Euler-Rodrigues formula

In mathematics and mechanics , the Euler-Rodrigues formula according to Leonhard Euler and Olinde Rodrigues is used to describe a rotation in three dimensions. Defined with four Euler parameters for which applies ${\displaystyle a,b,c,d}$ ${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}$

${\displaystyle Q:={\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-ad)&2(bd+ac)\\2(bc+ad)&a^{2}+c^{2}-b^{2}-d^{2}&2(cd-ab)\\2(bd-ac)&2(cd+ab)&a^{2}+d^{2}-b^{2}-c^{2}\end{pmatrix}}}$

a rotation matrix . This formula is based on Rodrigues' formula , but uses a different parameterization.

The formula is used in flight simulators and computer games .

properties

symmetry

The parameters ( ) and ( ) describe the same rotation, which is due to the fact that they are always multiplied in pairs in the Q-matrix and thus the minus signs are neutralized. Apart from this symmetry, four parameters define the rotation matrix in a unique way. ${\displaystyle a,b,c,d}$ ${\displaystyle -a,-b,-c,-d}$

Vector formulation

From the parameters a can vector are formed. In superscript denotes the transposed matrix , so that a column vector. Then applies to all : ${\displaystyle b,c,d}$ ${\displaystyle {\vec {\varphi }}=(b,c,d)^{\top }}$ ${\displaystyle \top }$ ${\displaystyle {\vec {\varphi }}}$ ${\displaystyle {\vec {x}}}$

${\displaystyle Q{\vec {x}}={\vec {x}}+2a{\vec {\varphi }}\times {\vec {x}}+2{\vec {\varphi }}\times ({\vec {\varphi }}\times {\vec {x}}).}$

This is how the designation for a scalar parameter and a vector parameter is motivated . With the cross product matrix ${\displaystyle a}$ ${\displaystyle b,c,d}$

${\displaystyle [{\vec {\varphi }}]_{\times }={\begin{pmatrix}0&-d&c\\d&0&-b\\-c&b&0\end{pmatrix}}}$

shows up

${\displaystyle Q=E_{3}+2a[{\vec {\varphi }}]_{\times }+2[{\vec {\varphi }}]_{\times }[{\vec {\varphi }}]_{\times }=(2a^{2}-1)E_{3}+2a[{\vec {\varphi }}]_{\times }+2{\vec {\varphi }}{\vec {\varphi }}^{\top }}$

Inside is the identity matrix . This is created with the Euler parameters . For 180 ° rotations , and . ${\displaystyle E_{3}}$ ${\displaystyle {\vec {\varphi }}={\vec {0}}}$ ${\displaystyle (a,b,c,d)=(\pm 1,0,0,0)}$ ${\displaystyle a=0}$ ${\displaystyle |{\vec {\varphi }}|=1}$

Angle of rotation and axis of rotation

Each rotation in three dimensions is determined uniquely by a rotational angle and a rotational axis defined by a unit vector with is defined. Then the Euler parameters of the rotation are: ${\displaystyle \phi }$ ${\displaystyle {\vec {e}}=(e_{x},e_{y},e_{z})^{\top }}$ ${\displaystyle e_{x}^{2}+e_{y}^{2}+e_{z}^{2}=1}$

${\displaystyle a=\cos(\phi /2)}$

${\displaystyle b=\sin(\phi /2)e_{x}}$

${\displaystyle c=\sin(\phi /2)e_{y}}$

${\displaystyle d=\sin(\phi /2)e_{z}}$

If it increases by a full 360 ° rotation, the Euler parameters arise which - as noted above - represent the same rotation. ${\displaystyle \phi }$ ${\displaystyle (-a,-b,-c,-d)}$

So the vector parameter is here . With these parameters and the double angle functions , the Rodrigues formula for the rotation matrix is ​​created: ${\displaystyle {\vec {\varphi }}=\sin(\varphi /2){\hat {e}}}$

${\displaystyle Q=E_{3}+\sin(\varphi )[{\hat {e}}]_{\times }+(1-\cos \phi )[{\hat {e}}]_{\times }[{\hat {e}}]_{\times }}$

Parameters of a rotation matrix

If the rotation matrix is given and the Euler parameters are sought, then they are obtained as follows. If only has positive diagonal elements, then is ${\displaystyle Q}$ ${\displaystyle Q}$

${\displaystyle \operatorname {Sp} (Q)=3a^{2}-b^{2}-c^{2}-d^{2}=4a^{2}-1\quad \rightarrow \quad a=\pm {\frac {1}{2}}{\sqrt {\operatorname {Sp} (Q)+1}}}$

The remaining parameters arise from

${\displaystyle q_{i}={\frac {Q_{kj}-Q_{jk}}{4a}}}$

With

i	1	2	3
q i	b	c	d
j	2	3	1
k	3	1	2

If there are partially negative diagonal elements, then let the largest diagonal element be and ${\displaystyle Q_{ii}}$

${\displaystyle q_{i}={\frac {1}{2}}{\sqrt {1+2Q_{ii}-\operatorname {Sp} (Q)}}}$

With this value and from the above table is determined ${\displaystyle j,k}$

${\displaystyle a=\pm {\frac {Q_{kj}-Q_{jk}}{4q_{i}}},\quad q_{j}={\frac {Q_{ji}+Q_{ij}}{4q_{i}}},\quad q_{k}={\frac {Q_{ki}+Q_{ik}}{4q_{i}}}}$

