Rodrigues formula

The Rodrigues formula , named after Olinde Rodrigues , is a formula for the exponential function of an antisymmetric 3 × 3 matrix, which describes a cross product in matrix form. It is:

${\displaystyle \exp([a]_{\times })=I+\sin(\|a\|){\begin{bmatrix}{\dfrac {a}{\|a\|}}\end{bmatrix}}_{\times }+(1-\cos(\|a\|)){\begin{bmatrix}{\dfrac {a}{\|a\|}}\end{bmatrix}}_{\times }^{2}}$

The 3 × 3 identity matrix denotes . ${\displaystyle I}$

Its main application is that the result describes a rotation around the axis with an angle as a matrix. ${\displaystyle a}$ ${\displaystyle \|a\|}$

Derivation

The exponential function can be represented in an infinite series that converges absolutely for all values ​​from as: ${\displaystyle \mathbb {R} }$

${\displaystyle \exp(x)=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$

The equation can also be used for any square matrices . One that is suitable for this because of its special properties is the matrix of the cross product . For three-dimensional, real space it reads : ${\displaystyle \mathbb {R} ^{3}}$

${\displaystyle {\vec {a}}={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}\qquad [a]_{\times }={\begin{pmatrix}0&-a_{3}&a_{2}\\a_{3}&0&-a_{1}\\-a_{2}&a_{1}&0\end{pmatrix}}}$

The following formula is obtained by multiplying:

${\displaystyle [a]_{\times }^{3}=[a]_{\times }\cdot [a]_{\times }\cdot [a]_{\times }=-\|a\|^{2}\cdot [a]_{\times }}$,

where is the length of the vector . This means that the powers of the matrix can always be reduced to powers with exponents less than 3. Therefore, these matrices are suitable for insertion into power series. ${\displaystyle \|a\|={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}}$ ${\displaystyle a}$ ${\displaystyle [a]_{\times }}$

There are also Taylor expansions for sine and cosine . They are:

${\displaystyle \sin(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n+1}}{(2n+1)!}}={\frac {x}{1!}}-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots }$

${\displaystyle \cos(x)=\sum _{n=0}^{\infty }(-1)^{n}{\frac {x^{2n}}{(2n)!}}=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots }$

These equations can be combined: terms with an even exponent can be replaced by the cosine expansion and terms with an odd exponent by the sine expansion. After a few simplifications, the Rodrigues equation is obtained.

properties

Be . Then: ${\displaystyle R(a)=\exp([a]_{\times })}$

${\displaystyle R(-a)\,=\,R(a)^{-1}\,=\,R(a)^{T}}$

${\displaystyle R(a)\cdot a=a}$

application

The Rodrigues formula plays a role especially in robotics and computer graphics . There is always a coordinate system, defined by , in which the following applies to a vector : ${\displaystyle \left(e_{1},e_{2},{\dfrac {a}{\|a\|}}\right)}$ ${\displaystyle {\vec {a}}}$

${\displaystyle {\vec {a}}={\begin{pmatrix}0\\0\\\alpha \end{pmatrix}}}$

This means that the matrix represents a rotation around the axis . The angle of rotation is included , i.e. the length of the vector. ${\displaystyle \exp([a]_{\times })}$ ${\displaystyle {\dfrac {a}{\|a\|}}}$ ${\displaystyle \|a\|}$

literature

• MEH Ismail: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge UK 2005, ISBN 0-521-78201-5 .
• O. Faugeras: Three-Dimensional Computer Vision - A Geometric Viewpoint. MIT Press, Cambridge MA 1993, ISBN 0-262-06158-9 .