Main geodetic task

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The main geodetic tasks in geodesy are two important types of coordinate transformation , namely those from right-angled to polar coordinates and vice versa.

First and second main task

The 1st main task (polar ⇒ rectangular) corresponds to the transfer of measurements ( direction and distance ) to the plane coordinate system of a plan or map .

In addition to the plane coordinates of a map (x, y), the geographic or geodetic coordinates on the earth's ellipsoid are also to be understood as right-angled coordinates , because these coordinate lines ( latitude and longitude ) intersect at right angles on the earth's surface .

The 2nd main task (right-angled ⇒ polar coordinates) corresponds to e.g. B. the calculation of the direction and distance between two survey points .

The position of surveying or other fixed points are usually given as Gauß-Krüger or UTM coordinates in meters ( x, y or Northern, Eastern ).

Point distances and accuracy

If the mutual distances D of two points - or their coordinate differences dx, dy - are not greater than about 5 km, then they can be converted directly into the other type of coordinate:

with the distance and the direction angle (azimuth) .

Up to distances of a few kilometers, this is approximately centimeters accurate. For longer distances, the projection distortion must be taken into account, and from around 20 km you have to switch to more complicated formulas:

Depending on the complexity of the computing surface used (plane, sphere, reference ellipsoid , geoid ), the mathematical formulation of the main tasks must therefore be designed differently. In the level - such as those for the measurement of land and large-scale maps is enough - it is limited to level trigonometric functions , as in the above formula example.

The analog task on the globe already requires a few lines of formulas from spherical trigonometry , while the exact solution of the task on the earth ellipsoid even requires a formula apparatus from about 1 page. In geodesy , geophysics or long-range navigation , such calculations are unavoidable on twice curved surfaces . The same task on the geoid or on more complexly shaped celestial bodies such as Mars or some minor planets can even only be solved iteratively . Several mathematicians active in geodesy in the last few centuries (for example Gauß , Bessel , Legendre , Laplace , Hilbert ) or more recently Grafarend and others have developed appropriate solutions for this.

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