International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.5, pp. 7583

1.5.1. Origin shift and change of the basis^{1}
Let a coordinate system be given with a basis and an origin O. Referred to this coordinate system, the column of coordinates of a point X is and the corresponding vector is . Referred to a new coordinate system, specified by the basis and the origin , the column of coordinates of the point X is . Let be the column of coefficients for the vector p from the old origin O to the new origin , see Fig. 1.5.1.1.


The coordinates of the points X (or Y) with respect to the old origin O are x (y), and with respect to the new origin they are . From the diagram one reads and . 
For the columns holds, i.e.This can be written in the formalism of matrix–column pairs (cf. Section 1.2.2.3 for details of the matrix–column formalism) aswhere represents the translation corresponding to the vector p of the origin shift.
The vector r determined by the points X and Y (also known as a `distance vector'), (cf. Fig. 1.5.1.1), and thus with coefficientsshows a different transformation behaviour under the origin shift. From the diagram one reads the equations , , , and thus i.e. the vector coefficients of r are not affected by the origin shift.
Example
The description of a crystal structure is closely related to its spacegroup symmetry: different descriptions of the underlying space group, in general, result in different descriptions of the crystal structure. This example illustrates the comparison of two structure descriptions corresponding to different origin choices of the space group.
To compare the two structures it is not only necessary to apply the originshift transformation but also to adjust the selection of the representative atoms of the two descriptions.
In the Inorganic Crystal Structure Database (2012) (abbreviated as ICSD) one finds the following two descriptions of the mineral zircon ZrSiO_{4}:
In order to compare the different structure descriptions, the atomic coordinates of the origin choice 1 description are to be transformed to `Origin at centre 2/m', i.e. origin choice 2.
Origin choice 2 has coordinates referred to origin choice 1. Therefore, the change of coordinates consists of subtracting from the origin choice 1 values, i.e. leave the x coordinate unchanged, add to the y coordinate and subtract from the z coordinate [cf. equation (1.5.1.1)].
The transformed and normalized coordinates (so that ) are
A change of the basis is described by a (3 × 3) matrix:The matrix P relates the new basis to the old basis according toThe matrix P is often referred to as the linear part of the coordinate transformation and it describes a change of direction and/or length of the basis vectors. It is preferable to choose the matrix P in such a way that its determinant is positive: a negative determinant of P implies a change from a righthanded coordinate system to a lefthanded coordinate system or vice versa. If det(P) = 0, then the new vectors are linearly dependent, i.e. they do not form a complete set of basis vectors.
For a point X (cf. Fig. 1.5.1.1), the vector isBy inserting equation (1.5.1.4) one obtainsori.e. or , which is often written asHere the inverse matrix is designated by Q, while o is the (3 × 1) column vector with zero coefficients. [Note that in equation (1.5.1.4) the sum is over the row (first) index of P, while in equation (1.5.1.5), the sum is over the column (second) index of Q.]
A selected set of transformation matrices P and their inverses that are frequently used in crystallographic calculations are listed in Table 1.5.1.1 and illustrated in Figs. 1.5.1.2 to 1.5.1.10.
Example
Consider an Fcentred cell with conventional basis and a corresponding primitive cell with basis , cf. Fig. 1.5.1.4. The transformation matrix P from the conventional basis to a primitive basis can either be deduced from Fig. 1.5.1.4 or can be read directly from Table 1.5.1.1: = , , , which in matrix notation isThe inverse matrix is also listed in Table 1.5.1.1 or can be deduced from Fig. 1.5.1.4. It is the matrix that describes the conventional basis vectors by linear combinations of : = , = , = , orCorrespondingly, the point coordinates transform asFor example, the coordinates of the end point of with respect to the conventional basis become in the primitive basis, the centring point of the plane becomes the end point of etc.

Bodycentred cell I with a_{I}, b_{I}, c_{I} and a corresponding primitive cell P with a_{P}, b_{P}, c_{P}. The origin for both cells is O. A cubic I cell with lattice constant can be considered as a primitive rhombohedral cell with and (rhombohedral axes) or a triple hexagonal cell with and (hexagonal axes). 

Facecentred cell F with a_{F}, b_{F}, c_{F} and a corresponding primitive cell P with a_{P}, b_{P}, c_{P}. The origin for both cells is O. A cubic F cell with lattice constant can be considered as a primitive rhombohedral cell with and (rhombohedral axes) or a triple hexagonal cell with and (hexagonal axes). 

Tetragonal lattices, projected along . (a) Primitive cell P with a, b, c and the Ccentred cells with and with . The origin for all three cells is the same. (b) Bodycentred cell I with a, b, c and the Fcentred cells with and with . The origin for all three cells is the same. The fractions indicate the height of the lattice points along the axis of projection. 

