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48-551: The Stübenwasen ( 1,386 m above  sea level (NHN) ) is the sixth highest mountain in the Black Forest after the Feldberg ( 1,493 m above NHN ), Baldenweger Buck ( 1,461 m above NHN ), Seebuck ( 1,448 m above NHN ), Herzogenhorn ( 1,415 m above NHN ) and the Belchen ( 1,414 m above NHN ). It is

96-644: A geocentric coordinate frame. One such frame is WGS84 , as well as frames by the International Earth Rotation and Reference Systems Service ( IERS ). GNSS receivers have almost completely replaced terrestrial instruments for large-scale base network surveys. To monitor the Earth's rotation irregularities and plate tectonic motions and for planet-wide geodetic surveys, methods of very-long-baseline interferometry (VLBI) measuring distances to quasars , lunar laser ranging (LLR) measuring distances to prisms on

144-406: A "reference frame" for the same. The ISO term for a datum transformation again is a "coordinate transformation". General geopositioning , or simply positioning, is the determination of the location of points on Earth, by myriad techniques. Geodetic positioning employs geodetic methods to determine a set of precise geodetic coordinates of a point on land, at sea, or in space. It may be done within

192-566: A coordinate system ( point positioning or absolute positioning ) or relative to another point ( relative positioning ). One computes the position of a point in space from measurements linking terrestrial or extraterrestrial points of known location ("known points") with terrestrial ones of unknown location ("unknown points"). The computation may involve transformations between or among astronomical and terrestrial coordinate systems. Known points used in point positioning can be GNSS continuously operating reference stations or triangulation points of

240-420: A country, usually documented by national mapping agencies. Surveyors involved in real estate and insurance will use these to tie their local measurements. In geometrical geodesy, there are two main problems: The solutions to both problems in plane geometry reduce to simple trigonometry and are valid for small areas on Earth's surface; on a sphere, solutions become significantly more complex as, for example, in

288-418: A higher-order network. Traditionally, geodesists built a hierarchy of networks to allow point positioning within a country. The highest in this hierarchy were triangulation networks, densified into the networks of traverses ( polygons ) into which local mapping and surveying measurements, usually collected using a measuring tape, a corner prism , and the red-and-white poles, are tied. Commonly used nowadays

336-416: A large extent, Earth's shape is the result of rotation , which causes its equatorial bulge , and the competition of geological processes such as the collision of plates , as well as of volcanism , resisted by Earth's gravitational field. This applies to the solid surface, the liquid surface ( dynamic sea surface topography ), and Earth's atmosphere . For this reason, the study of Earth's gravitational field

384-509: A physical (real-world) realization of a coordinate system used for describing point locations. This realization follows from choosing (therefore conventional) coordinate values for one or more datum points. In the case of height data, it suffices to choose one datum point — the reference benchmark, typically a tide gauge at the shore. Thus we have vertical datums, such as the NAVD 88 (North American Vertical Datum 1988), NAP ( Normaal Amsterdams Peil ),

432-482: A projection is UTM ( Universal Transverse Mercator ). Within the map plane, we have rectangular coordinates x and y . In this case, the north direction used for reference is the map north, not the local north. The difference between the two is called meridian convergence . It is easy enough to "translate" between polar and rectangular coordinates in the plane: let, as above, direction and distance be α and s respectively, then we have The reverse transformation

480-481: A series expansion — see, for example, Vincenty's formulae . As defined in geodesy (and also astronomy ), some basic observational concepts like angles and coordinates include (most commonly from the viewpoint of a local observer): The reference surface (level) used to determine height differences and height reference systems is known as mean sea level . The traditional spirit level directly produces such (for practical purposes most useful) heights above sea level ;

528-543: A single global, geocentric reference frame that serves as the "zero-order" (global) reference to which national measurements are attached. Real-time kinematic positioning (RTK GPS) is employed frequently in survey mapping. In that measurement technique, unknown points can get quickly tied into nearby terrestrial known points. One purpose of point positioning is the provision of known points for mapping measurements, also known as (horizontal and vertical) control. There can be thousands of those geodetically determined points in

