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 + p1a+p2b+p3c


[*] 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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