Transformation of vector coefficients

It has already been demonstrated, in Section 1.4, that point coordinates and vector coefficients display a different behaviour when the coordinate origin is shifted. The same happens when a translation is applied to a pair of points. The coordinates of the points will be changed according to

$$\tilde{\mbox{\textit{\textbf{x}}}}=(\mbox{\textit{\textbf{I,\... ...bf{y}}}=\mbox{\textit{\textbf{y}}}+\mbox{\textit{\textbf{t}}}.$$

However, the distance between the points will be invariant:

$$\tilde{\mbox{\textit{\textbf{y}}}}-\tilde{\mbox{\textit{\text... ...f{t}}})=\mbox{\textit{\textbf{y}}}-\mbox{\textit{\textbf{x}}}.$$

Distances are absolute values of vectors, see Section 1.6. Usually point coordinates and vector coefficients are described by the same kind of $(3 \times 1)$ columns and are difficult to distinguish. It is a great advantage of the augmented columns to provide a clear distinction between these quantities.

If $\mbox{$\mos{x}$}_x$ and $\mbox{$\mos{x}$}_y$ are the augmented columns of coordinates of the points $X$ and $Y$,

$$\mbox{$\mos{x}$}_x=\left( \begin{array}{c} x_1\\ x_2\\ x_3\\ ... ...{c} y_1-x_1\\ y_2-x_2\\ y_3-x_3\\ \hline0 \end{array} \right)$$

is the augmented column $\mbox{$\mos{r}$}$ of the coefficients of the distance vector r between $X$ and $Y$. The last coefficient of $\mbox{$\mos{r}$}$ is zero, because of $1-1=0$. It follows that columns of vector coefficients are augmented in another way than columns of point coordinates.

Let T be a translation, (I,t) its matrix-column pair, $\mbox{$\mos{T}$}$ its augmented matrix, r the $(3 \times 1)$ column of coefficients of the distance vector r between $X$ and $Y$, and $\mbox{$\mos{r}$}$ the augmented column of r. Then,

$$\tilde{\mbox{$\mos{r}$}}=\mbox{$\mos{T}$}\mbox{$\mos{r}$}\ \... ...egin{array}{c} r_1\\ r_2\\ r_3\\ \hline0 \end{array} \right).$$ (4.4.1)

When using augmented columns and matrices, the coefficients of t are multiplied with the last coefficient 0 of the $\mbox{$\mos{r}$}$ column and thus become ineffective.

This behaviour is valid not only for translations but holds in general for affine mappings, and thus for isometries and crystallographic symmetry operations:

$$\mbox{\textit{\textbf{y}}}-\mbox{\textit{\textbf{x}}} \righta... ...{\textit{\textbf{y}}}-\mbox{\textit{\textbf{x}}})\ \mbox{ or }$$

$$\tilde{\mbox{$\mos{r}$}}=\mbox{$\mos{W}$}\,\mbox{$\mos{r}$} ... ...+0\,\mbox{\textit{\textbf{w}}}=\mbox{\textit{\textbf{W\,r}}}.$$ (4.4.2)

Whereas point coordinates are transformed by $\tilde{\mbox{\textit{\textbf{x}}}}=(\mbox{\textit{\textbf{W,\,w}}})\mbox{\textit{\textbf{x}}}=\mbox{\textit{\textbf{W\,x}}}+\mbox{\textit{\textbf{w}}}$, vector coefficients r are affected only by the matrix part W:

$$\tilde{\mbox{\textit{\textbf{r}}}}=(\mbox{\textit{\textbf{W,\,w}}})\mbox{\textit{\textbf{r}}}=\mbox{\textit{\textbf{W\,r}}}.$$ (4.4.3)

In other words: if (W,w) describes an affine mapping (isometry, crystallographic symmetry operation) in point space, then W describes the corresponding mapping in vector space. For vector coefficients, the column part w does not contribute to the mapping. This is valid for any vector, e.g., also for the basis vectors of the coordinate system.

Note that $\tilde{\mbox{\textit{\textbf{y}}}}-\tilde{\mbox{\textit{\textbf{x}}}} =\mbox{\textit{\textbf{W}}}(\mbox{\textit{\textbf{y}}}-\mbox{\textit{\textbf{x}}})$ is different from $(\widetilde{\mbox{\textit{\textbf{y}}}-\mbox{\textit{\textbf{x}}}})=\mbox{\text... ...ox{\textit{\textbf{y}}}-\mbox{\textit{\textbf{x}}}) +\mbox{\textit{\textbf{w}}}$. The latter expression describes the image point $\tilde{Z}$ of the point $Z$ with the coordinates $\mbox{\textit{\textbf{z}}}=\mbox{\textit{\textbf{y}}}-\mbox{\textit{\textbf{x}}}$.

Copyright © 2002 International Union of Crystallography