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576-454: Is approximately the same as the direction of the plumbline, i.e., local gravity, which is also the normal to the geoid surface. For this reason, astronomical position determination – measuring the direction of the plumbline by astronomical means – works reasonably well when one also uses an ellipsoidal model of the figure of the Earth. One geographical mile, defined as one minute of arc on the equator, equals 1,855.32571922 m. One nautical mile

624-466: Is GPS, except for specialized measurements (e.g., in underground or high-precision engineering). The higher-order networks are measured with static GPS , using differential measurement to determine vectors between terrestrial points. These vectors then get adjusted in a traditional network fashion. A global polyhedron of permanently operating GPS stations under the auspices of the IERS is the basis for defining

672-517: Is a vertical datum used in Germany. In geographical terms, NHN is the reference plane for the normal height of a topographical eminence height above mean sea level used in the 1932 German Mean Height Reference System ( Deutsches Haupthöhennetz ). The plane is in the shape of a quasi- geoid . The reference height is a geodetic , fixed point on the New Church of St. Alexander at Wallenhorst in

720-750: Is accessible in summer on the footpath between Feldberg and Notschrei , and in winter on the Stübenwasenspur . In addition lifts run from Todtnauberg up the Stübenwasen, the Stübenwasen Lift (1,000 m long, from 1,100 m to 1,350 m ) and the Stübenwasen Summit Lift ( Gipfellift ) (400 m long, from 1,330 m to 1,370 m ). Normalh%C3%B6hennull Normalhöhennull ( German pronunciation: [nɔʁmaːlˈhøːənˌnʊl] , "standard elevation zero") or NHN

768-518: Is called physical geodesy . The geoid essentially is the figure of Earth abstracted from its topographical features. It is an idealized equilibrium surface of seawater , the mean sea level surface in the absence of currents and air pressure variations, and continued under the continental masses. Unlike a reference ellipsoid , the geoid is irregular and too complicated to serve as the computational surface for solving geometrical problems like point positioning. The geometrical separation between

816-728: Is called the height anomaly or quasi-geoid height. Since 1 January 2000 the whole of Germany has changed its height system over to normal heights based on the datum of the Amsterdam Ordnance Datum , known as the German Mean Height Reference System, DHHN92. At the same time, the new NHN is the basis of the United European Levelling Net (UELN), formerly known as the Reseau Européen Unifié de Nivellement or REUN , which standardises

864-416: Is described by (apparent) sidereal time , which accounts for variations in Earth's axial rotation ( length-of-day variations). A more accurate description also accounts for polar motion as a phenomenon closely monitored by geodesists. In geodetic applications like surveying and mapping , two general types of coordinate systems in the plane are in use: One can intuitively use rectangular coordinates in

912-399: Is given by: In geodesy, point or terrain heights are " above sea level " as an irregular, physically defined surface. Height systems in use are: Each system has its advantages and disadvantages. Both orthometric and normal heights are expressed in metres above sea level, whereas geopotential numbers are measures of potential energy (unit: m s ) and not metric. The reference surface is

960-578: Is off by 200 ppm in the current definitions). This situation means that one kilometre roughly equals (1/40,000) * 360 * 60 meridional minutes of arc, or 0.54 nautical miles. (This is not exactly so as the two units had been defined on different bases, so the international nautical mile is 1,852 m exactly, which corresponds to rounding the quotient from 1,000/0.54 m to four digits). Various techniques are used in geodesy to study temporally changing surfaces, bodies of mass, physical fields, and dynamical systems. Points on Earth's surface change their location due to

1008-457: Is one minute of astronomical latitude. The radius of curvature of the ellipsoid varies with latitude, being the longest at the pole and the shortest at the equator same as with the nautical mile. A metre was originally defined as the 10-millionth part of the length from the equator to the North Pole along the meridian through Paris (the target was not quite reached in actual implementation, as it

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1056-421: Is purely geometrical. The mechanical ellipticity of Earth (dynamical flattening, symbol J 2 ) can be determined to high precision by observation of satellite orbit perturbations . Its relationship with geometrical flattening is indirect and depends on the internal density distribution or, in simplest terms, the degree of central concentration of mass. The 1980 Geodetic Reference System ( GRS 80 ), adopted at

