International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 5.1, pp. 7885
https://doi.org/10.1107/97809553602060000510 Chapter 5.1. Transformations of the coordinate system (unitcell transformations)^{a}Institut für Kristallographie, RheinischWestfälische Technische Hochschule, Aachen, Germany Transformations of the coordinate system are useful when nonconventional descriptions of a crystal structure are considered, for instance in the study of relations between different structures, of phase transitions and of group–subgroup relations. Unitcell transformations occur particularly frequently when different settings or cell choices of monoclinic, orthorhombic or rhombohedral space groups are to be compared or when `reduced cells' are derived. In this chapter, matrix notation is used to describe a general transformation. Selected frequently occurring transformation matrices are tabulated and illustrated by diagrams. 
There are two main uses of transformations in crystallography.
Throughout this volume, matrices are written in the following notation:


Here the crystal structure is considered to be at rest, whereas the coordinate system and the unit cell are changed. Specifically, a point X in a crystal is defined with respect to the basis vectors a, b, c and the origin O by the coordinates x, y, z, i.e. the position vector r of point X is given by The same point X is given with respect to a new coordinate system, i.e. the new basis vectors a′, b′, c′ and the new origin O′ (Fig. 5.1.3.1), by the position vector In this section, the relations between the primed and unprimed quantities are treated.

General affine transformation, consisting of a shift of origin from O to O′ by a shift vector p with components and and a change of basis from a, b to a′, b′. This implies a change in the coordinates of the point X from x, y to x′, y′. 
The general transformation (affine transformation) of the coordinate system consists of two parts, a linear part and a shift of origin. The matrix P of the linear part and the column matrix p, containing the components of the shift vector p, define the transformation uniquely. It is represented by the symbol (P, p).

For a pure origin shift, the basis vectors do not change their lengths or orientations. In this case, the transformation matrix P is the unit matrix I and the symbol of the pure shift becomes (I, p).
Also, the inverse matrices of P and p are needed. They are and The matrix q consists of the components of the negative shift vector q which refer to the coordinate system a′, b′, c′, i.e. Thus, the transformation (Q, q) is the inverse transformation of (P, p). Applying (Q, q) to the basis vectors a′, b′, c′ and the origin O′, the old basis vectors a, b, c with origin O are obtained.
For a twodimensional transformation of a′ and b′, some elements of Q are set as follows: and .
The quantities which transform in the same way as the basis vectors a, b, c are called covariant quantities and are written as row matrices. They are:
the Miller indices of a plane (or a set of planes), (hkl), in direct space and the coordinates of a point in reciprocal space, h, k, l.
Both are transformed by Usually, the Miller indices are made relative prime before and after the transformation.
The quantities which are covariant with respect to the basis vectors a, b, c are contravariant with respect to the basis vectors of reciprocal space.
The basis vectors of reciprocal space are written as a column matrix and their transformation is achieved by the matrix Q: The inverse transformation is obtained by the inverse matrix : These transformation rules apply also to the quantities covariant with respect to the basis vectors and contravariant with respect to a, b, c, which are written as column matrices. They are the indices of a direction in direct space, [uvw], which are transformed by In contrast to all quantities mentioned above, the components of a position vector r or the coordinates of a point X in direct space x, y, z depend also on the shift of the origin in direct space. The general (affine) transformation is given by
Example
If no shift of origin is applied, i.e. , the position vector r of point X is transformed by In this case, , i.e. the position vector is invariant, although the basis vectors and the components are transformed. For a pure shift of origin, i.e. , the transformed position vector r′ becomes Here the transformed vector r′ is no longer identical with r.
It is convenient to introduce the augmented matrix which is composed of the matrices Q and q in the following manner (cf. Chapter 8.1 ): with o the row matrix containing zeros. In this notation, the transformed coordinates x′, y′, z′ are obtained by The inverse of the augmented matrix is the augmented matrix which contains the matrices P and p, specifically, The advantage of the use of matrices is that a sequence of affine transformations corresponds to the product of the corresponding matrices. However, the order of the factors in the product must be observed. If is the product of n transformation matrices , the sequence of the corresponding inverse matrices is reversed in the product
The following items are also affected by a transformation:


Monoclinic centred lattice, projected along the unique axis. Origin for all cells is the same. (a) Unique axis b. Cell choice 1: Ccentred cell . Cell choice 2: Acentred cell . Cell choice 3: Icentred cell . (b) Unique axis c. Cell choice 1: Acentred cell . Cell choice 2: Bcentred cell . Cell choice 3: Icentred cell . (c) Unique axis a. Cell choice 1: Bcentred cell . Cell choice 2: Ccentred cell . Cell choice 3: Icentred cell . 

Bodycentred cell I with a, b, c and a corresponding primitive cell P with a′, b′, c′. Origin for both cells 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, b, c and a corresponding primitive cell P with a′, b′, c′. Origin for both cells 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 . Origin for all three cells is the same. (b) Bodycentred cell I with a, b, c and the Fcentred cells with and with . Origin for all three cells is the same. 

Unit cells in the rhombohedral lattice: same origin for all cells. The basis of the rhombohedral cell is labelled a, b, c. 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 . (a) Obverse setting of triple hexagonal cell in relation to the primitive rhombohedral cell a, b, c. (b) Reverse setting of triple hexagonal cell in relation to the primitive rhombohedral cell a, b, c.(c) Primitive rhombohedral cell (   lower edges), a, b, c in relation to the three triple hexagonal cells in obverse setting ; ;. Projection along c′. (d) Primitive rhombohedral cell (   lower edges), a, b, c 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 ; ; . 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 ; ; . 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. 5.1.3.6c) and the three centred monoclinic cells. (a) Ccentred cells with ; with ; and with . The unique monoclinic axes are and , respectively. Origin for all four cells is the same. (b) Acentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. Origin for all four cells is the same. 

Rhombohedral lattice with primitive rhombohedral cell a, b, c and the three centred monoclinic cells. (a) Ccentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. Origin for all four cells is the same. (b) Acentred cells with ; with ; and with . The unique monoclinic axes are , and , respectively. Origin for all four cells is the same. 