Unit cells in the rhombohedral lattice: same origin for all cells. The basis of the rhombohedral cell is labelled a_{rh}, b_{rh}, c_{rh}. Two settings of the triple hexagonal cell are possible with respect to a primitive rhombohedral cell: The obverse setting with the lattice points 0, 0, 0; ; has been used in International Tables since 1952. Its general reflection condition is . The reverse setting with lattice points 0, 0, 0; ; was used in the 1935 edition. Its general reflection condition is . The fractions indicate the height of the lattice points along the axis of projection. (a) Obverse setting of triple hexagonal cell in relation to the primitive rhombohedral cell a_{rh}, b_{rh}, c_{rh}. (b) Reverse setting of triple hexagonal cell in relation to the primitive rhombohedral cell a_{rh}, b_{rh}, c_{rh}. (c) Primitive rhombohedral cell (   lower edges), a_{rh}, b_{rh}, c_{rh} in relation to the three triple hexagonal cells in obverse setting ; ; . Projection along c′. (d) Primitive rhombohedral cell (   lower edges), a_{rh}, b_{rh}, c_{rh} in relation to the three triple hexagonal cells in reverse setting ; ; . Projection along c′. 

Hexagonal lattice projected along . Primitive hexagonal cell P with a, b, c and the three Ccentred (orthohexagonal) cells ; ; . The origin for all cells is the same. 

Hexagonal lattice projected along . Primitive hexagonal cell P with a, b, c and the three triple hexagonal cells H with ; ; . The origin for all cells is the same. 

Rhombohedral lattice with a triple hexagonal unit cell a, b, c in obverse setting (i.e. unit cell a_{1}, b_{1}, c in Fig. 1.5.1.6c) and the three centred monoclinic cells. (a) Ccentred cells with ; with ; and with . The unique monoclinic axes are and , respectively. The origin for all four cells is the same. (b) Acentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. The origin for all four cells is the same. The fractions indicate the height of the lattice points along the axis of projection. 

Rhombohedral lattice with primitive rhombohedral cell a_{rh}, b_{rh}, c_{rh} and the three centred monoclinic cells. (a) Ccentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. The origin for all four cells is the same. (b) Acentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. The origin for all four cells is the same. The fractions indicate the height of the lattice points along the axis of projection. 
A general change of the coordinate system involves both an origin shift and a change of the basis. Such a transformation of the coordinate system is described by the matrix–column pair , where the (3 × 3) matrix P relates the new basis to the old one according to equation (1.5.1.4). The origin shift is described by the shift vector . The coordinates of the new origin with respect to the old coordinate system are given by the (3 × 1) column .
The general coordinate transformation can be performed in two consecutive steps. Because the origin shift p refers to the old basis , it has to be applied first (as described in Section 1.5.1.1), followed by the change of the basis (cf. Section 1.5.1.2):Here, I is the threedimensional unit matrix and o is the (3 × 1) column matrix containing only zeros as coefficients.
The formulae for the change of the point coordinates from x to uses , i.e.The effect of a general change of the coordinate system on the coefficients of a vector r is reduced to the linear transformation described by P, as the vector coefficients are not affected by the origin shift [cf. equation (1.5.1.3)].
Hereafter, the data for the matrix–column pairare often written in the following concise form:The concise notation of the transformation matrices is widely used in the tables of maximal subgroups of space groups in International Tables for Crystallography Volume A1 (2010), where describes the relation between the conventional bases of a group and its maximal subgroups. For example, the expression (cf. the table of maximal subgroups of , No. 111, in Volume A1) stands forNote that the matrix elements of P in equation (1.5.1.8) are read by columns since they act on the row matrices of basis vectors, and not by rows, as in the shorthand notation of symmetry operations which apply to column matrices of coordinates (cf. Section 1.2.2.1 ).
References
Inorganic Crystal Structure Database (2012). Release 2012/2. Fachinformationszentrum Karlsruhe and National Institute of Standards and Technology. http://www.fizkarlsruhe.de/icsd.html . (Abbreviated as ICSD.)International Tables for Crystallography (2010). Vol. A1, Symmetry Relations between Space Groups. Edited by H. Wondratschek & U. Müller, 2nd ed. Chichester: John Wiley & Sons.
Krstanovic, I. R. (1958). Redetermination of the oxygen parameters in zircon (ZrSiO_{4}). Acta Cryst. 11, 896–897.
Wyckoff, R. W. G. & Hendricks, S. B. (1927). Die Kristallstruktur von Zirkon und die Kriterien fuer spezielle Lagen in tetragonalen Raumgruppen. Z. Kristallogr. Kristallgeom. Kristallphys. Kristallchem. 66, 73–102.