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Motivation

Any isometry may be the symmetry operation of some object, e.g. of the whole space, because it maps the whole space onto itself. However, if the object is a crystal pattern, due to its periodicity not every rotation, roto-inversion, etc. can be a symmetry operation of this pattern. There are certain restrictions which are well known and which are taught in the elementary courses of crystallography.

How can these symmetry operations be described analytically ? Having chosen a coordinate system with a basis and an origin, each point of space can be represented by its column of coordinates. A mapping is then described by the instruction, in which way the coordinates $\tilde{\mbox{\textit{\textbf{x}}}}$ of the image point $\tilde{X}$ can be obtained from the coordinates x of the original point X:

$\tilde{x}_1=f_1(x_1, x_2, x_3)$, $\tilde{x}_2=f_2(x_1, x_2, x_3)$, $\tilde{x}_3=f_3(x_1, x_2, x_3)$.

The functions $f_1, f_2, \mbox{and} f_3$ are not restricted for an arbitrary mapping. However, for an affine mapping the functions $f_i$ are very simple: An affine mapping $X \longrightarrow \tilde{X}$ is always represented in the form

$$\begin{array}{rcl} \tilde{x}_1 & = & A_{11}x_1+A_{12}x_2+A_... ...tilde{x}_3 & = & A_{31}x_1+A_{32}x_2+A_{33}x_3+a_3 \end{array}$$ (2.2.1)

A second mapping which brings the point $\tilde{X} \longrightarrow \hspace{0.5mm}\tilde{\hspace{-0.5mm}\tilde{X}}$ is then represented by

$$\begin{array}{rcl} \tilde{\tilde{x}}_1 & = & B_{11}\tilde{x... ...2+B_{33} \tilde{x}_3+b_3 \end{array}, \hspace{3em}\mbox{or}$$ (2.2.2)

$$\begin{array}{l} \hspace{-1em}\begin{array}{r@{\hspace{0.1e... ...(A_{31}x_1+A_{32}x_2+A_{33}x_3+a_3)+b_3. \end{array}\end{array}$$ (2.2.3)

The equations (2.2.3) may be rearranged in the following way:

$$\begin{array}{l} \hspace{-1.5em}\begin{array}{r@{\hspace{0.2... ..._3+B_{31}a_1+B_{32}a_2+B_{33}a_3 + b_3. \end{array}\end{array}$$ (2.2.4)

Although straightforward, one will agree that this is not a comfortable way to describe and solve the problem of combining mappings. Matrix formalism does nothing else than to formalize what is being done in equations (2.2.1) to (2.2.4), and to describe this procedure in a kind of shorthand notation, called the matrix notation:

$$% latex2html id marker 3146\mbox{Equation (\ref{afful}) is... ...textbf{x}}}+\mbox{\textit{\textbf{a}}};\hfill \rule{3cm}{0cm}$$ (2.2.5)

$$% latex2html id marker 3158\mbox{Equation (\ref{douaf}) is... ...xtbf{x}}}}+\mbox{\textit{\textbf{b}}}; \hfill \rule{3cm}{0cm}$$ (2.2.6)

$$% latex2html id marker 3170\mbox{Equation (\ref{allaf}) is... ...f{a}}})+ \mbox{\textit{\textbf{b}}};\hfill \rule{16.5mm}{0cm}$$ (2.2.7)

$$% latex2html id marker 3184\mbox{Equation (\ref{alafr}) is... ...tbf{a}}}+\mbox{\textit{\textbf{b}}}.\hfill \rule{16.5mm}{0cm}$$ (2.2.8)

The matrix notation for mappings will be dealt with in more detail in Sections 4.1 and 4.2. In the next section the matrix formalism will be introduced.

Copyright © 2002 International Union of Crystallography