Cartesian coordinate system (geodesy)
In Geodesy are left-handed Cartesian coordinate systems used. The x-axis ( abscissa ) is regarded as the main axis, the y-axis (ordinate) is obtained by rotating the x-axis by 100 gon (90 °) clockwise around the coordinate origin. The "geodetically positive" direction of rotation is clockwise and not counterclockwise like the "mathematically positive" direction.
Compared to right-handed Cartesian coordinate systems in mathematics, the x- and y-axes are swapped: the x-axis usually points upwards in maps and plans, the y-axis to the right. For national coordinates , the x-axis points north and the y-axis points east.
The height as the third coordinate (also called applicate ) was determined and verified separately from the position coordinates for a long time, if at all. Because of this separation of position and height, there was no need for three-dimensional calculations. However, to the extent that three-dimensional spatial references gain in importance in geodesy, for example through satellite positioning , the importance of three-dimensional coordinate systems also increases.
With local coordinate systems, i. H. For coordinate systems that are not (for the time being) connected to a nationwide reference system , the x-axis and the zero point are appropriately chosen. For example, it can be the main axis of a structure or a polygon side and does not have to point north. The y-axis points to the right from this axis.
To avoid negative coordinates, positive values can be added to the coordinates, which shifts the origin of the coordinates. In the case of measurement lines in the orthogonal method, positive ordinates mean that a point is to the right of the measurement line, points with negative ordinates are to the left.
Origin of coordinates
For the longitude of the fundamental point or the central meridian of a transversal Mercator projection , instead of a coordinate value 0 - depending on the extent of the area to be mapped and other practical considerations - an arbitrary value is set ( "false easting" , see Fig.). In this way, a positive “easting value” (y-value) is obtained for each point that can be represented.
Since the north-south direction (“high value”, x-value, “false northing” ) is used accordingly, there is usually a restriction to the first quadrant of the coordinate system: all quadrants are defined, but practically only the coordinates of the first quadrant used.
Easting value (y-value)
The right value , also denoted by y , is the distance of a point from the (here vertically running) abscissa or x-axis in planar Cartesian coordinate systems related to the earth's surface. The Easting there is distance to the nearest central meridian and thus corresponds to the English "easting".
For better handling in practice, negative legal values (for areas west of the abscissa or the reference meridian) are avoided by arbitrarily setting a defined legal value instead of zero (i.e. the central meridian) (referred to as "false easting" in English-speaking countries, see above).
For example, the origin of the coordinates of the Swiss national survey was shifted 600 km to the west in the Bordeaux area in order to avoid confusion between right and north values : Coordinate values below 400,000 m must be high values, values above are always right values. In this way, there is no need to define a sequence of the coordinate components and interchanges can be recognized on the basis of the values.
Example in the Gauß-Krüger system (with 500 km false easting): R 4541238. R makes it clear that it is the easting value. The first number (in this case 4) represents the code number for the respective longitude, in this case (4 · 3 = 12) for the central meridian 12 ° E. The remaining numbers now indicate in meters how far the point is from the central meridian after subtracting the 500 km: 541238-500000 = 41238. The line we are looking for (a point is only obtained with a corresponding high value) is 41.238 km east of longitude 12 ° E.
A false-easting of 500 km for the central meridian as in the UTM coordinate system ensures that the entire valid range of the easting value (6 digits) is between 100,000 and 900,000.
The Finnish YKJ system shifts the origin of coordinates by 3,500 km to the west in order to always receive 7-digit coordinates horizontally and vertically.
High value (x-value)
The high value , also denoted by x , is the distance between a point measured in the north direction and its base point on the horizontally running baseline of the coordinate system (here the y-axis). The term high value corresponds to the English northing . By correspondingly selected positions of the y-axis (e.g. for Europe on the equator ), positive high values can also always be achieved.
Right and north values form the two-dimensional coordinates of a point.
Due to the spherical shape of the earth, Cartesian coordinate systems can only map areas of limited extent practically without distortions ( true to length ). Historically, they were based on a local fundamental point of trigonometric land surveys and were stretched along a normal perpendicular (zero length, central meridian), which typically, but not necessarily, corresponded to the meridian of the fundamental point. Modern coordinate systems use longitudes as reference or central meridians, the degree values of which are divisible by 3.
Practical applications of Cartesian coordinate systems in geodesy are
- Soldner coordinate networks
- Gauss-Krüger coordinate systems
- the UTM coordinate system
- the Austrian federal registration network
- the Swiss national coordinates .
- Find coordinates and display them in different systems.