Bilbao Crystallographic Server Transformation matrix

The transformation matrix

The relation between an arbitrary setting of a space group (given by a set of basis vectors (a, b, c) and an origin O) and a reference (default) coordinate system, defined by the set (a', b', c') and the origin O', is determined by a (3x4) matrix - column pair (P,p). The (3x3) linear matrix P

P =
P11 P12 P13
P21 P22 P23
P31 P32 P33

describes the transformation of the row of basis vectors (a, b, c) to the reference basis vectors (a', b', c').

a' = P11a + P21b + P31c
b' = P12a + P22b + P32c
c' = P13a + P23b + P33c

which is often written as

(a', b', c') = (a, b, c)P

The (3x1) column p = (p1, p2, p3) determines the origin shift of O' with respect the origin O:

O' = O + p

Transformation matrix for a group-subgroup pair G>H

The transformation matrix pair (P,p) of a group-subgroup chain G > H used by the programs on Bilbao Crystallographic Server always describes the transformation from the reference (default) coordinate system of the group G to that of the subgroup H. Let (a,b,c)G be the row of the basis vectors of G and (a',b',c')H the basis row of H. The basis (a',b',c') of H is expressed in the basis (a,b,c)G by the system of equations:

(a', b', c')H = (a, b, c)G P

The column p describes the origin shift between the default origin OG of G to that of H, O'H;

O'H = OG + p

Concise form of the transformation matrix (P,p)

In some of the applications on the Bilbao Crystallographic Server the data on the matrix-column pair (P,p) is listed in the following concise form:

P11a + P21b + P31c, P12a + P22b + P32c, P13a + P23b + P33c ; p1, p2, p3

Note: As the bases (a', b', c') and (a, b, c) are written as rows, in each of the sums a column of the matrix P is listed. The matrix part of concise form of (P,p) is left empty if there is no change of basis, i.e. if P is (3x3) unit matrix. The 'origin shift' part is empty is there is no origin shift, i.e. p is a column consisting of zeros.

Example: b,c,a ; 0,1/4,1/4

The first column of the linear part of the transformation matrix is (0 1 0). The second column of the linear part of the transformation matrix is (0 0 1). The third column of the linear part of the transformation matrix is (1 0 0) or

the linear part of the transformation matrix is:
0 0 1
1 0 0
0 1 0
and the origin shift is:
0 1/4 1/4

[*] For more information: International Tables for Crystallography. Vol. A, Space Group Symmetry. Ed. Theo Hahn (3rd ed.), Dordrecht, Kluwer Academic Publishers, Section "Transformations in crystallography", 1995.

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