Difference between revisions of "Affine isomorphism"
From Online Dictionary of Crystallography
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| − | Each symmetry operation of crystallographic group in ''E''<sup>3</sup> may be represented by a 3×3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub> = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub> = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' | + | <font color="red">Affiner Isomorphismus</font> (''Ge''). <font color="green">Isomorfismo afín</font> (''Sp''). |
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| + | Each symmetry operation of a crystallographic group in ''E''<sup>3</sup> may be represented by a 3×3 matrix '''W''' (the ''linear part'') and a vector '''w'''. Two crystallographic groups ''G''<sub>1</sub> = {('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)} and ''G''<sub>2</sub> = {('''W'''<sub>2''i''</sub>,'''w'''<sub>2''i''</sub>)} are called '''affine isomorphic''' if there exists a non-singular 3×3 matrix '''A''' and a vector '''a''' such that: | ||
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| − | ''G''<sub>2</sub> = {('''A''','''a''')('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)('''A''','''a''')<sup>-1</sup>} | + | ''G''<sub>2</sub> = {('''A''','''a''')('''W'''<sub>1''i''</sub>,'''w'''<sub>1''i''</sub>)('''A''','''a''')<sup>-1</sup>}. |
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Two crystallographic groups are affine isomorphic if and only if their arrangement of symmetry elements may be mapped onto each other by an [[affine mapping]] of ''E''<sup>3</sup>. Two affine isomorphic groups are always [[Group isomorphism|isomorphic]]. | Two crystallographic groups are affine isomorphic if and only if their arrangement of symmetry elements may be mapped onto each other by an [[affine mapping]] of ''E''<sup>3</sup>. Two affine isomorphic groups are always [[Group isomorphism|isomorphic]]. | ||
Latest revision as of 17:50, 8 November 2017
Affiner Isomorphismus (Ge). Isomorfismo afín (Sp).
Each symmetry operation of a crystallographic group in E3 may be represented by a 3×3 matrix W (the linear part) and a vector w. Two crystallographic groups G1 = {(W1i,w1i)} and G2 = {(W2i,w2i)} are called affine isomorphic if there exists a non-singular 3×3 matrix A and a vector a such that:
G2 = {(A,a)(W1i,w1i)(A,a)-1}.
Two crystallographic groups are affine isomorphic if and only if their arrangement of symmetry elements may be mapped onto each other by an affine mapping of E3. Two affine isomorphic groups are always isomorphic.