Cartesian coordinate system

A Cartesian coordinate system is an orthogonal coordinate system . It is named after the Latinized name Cartesius of the French mathematician René Descartes , who made the concept of "Cartesian coordinates" known. In two- and three-dimensional space, it is the most frequently used coordinate system, as many geometrical facts can be described clearly and clearly in this.

Axis orientation and direction of rotation of geodetic and mathematical coordinate systems

The coordinate system in two-dimensional space

As a rule: right-handed Cartesian coordinate systems

Flat (2-dimensional) Cartesian coordinate system with 2 points and their coordinates and

{\ displaystyle P (5 | 3)}

{\ displaystyle Q (-4 | 2)}

The two directional axes are orthogonal to one another, i.e. they intersect at a 90 ° angle. The coordinate lines are straight lines at a constant distance from one another. Assuming mathematical right-handedness , the horizontal axis is called the abscissa axis (from Latin linea abscissa "cut line ") or the right axis. The vertical axis is called the ordinate axis (from Latin linea ordinata, "ordered line") or vertical axis.

The variables and are often used in mathematics to denote the coordinates, for example when straight lines or curves are described by equations. One then speaks of the -axis instead of the abscissa axis and the -axis instead of the ordinate axis. The or value of a point is called the abscissa or ordinate. Sometimes the coordinate axes are also called the abscissa or ordinate for short (see also -axis and -axis or dependent and independent variables ). ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle y}$ ${\ displaystyle x}$

As a donkey bridge , you can remember that the terms in front and in the back in the alphabet always belong together: the abscissa and ordinate. Another mnemonic: The O rdinatenachse shows (for positive values) to o ben - the x-axis must also (with positive show values) to the right. ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle y}$ ${\ displaystyle x}$

The point where the two axes meet is called the coordinate origin or origo (Latin for "origin"). ${\ displaystyle O (0 \ mid 0)}$

For a point with the coordinates and one writes or also . ${\ displaystyle P}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle P (x | y)}$ ${\ displaystyle P = (x, y)}$

Left-handed Cartesian coordinate systems

In geodesy , the XY coordinate axes are swapped, and geodetic coordinate systems are sometimes limited to the first quadrant in order to avoid negative values. For this purpose, the zero point of the coordinate system is fictitiously shifted to the southwest, outside of the mapping area, using addition constants, so that only positive coordinate values occur (example: Soldner-Berlin coordinate system). Left-handed coordinate systems can also be found in areas such as economics, where, for example, the dependent size of the supply , price-sales or demand function is usually not plotted on the vertical but transverse axis, while the independent one is instead on the vertical axis.

Left-handed coordinate systems are also commonly used in computer graphics . Most 2D systems use the upper left corner as (0,0).

Coordinate systems with more than two dimensions

In three-dimensional space there is also a third axis, the spatial axis ( axis, not shown here), called Applikate (in geography : Kote ). Most of the time the - and -axis lie in the plane, and the -axis is used to display the height. The eight parts of space formed by the coordinate planes are called octants . Graphically, points here result in a point cloud . ${\ displaystyle z}$ ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$

As in the two-dimensional case, in three-dimensional geodetic coordinate systems - and - axis are swapped, while the - axis points upwards, as in the mathematical coordinate system. ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle z}$

In the generalization, mathematics provides for higher-dimensional spaces (see: 4D ). For example, the axis for the expansion in the fourth spatial dimension is sometimes referred to as the -axis, the directions of expansion as ana ("above") and kata ("below"). ${\ displaystyle w}$

Exemplary applications

In physics, the right axis is often used to represent time as an independent variable; of it is then spoken of as the time or axis , while the vertical axis represents the time-variable variable, e.g. B. the distance covered or the speed , represented and accordingly referred to as - or - axis. ${\ displaystyle t}$ ${\ displaystyle t}$ ${\ displaystyle s}$ ${\ displaystyle v}$ ${\ displaystyle s}$ ${\ displaystyle v}$

Three-dimensional coordinate systems allow, for example, the representation of two-dimensional statistical distributions in which the height axis indicates the probability or density function.

One of the most frequent applications of 3-axis coordinate systems is likely to be the application for spatial detection and description, e.g. B. in construction, surveying and navigation. In navigation , for example, they are used to localize an object using GPS . This is followed by the description of the spatial orientation of objects using angles, which is often implemented with the help of roll, pitch and yaw angles ( RPY angles ). Earth-related coordinate systems are often considered to be approximately Cartesian, since they are actually spherical coordinate systems. In most cases, the use delivers very useful values for the application with relatively short distances - the deviations from the approximation are typically several orders of magnitude smaller than the measurement accuracy required for the application or achievable in practice based on other factors.

history

In Definition 4 of the Konika, Apollonios writes of parallels that are "drawn in an orderly manner" to the diameter of a conic section. The Greek term for “ordered”, tetagmenos , is rendered in Latin as ordinatim . That is the origin of the word ordinate .

The first known use of the words abscissa and ordinate can be found in a letter from Gottfried Wilhelm Leibniz to Henry Oldenburg on August 27, 1676.

Synthetic geometry

The term Cartesian coordinate system is generalized for planes in synthetic geometry : There, an affine coordinate system is called Cartesian if the unit points are adjacent corners in a square with a center . ${\ displaystyle (O, E_ {1}, E_ {2})}$ ${\ displaystyle E_ {1}, E_ {2}}$ ${\ displaystyle O}$

Web links

Explanation with images on mathe-online.at
Eric W. Weisstein : Cartesian Coordinates . In: MathWorld (English).
A. B. Ivanov: Cartesian orthogonal coordinate system . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Cartesian coordinates . In: PlanetMath . (English)

Individual evidence

^ Duden, the large foreign dictionary, Mannheim & Leipzig, 2000, ISBN 3-411-04162-5 .
^ Helmuth Gericke : Mathematics in Antiquity, Orient and Occident. Marix Verlag, Wiesbaden 2005, ISBN 3-937715-71-1 , p. 132.
↑ Christoph J. Scriba, Peter Schreiber: 5000 years of geometry. 2nd Edition. Springer, 2005, ISBN 3-540-22471-8 , p. 331.

[duden-1] Duden, the large foreign dictionary, Mannheim & Leipzig, 2000, ISBN 3-411-04162-5 .

[2] Helmuth Gericke : Mathematics in Antiquity, Orient and Occident. Marix Verlag, Wiesbaden 2005, ISBN 3-937715-71-1 , p. 132.

[3] Christoph J. Scriba, Peter Schreiber: 5000 years of geometry. 2nd Edition. Springer, 2005, ISBN 3-540-22471-8 , p. 331.