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Special coordinate systems: Cartesian coordinates

There are different types of coordinate systems. Coordinate systems with
straight lines as axes as introduced in Section 1.1 are called
*parallel coordinates*.
In physics polar coordinates in the plane and
cylindrical or spherical coordinates in the space are used frequently
depending on the kind of problems.

In general those coordinates are chosen in which the
solution of the given problem is expected to cause the least difficulties. We
shall consider mainly parallel coordinates. Such coordinate
systems are of utmost importance for crystallography due to the
periodicity of the crystals. In this section a special system with parallel
coordinates will be defined which is used frequently in physics, also in
crystal physics, and in mathematics. It is applied in Section 1.6.
In crystallography, mostly special crystallographic coordinate systems are used.

**Definition** (D 1.2.1) A coordinate system with 3 coordinate axes
perpendicular to each other and lengths is called a
*Cartesian coordinate system*.

Referring the points to a Cartesian coordinate system simplifies many formulae,
*e.g.* for the determination of distances between points and of angles
between lines, and thus makes such calculations particularly easy, cf.
Sections 1.6 and 2.6. On the other hand, the
description of the symmetry of crystals, in particular of the translational
symmetry (also in reciprocal space) becomes quite involved when using
Cartesian coordinates. With the exception of crystal physics, the
disadvantages of Cartesian coordinates outweigh their advantages when dealing
with crystallographic problems.

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