Bilbao Crystallographic Server arrow Transformation matrix

The transformation matrix

The transformation matrix P = (P,p) relating two coordinate systems with basis vectors a, b, c and a', b', c' has two parts: The new basis vectors a', b', c' are related with the basis vectors a, b, c by

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

And if the new origin has the coordinates p1, p2, p3 in the coordinate system with basis vectors a, b, c, then

the rotational part (P) of the transformation matrix is:
P11 P12 P13
P21 P22 P23
P31 P32 P33
and the origin shift (p) is
p1 p2 p3
For example if:
a' = b; b' = c + 1/4; c' = a + 1/4
the first column of the rotational part of the transformation matrix is (0 1 0) and the first element of the origin shift is 0;
the second column of the rotational part of the transformation matrix is (0 0 1) and the second element of the origin shift is 1/4 ( or 0.25);
the third column of the rotational part of the transformation matrix is (1 0 0) and the third element of the origin shift is 1/4 ( or 0.25);
or
the rotational 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


abc and xyz matrix notations

A (3x4) transformation matrix (P,p) with rotational part P and the origin shift p can be represented by using the concise form:

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

where it is understood that the abc entries should be read in columns.

As an example, a transformation matrix given as

-2/3a-1/3b+2/3c,-b,2a+b;1/6,1/3,1/3

can be written as:

  -2/3 0 2 1/6
(P,p) = -1/3 -1 1 1/3
  2/3 0 0 1/3


On BCS, we also use the xyz notation likewise to represent transformation matrixes. But, this time, it should be interpreted in rows. Therefore, the example matrix above would be represented in the xyz notation as:

-2/3x+2z,-1/3x-y+z,2/3x;1/6,1/3,1/3


This time we used the concise form:

P11x+P21y+P31z,P12x+P22y+P32z,P13x+P23y+P33z;p1,p2,p3

[*] 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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