-k_{1}\\-k_{2}&k_{1}&0\end{array}}\right]}$.

Note that ${\displaystyle [\mathbf {k} ]_{\times }}$ satisfies ${\displaystyle [\mathbf {k} ]_{\times }\mathbf {v} =\mathbf {k} \times \mathbf {v} }$ for all ${\displaystyle \mathbf {v} }$.

### The general case

The operator

"Rotation with the angle ${\displaystyle \theta }$ around a specified axis"

discussed above is an orthogonal mapping and its matrix relative to any base vector system is therefore an orthogonal matrix . Furthermore its determinant has the value 1. A non-trivial fact is the opposite, i.e. that for any orthogonal linear mapping in ${\displaystyle R^{3}}$ having determinant = 1 there exist base vectors

${\displaystyle {\hat {e}}_{1}\ ,\ {\hat {e}}_{2}\ ,\ {\hat {e}}_{3}}$

such that the matrix takes the "canonical form"

${\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}}$

for some value of ${\displaystyle \theta }$.

In fact, if a linear operator has the orthogonal matrix

${\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}}}$

relative some base vector system

${\displaystyle {\hat {f}}_{1}\ ,\ {\hat {f}}_{2}\ ,\ {\hat {f}}_{3}}$

and this matrix is symmetric, the "Symmetric operator theorem" valid in ${\displaystyle R^{n}}$ (any dimension) applies saying

that it has n orthogonal eigenvectors. This means for the 3-dimensional case that there exists a coordinate system

${\displaystyle {\hat {e}}_{1}\ ,\ {\hat {e}}_{2}\ ,\ {\hat {e}}_{3}}$

such that the matrix takes the form

${\displaystyle {\begin{bmatrix}B_{11}&0&0\\0&B_{22}&0\\0&0&B_{33}\end{bmatrix}}}$

As it is an orthogonal matrix these diagonal elements ${\displaystyle B_{ii}}$ are either 1 or 1. As the determinant is 1 these elements are either all 1 or one of the elements is 1 and the other two are 1.

In the first case it is the trivial identity operator corresponding to ${\displaystyle \theta =0}$.

In the second case it has the form

${\displaystyle {\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}}$

if the basevectors are numbered such that the one with eigenvalue 1 has index 3. This matrix is then of the desired form for ${\displaystyle \theta =\pi }$.

If the matrix is un-symmetric, the vector

${\displaystyle {\bar {E}}=\alpha _{1}\ {\hat {f}}_{1}+\alpha _{2}\ {\hat {f}}_{2}+\alpha _{3}\ {\hat {f}}_{3}}$

where

${\displaystyle \alpha _{1}={\frac {A_{32}-A_{23}}{2}}}$
${\displaystyle \alpha _{2}={\frac {A_{13}-A_{31}}{2}}}$
${\displaystyle \alpha _{3}={\frac {A_{21}-A_{12}}{2}}}$

is non-zero. This vector is an eigenvector with eigenvalue

${\displaystyle \lambda =1}$

Setting

${\displaystyle {\hat {e}}_{3}={\frac {\bar {E}}{|{\bar {E}}|}}}$

and selecting any two orthogonal unit vectors in the plane orthogonal to ${\displaystyle {\hat {e}}_{3}}$:

${\displaystyle {\hat {e}}_{1}\ ,\ {\hat {e}}_{2}}$

such that

${\displaystyle {\hat {e}}_{1}\ ,\ {\hat {e}}_{2},\ {\hat {e}}_{3}}$

form a positively oriented triple, the operator takes the desired form with

${\displaystyle \cos \theta ={\frac {A_{11}+A_{22}+A_{33}-1}{2}}}$
${\displaystyle \sin \theta =|{\bar {E}}|}$

The expressions above are in fact valid also for the case of a symmetric rotation operator corresponding to a rotation with ${\displaystyle \theta =0}$ or ${\displaystyle \theta =\pi }$. But the difference is that for ${\displaystyle \theta =\pi }$ the vector

${\displaystyle {\bar {E}}=\alpha _{1}\ {\hat {f}}_{1}+\alpha _{2}\ {\hat {f}}_{2}+\alpha _{3}\ {\hat {f}}_{3}}$

is zero and of no use for finding the eigenspace of eigenvalue 1, i.e. the rotation axis.

