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''' | + | 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]]. | ||
Revision as of 11:10, 12 May 2017
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.