Coordinate system
A coordinate system (Mathematical symbol: KOS ) is used to uniquely identify the location of points and objects in a geometric space.
Use
Coordinate systems are tools of mathematics to describe positions in space. They are used in many sciences and in technology. In everyday coordinate systems are frequently used:
- Longitude and latitude form a geographic coordinate system of the earth.
- In game notations as in Battleship or the chessboard and in spreadsheets fields are denoted by coordinates as B3.
- Hiking maps and city plans are usually divided into coordinate squares. In the squares of the city of Mannheim downtown streets form the coordinate system.
- The location of fire hydrants is described by an outgoing sign of the coordinate system.
Mannheimer Street Sign
Chessboard
Map
Coordinate
A coordinate is one of a plurality of numbers, which is indicative of the position of a point in a plane or in a room. Each of the required dimensions for the description is expressed by a coordinate. If a location is described by two coordinates, for example, on the map, it is called a " coordinate pair ". The technical term for the coordinate, in the meaning " position statement " in the 18th century from the word ordinate (vertical axis ) was formed.
Coordinate origin, Pol
The coordinate origin (Mathematical symbol: KOU ) denotes the point in a coordinate system or a map on which all coordinates assume the value zero. It is also called zero point, or with polar coordinates "pole".
Through the origin often, but not necessarily run the coordinate axes, see also axis label. Geographic coordinate systems have no coordinate axis.
Mathematical Foundations
The position of a point in space is uniquely determined in the chosen coordinate system by specifying numerical values or variable values , the coordinates. Accordingly, the position of a certain number of points by object (line, curve, surface, body ) indicate on their coordinates.
The number of values required to describe the dimension of the space. In this sense, is a plane as a two-dimensional space. We group all the coordinate of an n- dimensional space then also as an n- tuple of real values ( generally, elements of the underlying body ) on.
The coordinate systems most commonly used - this is especially true for school mathematics - are the Cartesian coordinate system, the general affine coordinate system and the polar coordinate systems.
In projective spaces, a point by its coordinates in the reference is displayed on a projective coordinate system. These coordinates are also referred to as a homogeneous coordinate and used in this form for "ordinary" point, which may also be described in Cartesian coordinates or affine. Here, an additional " homogenizing " coordinate required, a point in an n- dimensional space is thus defined by a homogeneous coordinate.
Different Coordinate Systems
The position of a point in space can be represented in different coordinate systems. The position is expressed by coordinates. Depending on the coordinate system used, the same point has different coordinate values.
Sphere and plane in space
For symmetrical systems in which one dimension is the same everywhere, can be achieved by displaying in a suitable coordinate system, that individual coordinates will remain constant. For example, it is enough to establish a position on the earth's surface the specification of only two coordinates (longitude and latitude), as the third coordinate is determined by the radius of the earth. If, however, in addition, the height of a point to be described, it must be recognized as a third coordinate, in addition. For additional height reference surface is needed.
Round body, such as the Earth or other celestial bodies are described by spherical polar coordinates ( spherical coordinates ). ( Special feature: coordinate singularity )
A plane in space is described by Cartesian coordinates: two coordinates are variable, the third is ( without loss of generality ) determined by the distance of the plane from the origin.
Straight, crooked and orthogonal
We distinguish between linear ( affine ) and curvilinear coordinate systems. In addition, if coordinates are lines perpendicular at every point to each other, such is called orthogonal coordinate systems.
Transformations between coordinate systems
The transformation between different coordinate systems is done by coordinate transformation. The different numerical values of the n-tuple describes the same position in space. In the transition from linear ( affine ) coordinates to curvilinear coordinates is to calculate quantities such as volume apply the functional determinant ( Jacobian ).
Special coordinate systems
The surrounding us space is often modeled in mathematics and physics as a three-dimensional Euclidean space. If true, the Newtonian law of inertia of classical physics for this space, it is called an inertial frame.
Often a spatial dimension can be neglected, so that only a two dimensional space is considered. Including the time the four-dimensional Minkowski space theory of relativity arises.
These spaces can be described by Cartesian coordinates, which are affine ( linear ) coordinates, which are measured along mutually perpendicular axes.
In the description in polar coordinates, the distance from a specified origin and angle are used as the coordinate axes at given. Again, the coordinate axes are orthogonal.
Other coordinate systems are defined in terms of geometric objects ( cylinder, cone section): cylindrical coordinates, hyperbolic coordinates.
Some only in specialist areas (eg geodesy, cartography, geography, remote sensing, astronomy) common coordinate systems are:
- Geodetic coordinate system
- Geographic Coordinate System
- Gauss -Krüger coordinate system
- UTM coordinate system UTM reference system also MGRS
Mathematical considerations
In a ( finite-dimensional ) vector space, a coordinate system is automatically given by a base. The coefficients of the basis vectors can be seen as coordinates. A transformation between two base systems corresponds to a transformation between the corresponding coordinate systems.
When a transformation of a base is a linear map to another, which can be represented approximately by a matrix, the corresponding transformations of the coordinate systems are linear.
Right and left-handed coordinate systems
A coordinate system is distinguished not only by the standard, that is, the length " 1", the degree or Krummlinigkeit of the principal axes, that is, the coordinate axes and the angles between the coordinate axis, but also by the orientation and the direction of rotation of the coordinate system. Both properties together describe the relationship of the coordinate axes for rotational transformation of an axis to the other.
One differentiates between right - and left-handed coordinate systems, with right-handed coordinate systems have agreed a mathematically positive sense. To check whether a three-dimensional Cartesian coordinate system complies with this standard right-handed axis orientation, using the so-called three - finger rule of the right hand.