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

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.
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:


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. 