Derivation of coordinate triplets from symbols for symmetry operations

Fischer, W.; Koch, E.

doi:10.1107/97809553602060000523

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

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International Tables for Crystallography (2006). Vol. A. ch. 11.2, pp. 813-814

Section 11.2.2. Derivation of coordinate triplets from symbols for symmetry operations

W. Fischer^a and E. Koch^a ^*

^a Institut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: kochelke@mailer.uni-marburg.de

11.2.2. Derivation of coordinate triplets from symbols for symmetry operations

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A particular symmetry operation is uniquely described by its symbol, as introduced in Section 11.1.2 , and the coordinate system to which it refers. In the examples of the previous section, the symbols have been derived from the coordinate triplets representing the respective symmetry operations. Inversely, the pair (W, w) of the symmetry operation and the coordinate triplet of the image point can be deduced from the symbol.

(i) For all symmetry operations of space groups, the rotation parts W referring to conventional coordinate systems are listed in Tables 11.2.2.1 and 11.2.2.2 as matrices for point-group symmetry operations. For rotoinversions, the position of the inversion point at 0, 0, 0 is not explicitly given.

Table 11.2.2.1 | top | pdf |
Matrices for point-group symmetry operations and orientation of corresponding symmetry elements, referred to a cubic, tetragonal, orthorhombic, monoclinic, triclinic or rhombohedral coordinate system (cf. Table 2.1.2.1 )

Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W
1		$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\matrix{2 {\hbox to 12pt{}}0,0,z\cr \cr {\hbox to 13pt{}}[001]\cr}]$	$[\bar{x},\bar{y},z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{2 {\hbox to 12pt{}}0,y,0\cr \cr {\hbox to 13pt{}}[010]\cr}]$	$[\bar{x},y,\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}x,0,0\cr \cr {\hbox to 12pt{}}[100]\cr}]$	$[x,\bar{y},\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{3^{+} &x,x,x\cr \cr &[111]\cr}]$		$[\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[\bar{z},\bar{x},y]$	$[\pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[z,\bar{x},\bar{y}]$	$[\pmatrix{0 &0 &1\cr \bar{1} &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{3^{+} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[\bar{z},x,\bar{y}]$	$[\pmatrix{0 &0 &\bar{1}\cr 1 &0 &0\cr 0 &\bar{1} &0\cr}]$
$[\matrix{3^{-} &x,x,x\cr \cr &[111]\cr}]$		$[\pmatrix{0 &1 &0\cr 0 &0 &1\cr 1 &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr {\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[\bar{y},z,\bar{x}]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &1\cr \bar{1} &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr {\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[\bar{y},\bar{z},x]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr 1 &0 &0\cr}]$	$[\matrix{3^{-} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[y,\bar{z},\bar{x}]$	$[\pmatrix{0 &1 &0\cr 0 &0 &\bar{1}\cr \bar{1} &0 &0\cr}]$
			$[\matrix{2 {\hbox to 12pt{}}x,x,0\cr \cr{\hbox to 13pt{}}[110]\cr}]$	$[y,x,\bar{z}]$	$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}x,0,x\cr \cr{\hbox to 13.5pt{}}[101]\cr}]$	$[z,\bar{y},x]$	$[\pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr 1 &0 &0\cr}]$	$[\matrix{2 {\hbox to 11pt{}}0,y,y\cr \cr{\hbox to 14pt{}}[011]\cr}]$	$[\bar{x},z,y]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &1\cr0 &1 &0\cr}]$
			$[\matrix{2 {\hbox to 12pt{}}x,\bar{x},0\cr \cr{\hbox to 13pt{}}[1\bar{1}0]\cr}]$	$[\bar{y},\bar{x},\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 12pt{}}\bar{x},0,x\cr \cr{\hbox to 13.5pt{}}[\bar{1}01]\cr}]$	$[\bar{z},\bar{y},\bar{x}]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{2 {\hbox to 11pt{}}0,y,\bar{y}\cr \cr{\hbox to 13.5pt{}}[01\bar{1}]\cr}]$	$[\bar{x},\bar{z},\bar{y}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{4^{+} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[\bar{y},x,z]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{4^{+} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[z,y,\bar{x}]$	$[\pmatrix{0 &0 &1\cr 0 &1 &0\cr\bar{1} &0 &0\cr}]$	$[\matrix{4^{+} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[x,\bar{z},y]$	$[\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}]$
			$[\matrix{4^{-} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[y,\bar{x},z]$	$[\pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{4^{-} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[\bar{z},y,x]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr 1 &0 &0\cr}]$	$[\matrix{4^{-} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[x,z,\bar{y}]$	$[\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}]$
$[\matrix{\bar{1} &0,0,0\cr}]$	$[\bar{x},\bar{y},\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,0\cr \cr{\hbox to 13pt{}}[001]\cr}]$	$[x,y,\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,0,z\cr \cr{\hbox to 14pt{}}[010]\cr}]$	$[x,\bar{y},z]$	$[\pmatrix{1 &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 9pt{}}0,y,z\cr \cr{\hbox to 14pt{}}[100]\cr}]$	$[\bar{x},y,z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$
$[\matrix{\bar{3}^{+} &x,x,x\cr \cr&[111]\cr}]$	$[\bar{z},\bar{x},\bar{y}]$	$[\pmatrix{0 &0 &\bar{1}\cr \bar{1} &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[z,x,\bar{y}]$	$[\pmatrix{0 &0 &1\cr 1 &0 &0\cr 0 &\bar{1} &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[\bar{z},x,y]$	$[\pmatrix{0 &0 &\bar{1}\cr 1 &0 &0\cr 0 &1 &0\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[z,\bar{x},y]$	$[\pmatrix{0 &0 &1\cr \bar{1} &0 &0\cr 0 &1 &0\cr}]$
$[\matrix{\bar{3}^{-} &x,x,x\cr \cr&[111]\cr}]$	$[\bar{y},\bar{z},\bar{x}]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}x,\bar{x},\bar{x}\cr \cr{\hbox to 14pt{}}[1\bar{1}\bar{1}]\cr}]$	$[y,\bar{z},x]$	$[\pmatrix{0 &1 &0\cr 0 &0 &\bar{1}\cr 1 &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}\bar{x},x,\bar{x}\cr \cr{\hbox to 14pt{}}[\bar{1}1\bar{1}]\cr}]$	$[y,z,\bar{x}]$	$[\pmatrix{0 &1 &0\cr 0 &0 &1\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 6pt{}}\bar{x},\bar{x},x\cr \cr{\hbox to 13pt{}}[\bar{1}\bar{1}1]\cr}]$	$[\bar{y},z,x]$	$[\pmatrix{0 &\bar{1} &0\cr 0 &0 &1\cr 1 &0 &0\cr}]$
			$[\matrix{m {\hbox to 10pt{}}x,\bar{x},z\cr \cr{\hbox to 14pt{}}[110]\cr}]$	$[\bar{y},\bar{x},z]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 10pt{}}\bar{x},y,x\cr \cr{\hbox to 14pt{}}[101]\cr}]$	$[\bar{z},y,\bar{x}]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &1 &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,\bar{y}\cr \cr{\hbox to 13pt{}}[011]\cr}]$	$[x,\bar{z},\bar{y}]$	$[\pmatrix{1 &0 &0\cr 0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{m {\hbox to 10pt{}}x,x,z\cr \cr{\hbox to 14pt{}}[1\bar{1}0]\cr}]$		$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,x\cr \cr{\hbox to 14pt{}}[\bar{1}01]\cr}]$		$[\pmatrix{0 &0 &1\cr 0 &1 &0\cr 1 &0 &0\cr}]$	$[\matrix{m {\hbox to 10pt{}}x,y,y\cr \cr{\hbox to 13pt{}}[01\bar{1}]\cr}]$		$[\pmatrix{1 &0 &0\cr 0 &0 &1\cr 0 &1 &0\cr}]$
			$[\matrix{\bar{4}^{+} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[y,\bar{x},\bar{z}]$	$[\pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{4}^{+} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13.5pt{}}[010]\cr}]$	$[\bar{z},\bar{y},x]$	$[\pmatrix{0 &0 &\bar{1}\cr 0 &\bar{1} &0\cr 1 &0 &0\cr}]$	$[\matrix{\bar{4}^{+} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[\bar{x},z,\bar{y}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &1\cr 0 &\bar{1} &0\cr}]$
			$[\matrix{\bar{4}^{-} {\hbox to 5pt{}}0,0,z\cr \cr{\hbox to 14.5pt{}}[001]\cr}]$	$[\bar{y},x,\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{4}^{-} {\hbox to 6pt{}}0,y,0\cr \cr{\hbox to 13pt{}}[010]\cr}]$	$[z,\bar{y},\bar{x}]$	$[\pmatrix{0 &0 &1\cr 0 &\bar{1} &0\cr \bar{1} &0 &0\cr}]$	$[\matrix{\bar{4}^{-} {\hbox to 6pt{}}x,0,0\cr \cr{\hbox to 12pt{}}[100]\cr}]$	$[\bar{x},\bar{z},y]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &0 &\bar{1}\cr 0 &1 &0\cr}]$

