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.

Bilbao Crystallographic Server
For comments, please mail to