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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 $OA=OB=OC=1$ 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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