Points and their coordinates

A mathematical model of the space in which we live is the *point space*.
Its elements are points.
Objects in point space may be single points; finite sets of
points, *e.g.* the centers of the atoms of a molecule; infinite
discontinuous point sets, *e.g.* the centers of the atoms of an ideal
(infinitely extended and periodic) crystal pattern; continuous point sets
like straight lines, curves, planes, curved surfaces, to mention just a few
which play a role in crystallography.

In the following we restrict our considerations to the 3-dimensional space.
The transfer to the plane should be obvious. One can even extend the whole
concept to *n*-dimensional space with arbitrary dimension *n*.

In order to describe the objects in point space analytically, one
introduces a *coordinate system*. To achieve
this, one selects some point as the *origin* *O*. Then one
chooses three straight lines running through the origin and not lying in a
plane. They
are called the *coordinate axes* , , and
or , , and . On each of these lines a point different
from *O* is chosen marking the unit on that axis: on , on
, and on . An arbitrary point *P* is then described by its
coordinates , , or , , , see Fig.1.1.1:

**Definition** (D 1.1.3) The *parallel coordinates*
*x*, *y*, and *z* or , ,
and of an arbitrary point *P* are defined in the following way:

- The origin
*O*is the point with the coordinates 0,0,0. - One constructs the 3 planes through the point which are parallel to the pairs of axes and , and , and and , respectively. These 3 planes intersect the coordinate axes , , and in the points , , and , respectively.
- The fractions of the lengths
on the axis ,

on , and on are the coordinates of the point*P*.

In this way one assigns uniquely to each point a triplet of coordinates
and *vice versa*. In crystallography the coordinates are written
usually in
a column which is designated by a boldface-italics lower-case letter,
*e.g.*,

**Definition** (D 1.1.3) The set of all columns of three real numbers represents
all points of the point space and is called the *affine space*.

The affine space is not yet a good model for our physical space. In reality
one can measure distances and angles which is possible in the affine space
only after the introduction of a *scalar product*, see Sections
1.5 and 1.6. Such a space with a scalar product is the
fundament of the following considerations.

**Definition** (D 1.1.3) An affine space, for which a scalar product is defined,
is called a *Euclidean space*.

The coordinates of a point *P* depend on the position of *P* in
space as well as on the coordinate system. The coordinates of a fixed
point *P* are changed by another choice of the coordinate axes but
also by another choice of
the origin. Therefore, the comparison of two points by their columns of
coordinates is only possible if the coordinate system is the same to which
these points are referred. Two points are equal if and only if their columns
of coordinates agree in all coordinates when referred to the same coordinate
system. If points are referred to different coordinate systems and if the
relations between these coordinate systems are known then one can recalculate
the coordinates of the points by a *coordinate transformation*
in order to refer them to the *same* coordinate system, see Subsection
5.3.3. Only after this transformation a comparison
of the coordinates is meaningful.

**Copyright © 2002 International Union of
Crystallography**