Calculation of the rotation matrix once with and once with and comparison with the given rotation matrix finally provides the sign of . ${\displaystyle a}$ ${\displaystyle -a}$ ${\displaystyle a}$

Linking two rotations

The combination of two rotations results in a rotation. From Euler parameters for the first rotation and for the second rotation , the combined rotation from the first rotation and the subsequent second rotation results from the Euler parameters ${\displaystyle a_{1},b_{1},c_{1},d_{1}}$ ${\displaystyle Q_{1}}$ ${\displaystyle a_{2},b_{2},c_{2},d_{2}}$ ${\displaystyle Q_{2}}$ ${\displaystyle Q_{2}Q_{1}}$

${\displaystyle a=a_{1}a_{2}-b_{1}b_{2}-c_{1}c_{2}-d_{1}d_{2}}$

${\displaystyle b=a_{1}b_{2}+b_{1}a_{2}-c_{1}d_{2}+d_{1}c_{2}}$

${\displaystyle c=a_{1}c_{2}+c_{1}a_{2}-d_{1}b_{2}+b_{1}d_{2}}$

${\displaystyle d=a_{1}d_{2}+d_{1}a_{2}-b_{1}c_{2}+c_{1}b_{2}}$.

Again , what can be confirmed by inserting applies here . The latter identity has over ${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1}$

${\displaystyle a^{2}+b^{2}+c^{2}+d^{2}=1=(a_{1}^{2}+b_{1}^{2}+c_{1}^{2}+d_{1}^{2})(a_{2}^{2}+b_{2}^{2}+c_{2}^{2}+d_{2}^{2})}$

a direct reference to Euler's four-squares theorem and the quaternions.

Connection with other constructs

Quaternions

→ Main article : "Rotations in three-dimensional space" in the article Quaternion

The Euler parameters can be viewed as components of a unit quaternion. The parameter is its real part and its imaginary part. With the unit quaternions , which consist of the Euler parameters of two rotations , the Euler parameters of the combined rotation can be elegantly calculated with the product of the quaternions: ${\displaystyle a}$ ${\displaystyle b,c,d}$ ${\displaystyle q_{1,2}=a_{1,2}+ib_{1,2}+jc_{1,2}+kd_{1,2}}$ ${\displaystyle Q_{1,2}}$ ${\displaystyle Q_{2}Q_{1}}$

${\displaystyle a+ib+jc+kd=q_{1}q_{2}}$

Here are and the complex imaginary units that are not commutatively linked with the Hamilton rules . For example is . ${\displaystyle i,j}$ ${\displaystyle k}$ ${\displaystyle i^{2}=j^{2}=k^{2}=ijk=-1}$ ${\displaystyle jk=-kj=i}$

Pauli matrices

The unitary 2 × 2 matrices

${\displaystyle {\begin{aligned}E_{2}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=\sigma _{0},&\quad \sigma _{x}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}=\mathrm {i} \sigma _{2}\\\sigma _{y}={\begin{pmatrix}0&\mathrm {i} \\\mathrm {i} &0\end{pmatrix}}=\mathrm {i} \sigma _{1},&\quad \sigma _{z}={\begin{pmatrix}\mathrm {i} &0\\0&-\mathrm {i} \end{pmatrix}}=\mathrm {i} \sigma _{3}\end{aligned}}}$

with the imaginary unit of the complex numbers are related to the Pauli matrices , which are used in the standard model of elementary particle physics and in quantum mechanics. ${\displaystyle i^{2}=-1}$ ${\displaystyle \sigma _{0,1,2,3}}$

The matrices transform in a similar way to the above Hamilton rules of the complex imaginary units of the quaternions: ${\displaystyle \sigma _{x,y,z}}$

${\displaystyle \sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=\sigma _{x}\sigma _{y}\sigma _{z}=-E_{2}}$

Correspondingly, these unitary 2 × 2 matrices can also be used to describe rotations. Details can be found in Quaternion , SU (2) and Spin group .

Using the Euler parameters, the unitary 2 × 2 matrix corresponding to a rotation is:

${\displaystyle U={\begin{pmatrix}\ \ \,a+\mathrm {i} d&b+\mathrm {i} c\\-b+\mathrm {i} c&a-\mathrm {i} d\end{pmatrix}}=a\,E_{2}+b\,\sigma _{x}+c\,\sigma _{y}+d\,\sigma _{z}=a\,\sigma _{0}+\mathrm {i} b\,\sigma _{2}+\mathrm {i} c\,\sigma _{1}+\mathrm {i} d\,\sigma _{3}}$

See also

• Spinor
• Orthogonal tensor

Individual evidence

1. ↑ Axel Volkwein: Numerical simulation of flexible rockfall protection systems . Ed .: Institute for Structural Analysis and Design, Swiss Federal Institute of Technology, Zurich . vdf Hochschulverlag AG, 2004, ISBN 978-3-7281-2986-4 ( limited preview in Google book search [accessed June 30, 2017]).