Interpretation of elements of the logarithm of a rotation matrix as rotation components around coordinate axes of a reference frame

Abstract

Crystal orientation is an important factor when we consider microstructures in materials. With respect to a reference frame, certain crystal orientation can be expressed by a rotation angle $\Phi$ around a unit vector $\mathbf{n}=\left( {h,\hbox { }k,\hbox { }l} \right)$. Partitioning of $\Phi$ into rotation components around coordinate axes of the reference frame is discussed. For a rotation matrix $\mathbf{R}$ corresponding to the axis/angle pair, its logarithm $\ln \mathbf{R}$ is a skew symmetric tensor with three independent elements, $h\Phi , k\Phi$ and $l\Phi$. It is shown that these elements can be interpreted to be sums of the divided rotation angles around the coordinate axes. The elements $h\Phi , k\Phi$ and $l\Phi$ of $\ln \mathbf{R}$ called the log angles can be used as the rotation components to evaluate crystal orientation in materials.

Appendices

Appendix 1

The rotation matrix $\mathbf{R}$ corresponding to the model shown in Fig. 1 is written by the successive rotations as

$$\begin{aligned} \mathbf{R}=\mathbf{MR}_x \left( \Phi \right) { }^t\mathbf{M}, \end{aligned}$$

(15)

where

$$\begin{aligned} \mathbf{R}_x \left( \Phi \right) =\left( {{\begin{array}{lll} 1&{}\quad 0&{} \quad 0 \\ 0&{}\quad {\cos \Phi }&{}\quad {-\sin \Phi } \\ 0&{} \quad {\sin \Phi }&{}\quad {\cos \Phi } \\ \end{array} }} \right) , \end{aligned}$$

(16)

$\mathbf{M}$ is the rotation matrix giving the transformation

$$\begin{aligned} \left( {{\begin{array}{l} h \\ k \\ l \\ \end{array} }} \right) =\mathbf{M}\left( {{\begin{array}{l} 1 \\ 0 \\ 0 \\ \end{array} }} \right) , \end{aligned}$$

(17)

and $^t\mathbf{M}$ is the transpose of $\mathbf{M}$. Since $\mathbf{M}$ is the orthogonal matrix with determinant 1, the elements of $\mathbf{R}$ given by (15) can be written as a function of h, k, l and $\Phi$. Calculating the right-hand-side of (15) from (16) and (17), we find it is the same with the right-hand-side of (1).

Appendix 2

The definition of Rodrigues’ vector $\mathbf{v}$ for the axis/angle pair $\mathbf{n}=\left( {h,\hbox { }k,\hbox { }l} \right) /\Phi$ is [10, 11, 18]

$$\begin{aligned} \mathbf{v}=\tan \left( {\Phi /2} \right) \mathbf{n}=\tan \left( {\Phi /2} \right) \left( {{\begin{array}{l} h \\ k \\ l \\ \end{array} }} \right) . \end{aligned}$$

Using Rodrigues’ vector $\mathbf{v}$, the rotation matrix $\mathbf{R}$ corresponding to this axis/angle pair is given by [11]

$$\begin{aligned} \mathbf{R}=\frac{1}{1+\,^t\mathbf{v v}}\left[ {\left( {1-\,^t\mathbf{v v}} \right) \mathbf{E}+2\mathbf{v}\, ^t\mathbf{v}+2{\hat{\mathbf{v}}}} \right] , \end{aligned}$$

where $^t\mathbf{v}$ is the transpose of $\mathbf{v}$ and ${\hat{\mathbf{v}}}$ the hat map of $\mathbf{v}$.