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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 $a$, $b$, and $c$ or $a_1$, $a_2$, and $a_3$. On each of these lines a point different from O is chosen marking the unit on that axis: $A$ on $a$, $B$ on $b$, and $C$ on $c$. An arbitrary point P is then described by its coordinates $x$, $y$, $z$ or $x_1$, $x_2$, $x_3$, see Fig.1.1.1:

Fig. 1.1.1 Point $P$ in a coordinate system {$O, a, b, c$}. The end points $A$, $B$, and $C$ of the arrows determine the different unit lengths on the lines $a$, $b$, and $c$, respectively. The coordinate points are $X_{\circ},\ Y_{\circ}$, and $Z_{\circ}$; the coordinates of $P$ are

$x=(OX_{\circ})/(OA)$,
$y=(OY_{\circ})/(OB)$, and
$z=(OZ_{\circ})/(OC).$

Definition (D 1.1.3) The parallel coordinates x, y, and z or $x_1$, $x_2$, and $x_3$ of an arbitrary point P are defined in the following way:

1. The origin O is the point with the coordinates 0,0,0.
2. One constructs the 3 planes through the point $P$ which are parallel to the pairs of axes $b$ and $c$, $c$ and $a$, and $a$ and $b$, respectively. These 3 planes intersect the coordinate axes $a$, $b$, and $c$ in the points $X_{\circ}$, $Y_{\circ}$, and $Z_{\circ}$, respectively.
3. The fractions of the lengths $(OX_{\circ})/(OA)=x$ on the axis $a$,
   $(OY_{\circ})/(OB)=y$ on $b$, and $(OZ_{\circ})/(OC)=z$ on $c$ 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.,

$$\mathit{P}:\rule{2em}{0ex} \mbox{\textit{\textbf{x}}}=\left( ... ...= \left( \begin{array}{c} x_1\\ x_2\\ x_3 \end{array} \right).$$

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