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Affine isomorphism

From Online Dictionary of Crystallography

Revision as of 14:02, 23 April 2007 by MassimoNespolo (talk | contribs) (See also)

Each symmetry operation of 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 is 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.

See also