Reference ellipsoid

Definition contradicts the obvious sense of the word as well as the source - Itu (talk) 23:29, March 11 2014 ( CET )

A reference ellipsoid is an oblate ellipsoid at the poles, usually a spheroid, which serves as a reference system for the calculation of surveying networks or direct indication of geographical coordinates. It is intended as a mathematical figure of the earth, the surface of constant height (see geoid ) approach, in which the historical development of regional level measurement to a global adjustment of the gravity field was.

History

As a scientifically accepted Earth model for the Greek natural philosophy was already considered the globe. First doubts about the exact spherical shape appeared in the 17th century; 1680 Isaac Newton was able to prove theoretically in a dispute with Giovanni Domenico Cassini and the Paris Academy that the Earth's rotation would cause a flattening at the poles, not the equator. The empirical evidence of this, which still suggested the opposite in the national survey in France by La Hire and Jacques Cassini (1683-1718), until the mid- 18th century, succeeded by Pierre Bouguer and Alexis -Claude Clairaut, as the measurements of the expeditions to Peru and Lapland (1735-1741) were unambiguously evaluated. This first accurate measurement of a degree also led to the definition of the meter as a 10 - millionth part of the earth's quadrant, which, however, to 0.022% was " too short " due to unavoidable small measurement error.

In the 19th century, many mathematicians and surveyors began to deal with the determination of Ellipsoiddimensionen. The determined values ​​of the equatorial radius varied even between 6376.9 ( Delambre 1810) and 6378.3 km ( Clarke 1880), while the widely accepted Bessel ellipsoid 6377.397 km gave ( the modern reference value is 6378.137 km ). The fact that the differences in the former measurement accuracy exceeded by five times, is due to the location of each measurement networks on different curved regions of the earth's surface (see deflection of the vertical ).

The values ​​of the Earth flattening varied less contrast - between 1:294 and 1:308, which ± 0,5 km in the polar axis means. Here Bessels value ( 1:299,15 ) was the best by far. Because of an increasing surveying networks " commuted " the result in the 20th century to about 1:298,3 a ( FRHelmert 1906 F.Krassowski 1940), which corresponds to 21.4 km difference between the equator and the polar axis, while the Hayford - ellipsoid with 1:297,0 fell significantly by the type of geophysical reduction of the series. Due to the large U.S. influence after the Second World War it was still chosen as the basis of the ED50 reference system while the " Eastern bloc " the Krassowski values ​​took the norm. The latter were confirmed in the 1970s by the satellite - world network and global multilateration ( runtime measurements on signals from quasars and geodetic satellites) than the better.

Reference ellipsoid in practice

Reference ellipsoids are used by surveyors for calculations on the Earth's surface and are relevant to other geosciences the most common reference system. Each regional administration and land surveying of a State requires such a reference ellipsoid to

• To create a national geodetic network (network expansion )
• Produce accurate maps and clearly define the state borders,
• Be able to calculate the position and shape of all land and buildings
• ( land register, etc. ) to guarantee and with a few thousand so-called fixed points of the geodetic network (triangulation, etc.) the boundary points and other rights.
• Since around 1985 it will " register" by digital information systems ( Geographic Information System, Land Information System, Environmental Information System, etc. ) which have also relied on the reference ellipsoid of the country.

Reference ellipsoid, in theory

Since the physical figure of the earth, the geoid, which by the irregularities of the earth's surface and gravity field of light waves, calculations are on a geometrically defined figure of the earth much easier. The objects to be measured are projected perpendicular to the ellipsoid and can be considered as a small scale even in a plane. For example, a Gauss -Krüger coordinate system is used.

With the amount of distance is given to the ellipsoid, perpendicular to its surface. This perpendicular line, however, differs by the so-called vertical deflection from the real plumb line as it would be a plumb bob. In measurements that are more accurate than some decimeters per kilometer, this effect has to be calculated and the measurements can be reduced by it. The deflections of the vertical can be in Central Europe depending on the terrain and geology 10-50 " and causes a difference between astronomical and ellipsoidal latitude and longitude (or ).

See also: Geodesics on an ellipsoid of

Conversion to geocentric Cartesian coordinates

In a geocentric rectangular reference system whose origin lies at the center of the ellipsoid and is aligned in the direction of the axis of rotation ( ) and the zero meridian ( ), then

With

Calculation of, and from Cartesian coordinates

The ellipsoidal length can be determined exactly as

Given the amount of which is h as

Although this relationship is exact, there is the formula

More suitable for practical calculations on because the error only depends on the square of the error in. The result is therefore more accurate to several orders of magnitude.

Be used for the calculation of needs on approximation methods. Because of the rotational symmetry of the problem is transferred to the XZ plane (). For the general case then X is replaced by.

The solder of the desired point on the ellipse is the rise. The elongated solder passes through the center M of the circle of curvature, which touches the ellipse in the nadir point. Are the coordinates of the center point

With

This is true

This is an iteration solution, as and t are into relationship. An obvious initial value would be

With this choice after one iteration to reach an accuracy of. That on the earth's surface results in a maximum error of 0.00000003 " arc seconds and the global maximum of the error ( in ) is 0.0018 ".

With favorable choice of and the maximum errors for points can be further reduced in the space. with

Is by a single insertion into the iteration formula of the angle ( for the parameters of the earth) to 0.0000001 " arc seconds exactly determined ( regardless of the value of h ).

Important reference ellipsoid

The shape and size of the ellipsoids used in different regions are generally characterized by their semi-major axis and the flattening (English flattening ) set. Furthermore, even those " fundamental point " centrally located to define on which the reference ellipsoid touches the geoid and thus gives it an unambiguous altitude. Both definitions together are called " geodetic datum ".

Even if two countries use the same ellipsoid (eg Germany and Austria the Bessel ellipsoid), they differ in this central point or fundamental point. Therefore, the coordinates of the common boundary points can differ by up to one kilometer.

The axes of the ellipsoids are depending on the region from which measurements were determined differ by up to 0.01 %. The increase in accuracy in determining the flattening ( difference of the ellipsoid axes approximately 21 km) is related to the launch of the first artificial satellite. This showed very significant perturbations with respect to the tracks that had been predicted. Based on the error could be calculated back and determine the flattening detail.

The table shows regional ellipsoids 1810-1906 and certain global earth ellipsoids 1924-1984 and the development of the knowledge of the mean equatorial radius and the Earth flattening.

The Bessel ellipsoid is ideally suited for Eurasia, so his 800 -meter " error " for geodesy Europe is low - much like the opposite 200 m of the Hayford ellipsoid (after John Fillmore Hayford ) for America.

For many Central European countries, the Bessel ellipsoid is important, also, the ellipsoids of Hayford and Krassowski ( spelling differently), and GPS survey the WGS84.

The results of Delambre and Schmidt are pioneering work and are based on limited measurements. On the other hand creates the big difference between Everest ( Asia) and Hayford ( America) by the geologically -related geoid curvature different continents. Part of this effect could eliminate Hayford by mathematical reduction of isostasy, so that one whose values ​​at that time for better kept than the European reference values ​​.