Table 11.2.2.2 | top | pdf |
Matrices for point-group symmetry operations and orientation of corresponding symmetry elements, referred to a hexagonal coordinate system (cf. Table 2.1.2.1 )

Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W	Symbol of symmetry operation and orientation of symmetry element	Transformed coordinates $[\tilde{x},\tilde{y},\tilde{z}]$	Matrix W
1		$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\matrix{3^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[\bar{y},x - y,z]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{3^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 15.5pt{}}[001]\cr}]$	$[y - x,\bar{x},z]$	$[\pmatrix{\bar{1} &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$
$[\matrix{2 {\hbox to 12pt{}}0,0,z\cr \cr{\hbox to 14pt{}}[001]\cr}]$	$[\bar{x},\bar{y},z]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{6^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$		$[\pmatrix{1 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{6^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 15.5pt{}}[001]\cr}]$		$[\pmatrix{0 &1 &0\cr \bar{1} &1 &0\cr 0 &0 &1\cr}]$
$[\matrix{2 {\hbox to 12pt{}}x,x,0\cr \cr{\hbox to 14pt{}}[110]\cr}]$	$[y,x,\bar{z}]$	$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14.5pt{}}x,0,0\cr \cr{\hbox to 15pt{}}[100]\cr}]$	$[x - y,\bar{y},\bar{z}]$	$[\pmatrix{1 &\bar{1} &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14pt{}}0,y,0\cr \cr{\hbox to 15pt{}}[010]\cr}]$	$[\bar{x},y - x,\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr \bar{1} &1 &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{2 {\hbox to 12pt{}}x,\bar{x},0\cr \cr{\hbox to 14pt{}}[1\bar{1}0]\cr}]$	$[\bar{y},\bar{x},\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14.5pt{}}x,2x,0\cr \cr{\hbox to 10.5pt{}}[120]\cr}]$	$[y - x,y,\bar{z}]$	$[\pmatrix{\bar{1} &1 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{2 {\hbox to 14pt{}}2x,x,0\cr \cr{\hbox to 10.5pt{}}[210]\cr}]$	$[x,x - y,\bar{z}]$	$[\pmatrix{1 &0 &0\cr 1 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{\bar{1} {\hbox to 12pt{}}0,0,0\cr}]$	$[\bar{x},\bar{y},\bar{z}]$	$[\pmatrix{\bar{1} &0 &0\cr 0 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{3}^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[y,y - x,\bar{z}]$	$[\pmatrix{0 &1 &0\cr \bar{1} &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{3}^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[x - y,x,\bar{z}]$	$[\pmatrix{1 &\bar{1} &0\cr 1 &0 &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{m {\hbox to 10pt{}}x,y,0\cr \cr{\hbox to 13.5pt{}}[001]\cr}]$	$[x,y,\bar{z}]$	$[\pmatrix{1 &0 &0\cr 0 &1 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{6}^{+} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 16pt{}}[001]\cr}]$	$[y - x,\bar{x},\bar{z}]$	$[\pmatrix{\bar{1} &1 &0\cr \bar{1} &0 &0\cr 0 &0 &\bar{1}\cr}]$	$[\matrix{\bar{6}^{-} {\hbox to 8pt{}}0,0,z\cr \cr{\hbox to 15.5pt{}}[001]\cr}]$	$[\bar{y},x - y,\bar{z}]$	$[\pmatrix{0 &\bar{1} &0\cr 1 &\bar{1} &0\cr 0 &0 &\bar{1}\cr}]$
$[\matrix{m {\hbox to 10pt{}}x,\bar{x},z\cr \cr{\hbox to 14.5pt{}}[110]\cr}]$	$[\bar{y},\bar{x},z]$	$[\pmatrix{0 &\bar{1} &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}x,2x,z\cr \cr{\hbox to 12pt{}}[100]\cr}]$		$[\pmatrix{\bar{1} &1 &0\cr 0 &1 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}2x,x,z\cr \cr{\hbox to 11.5pt{}}[010]\cr}]$		$[\pmatrix{1 &0 &0\cr 1 &\bar{1} &0\cr 0 &0 &1\cr}]$
$[\matrix{m {\hbox to 10pt{}}x,x,z\cr \cr{\hbox to 14.5pt{}}[1\bar{1}0]\cr}]$		$[\pmatrix{0 &1 &0\cr 1 &0 &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}x,0,z\cr \cr{\hbox to 16.5pt{}}[120]\cr}]$	$[x - y,\bar{y},z]$	$[\pmatrix{1 &\bar{1} &0\cr 0 &\bar{1} &0\cr 0 &0 &1\cr}]$	$[\matrix{m {\hbox to 12pt{}}0,y,z\cr \cr{\hbox to 16pt{}}[210]\cr}]$	$[\bar{x},y - x,z]$	$[\pmatrix{\bar{1} &0 &0\cr \bar{1} &1 &0\cr 0 &0 &1\cr}]$