Defining ${\displaystyle E_{4}}$ as ${\displaystyle \cos \theta }$ the matrix for the rotation operator is

${\displaystyle {\frac {1-E_{4}}{{E_{1}}^{2}+{E_{2}}^{2}+{E_{3}}^{2}}}{\begin{bmatrix}E_{1}E_{1}&E_{1}E_{2}&E_{1}E_{3}\\E_{2}E_{1}&E_{2}E_{2}&E_{2}E_{3}\\E_{3}E_{1}&E_{3}E_{2}&E_{3}E_{3}\end{bmatrix}}+{\begin{bmatrix}E_{4}&-E_{3}&E_{2}\\E_{3}&E_{4}&-E_{1}\\-E_{2}&E_{1}&E_{4}\end{bmatrix}}}$

provided that

${\displaystyle {E_{1}}^{2}+{E_{2}}^{2}+{E_{3}}^{2}>0}$

i.e. except for the cases ${\displaystyle \theta =0}$ (the identity operator) and ${\displaystyle \theta =\pi }$

## Quaternions

Quaternions are defined similar to ${\displaystyle E_{1}\ ,\ E_{2}\ ,\ E_{3}\ ,\ E_{4}}$ with the difference that the half angle ${\displaystyle {\frac {\theta }{2}}}$ is used instead of the full angle ${\displaystyle \theta }$.

This means that the first 3 components ${\displaystyle q_{1}\ ,\ q_{2}\ ,\ q_{3}\ }$ are components of a vector defined from

${\displaystyle q_{1}\ {\hat {f_{1}}}\ +\ q_{2}\ {\hat {f_{2}}}\ +\ \ q_{3}\ {\hat {f_{1}}}\ =\ \sin {\frac {\theta }{2}}\quad {\hat {e_{3}}}={\frac {\sin {\frac {\theta }{2}}}{\sin \theta }}\quad {\bar {E}}}$

and that the fourth component is the scalar

${\displaystyle q_{4}=\cos {\frac {\theta }{2}}}$

As the angle ${\displaystyle \theta }$ defined from the canonical form is in the interval

${\displaystyle 0\leq \theta \leq \pi }$

one would normally have that ${\displaystyle q_{4}\geq 0}$. But a "dual" representation of a rotation with quaternions is used, i.e.

${\displaystyle q_{1}\ ,\ q_{2}\ ,\ q_{3}\ ,\ q_{4}\ }$

and

${\displaystyle -q_{1}\ ,\ -q_{2}\ ,\ -q_{3}\ ,\ -q_{4}\ }$

are two alternative representations of one and the same rotation.

The entities ${\displaystyle E_{k}}$ are defined from the quaternions by

${\displaystyle E_{1}=2q_{4}q_{1}}$
${\displaystyle E_{2}=2q_{4}q_{2}}$
${\displaystyle E_{3}=2q_{4}q_{3}}$
${\displaystyle E_{4}={q_{4}}^{2}-({q_{1}}^{2}+{q_{2}}^{2}+{q_{3}}^{2})}$

Using quaternions the matrix of the rotation operator is

${\displaystyle {\begin{bmatrix}2({q_{1}}^{2}+{q_{4}}^{2})-1&2({q_{1}}{q_{2}}-{q_{3}}{q_{4}})&2({q_{1}}{q_{3}}+{q_{2}}{q_{4}})\\2({q_{1}}{q_{2}}+{q_{3}}{q_{4}})&2({q_{2}}^{2}+{q_{4}}^{2})-1&2({q_{2}}{q_{3}}-{q_{1}}{q_{4}})\\2({q_{1}}{q_{3}}-{q_{2}}{q_{4}})&2({q_{2}}{q_{3}}+{q_{1}}{q_{4}})&2({q_{3}}^{2}+{q_{4}}^{2})-1\\\end{bmatrix}}}$

## Numerical example

Consider the reorientation corresponding to the Euler angles ${\displaystyle \alpha =10^{\circ }\quad \beta =20^{\circ }\quad \gamma =30^{\circ }\quad }$ relative a given base vector system

${\displaystyle {\hat {f}}_{1}\ ,\ {\hat {f}}_{2},\ {\hat {f}}_{3}}$

Corresponding matrix relative to this base vector system is (see Euler angles#Matrix orientation)

${\displaystyle {\begin{bmatrix}0.771281&-0.633718&0.059391\\0.613092&0.714610&-0.336824\\0.171010&0.296198&0.939693\end{bmatrix}}}$

and the quaternion is

${\displaystyle (0.171010,\ -0.030154,\ 0.336824,\ 0.925417)}$

The canonical form of this operator

${\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{bmatrix}}}$

with ${\displaystyle \theta =44.537^{\circ }}$ is obtained with

${\displaystyle {\hat {e}}_{3}=(0.451272,-0.079571,0.888832)}$

The quaternion relative to this new system is then

${\displaystyle (0,\ 0,\ 0.378951,\ 0.925417)=(0,\ 0,\ \sin {\frac {\theta }{2}},\ \cos {\frac {\theta }{2}})}$

Instead of making the three Euler rotations

${\displaystyle 10^{\circ },20^{\circ },30^{\circ }}$

the same orientation can be reached with one single rotation of size ${\displaystyle 44.537^{\circ }}$ around ${\displaystyle {\hat {e}}_{3}}$

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## References

• Shilov, Georgi (1961), An Introduction to the Theory of Linear Spaces, Prentice-Hall, Library of Congress 61-13845.