Matrix–column pairs and (n + 1) × (n + 1) matrices
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2.4,
p.

[ doi:10.1107/97809553602060000791 ]
and are designated by open-face letters in this volume:
In order to write equation (

**1.2.2.3**) as with the augmented matrices, the columns and x also have to be extended to the augmented columns and . Equations (

**1.2.2.5** ...

Coordinate systems and coordinates
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2.2,
p.

[ doi:10.1107/97809553602060000791 ]
vectors.
Definition

**1.2.2.2.1**. A basis which consists of lattice vectors of a crystal pattern is called a lattice basis or a crystallographic basis .
Referred to a lattice basis, each lattice vector ...

The description of mappings
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2.3,
p.

[ doi:10.1107/97809553602060000791 ]
or matrix part, the column w is the translation part or column part of a mapping.
In equations (

**1.2.2.1**) and (

**1.2.2.3**), the coordinates are mixed with the quantities describing the mapping, designated ...

Isometries
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2.5,
p.

[ doi:10.1107/97809553602060000791 ]
References
Hahn, Th. &

**Wondratschek**,

**H**. (1994). Symmetry of Crystals. Introduction to International Tables for Crystallography, Vol. A . Sofia: Heron Press. Google Scholar
International Tables for Crystallography (2005). Vol ...

Vectors and vector coefficients
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2.6,
p.

[ doi:10.1107/97809553602060000791 ]
. Thus, the column of the coefficients of a vector is not augmented by `1' but by `0'. Therefore, when the point P is mapped onto the point by according to equation (

**1.2.2.3**), then the vector is mapped onto the vector by transforming ...

Origin shift and change of the basis
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2.7,
p.

[ doi:10.1107/97809553602060000791 ]
according to equations (

**1.2.2.1**) to (

**1.2.2.3**), and the column of vector coefficients is v, see Section

**1.2.2.6** . A new coordinate system may be introduced with the basis and the origin . Referred to the new coordinate system ...

Mappings and matrices
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2,
p.

[ doi:10.1107/97809553602060000791 ]
:
In order to write equation (

**1.2.2.3**) as with the augmented matrices, the columns and x also have to be extended to the augmented columns and . Equations (

**1.2.2.5**) and (

**1.2.2.6**) then become
The vertical ...

Crystallographic symmetry operations
Wondratschek, H.,

International Tables for Crystallography
(2011).
Vol. A1,
Section 1.2.2.1,
p.

[ doi:10.1107/97809553602060000791 ]
Definition

**1.2.2.1.1**. A mapping is called a motion, a rigid motion or an isometry if it leaves all distances invariant (and thus all angles, as well as the size and shape of an object). In this volume the term `isometry' is used ...