(ii) The location part $[{\bi w}_{l}]$ of w may easily be derived from $[({\bi W}, {\bi w}_{l}) \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr} = \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr},\quad i.e. \ {\bi w}_{l} = ({\bi I} - {\bi W}) \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr}]$ with $[x_{0},y_{0},z_{0}]$ being the coordinate triplet of the inversion point of a rotoinversion or the coordinate triplet of an arbitrary fixed point of any other symmetry operation. The intrinsic translation part $[{\bi w}_{g}]$ of w is given explicitly in the symbol of the symmetry operation, so that the translation part w is obtained as $[{\bi w} = {\bi w}_{g} + {\bi w}_{l} = \pmatrix{w_{1}\cr w_{2}\cr w_{3}\cr}.]$
(iii) The coordinate triplet $[\tilde{x},\tilde{y},\tilde{z}]$ corresponding to the symmetry operation is now given by $[\eqalign{\tilde{x} &= {W}_{11} x + {W}_{12} y + {W}_{13} z + {w}_{1}\cr \tilde{y} &= {W}_{21} x + {W}_{22} y + {W}_{23} z + {w}_{2} \cr \tilde{z} &= {W}_{31} x + {W}_{32} y + {W}_{33} z + {w}_{3}.\cr}]$

Example

$[4^{-}(0,0,{\textstyle{3 \over 4}})\ {\textstyle{1 \over 4}},\! - {\textstyle{1 \over 4}}, z]$ tetragonal system $[4^{-}\ 0,0,z \; \Rightarrow\; {\bi W} = \pmatrix{0 &1 &0\cr \bar{1} &0 &0\cr 0 &0 &1\cr}\hbox{ from Table 11.2.2.1}.\hfill]$ $[x_{0} = {\textstyle{1 \over 4}}, y_{0} = - {\textstyle{1 \over 4}}, z_{0} = 0]$ is a fixed point of $[4^{-}\ {\textstyle{1 \over 4}},\! -\! {\textstyle{1 \over 4}}, z]$ , i.e. a point on the screw axis. $[\eqalign{&{\bi w}_{l} = ({\bi I} - {\bi W}) \pmatrix{x_{0}\cr y_{0}\cr z_{0}\cr}\cr &\Rightarrow {\bi w}_{l} = \pmatrix{1 &\bar{1} &0\cr 1 &1 &0\cr 0 &0 &0\cr} \pmatrix{\hfill {1 \over 4}\cr - {1 \over 4}\cr \hfill0\cr} = \pmatrix{{1 \over 2}\cr 0\cr 0\cr};\cr &{\bi w} = {\bi w}_{g} + {\bi w}_{l}\cr &\Rightarrow {\bi w} = \pmatrix{0\cr 0\cr {3 \over 4}\cr} + \pmatrix{{1 \over 2}\cr 0\cr 0\cr} = \pmatrix{{1 \over 2}\cr 0\cr {3 \over 4}\cr}\cr &\Rightarrow \tilde{x} = y + {\textstyle{1 \over 2}},\quad \tilde{y} = - x,\quad \tilde{z} = z + {\textstyle{3 \over 4}}.}\hfill]$