1104-446: The geoid , an equigeopotential surface approximating the mean sea level as described above. For normal heights, the reference surface is the so-called quasi-geoid , which has a few-metre separation from the geoid due to the density assumption in its continuation under the continental masses. One can relate these heights through the geoid undulation concept to ellipsoidal heights (also known as geodetic heights ), representing

1152-527: The geoids within their areas of validity, minimizing the deflections of the vertical over these areas. It is only because GPS satellites orbit about the geocenter that this point becomes naturally the origin of a coordinate system defined by satellite geodetic means, as the satellite positions in space themselves get computed within such a system. Geocentric coordinate systems used in geodesy can be divided naturally into two classes: The coordinate transformation between these two systems to good approximation

1200-702: The tachymeter determines, electronically or electro-optically , the distance to a target and is highly automated or even robotic in operations. Widely used for the same purpose is the method of free station position. Commonly for local detail surveys, tachymeters are employed, although the old-fashioned rectangular technique using an angle prism and steel tape is still an inexpensive alternative. As mentioned, also there are quick and relatively accurate real-time kinematic (RTK) GPS techniques. Data collected are tagged and recorded digitally for entry into Geographic Information System (GIS) databases. Geodetic GNSS (most commonly GPS ) receivers directly produce 3D coordinates in

1248-461: The topographic surface of Earth — is also realizable. The locations of points in 3D space most conveniently are described by three cartesian or rectangular coordinates, X , Y , and Z . Since the advent of satellite positioning, such coordinate systems are typically geocentric , with the Z-axis aligned to Earth's (conventional or instantaneous) rotation axis. Before the era of satellite geodesy ,

1296-533: The Amsterdam Datum). The elevations differed — depending on location — by 0.06 to 0.16 m. As a result of new measurements as part of the changeover, however, variations of 0.59 m ( Zugspitze ) have surfaced. Older relief maps often show heights above the old reference planes. Current maps by the federal survey authorities are based on NHN. At the beginning of 2013 most of the federal states (except Berlin, Thuringia and Saxony-Anhalt) had complete coverage by

1344-485: The GRS 80 reference ellipsoid. The geoid is a "realizable" surface, meaning it can be consistently located on Earth by suitable simple measurements from physical objects like a tide gauge . The geoid can, therefore, be considered a physical ("real") surface. The reference ellipsoid, however, has many possible instantiations and is not readily realizable, so it is an abstract surface. The third primary surface of geodetic interest —

1392-523: The German state of Lower Saxony . The geopotential height of this point was calculated in 1986 as part of the United European Levelling Network (UELN), based on the Amsterdam Ordnance Datum . The NHN plane is a theoretical reference plane. It is derived by deducting normal heights from the normal plumb line . The difference between the resulting quasi-geoid and the reference ellipsoid

1440-611: The Kronstadt datum, the Trieste datum, and numerous others. In both mathematics and geodesy, a coordinate system is a "coordinate system" per ISO terminology, whereas the International Earth Rotation and Reference Systems Service (IERS) uses the term "reference system" for the same. When coordinates are realized by choosing datum points and fixing a geodetic datum, ISO speaks of a "coordinate reference system", whereas IERS uses

1488-580: The Moon, and satellite laser ranging (SLR) measuring distances to prisms on artificial satellites , are employed. Gravity is measured using gravimeters , of which there are two kinds. First are absolute gravimeter s, based on measuring the acceleration of free fall (e.g., of a reflecting prism in a vacuum tube ). They are used to establish vertical geospatial control or in the field. Second, relative gravimeter s are spring-based and more common. They are used in gravity surveys over large areas — to establish

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1536-671: The XVII General Assembly of the International Union of Geodesy and Geophysics ( IUGG ), posited a 6,378,137 m semi-major axis and a 1:298.257 flattening. GRS 80 essentially constitutes the basis for geodetic positioning by the Global Positioning System (GPS) and is thus also in widespread use outside the geodetic community. Numerous systems used for mapping and charting are becoming obsolete as countries increasingly move to global, geocentric reference systems utilizing

1584-414: The coordinate systems associated with a geodetic datum attempted to be geocentric , but with the origin differing from the geocenter by hundreds of meters due to regional deviations in the direction of the plumbline (vertical). These regional geodetic datums, such as ED 50 (European Datum 1950) or NAD 27 (North American Datum 1927), have ellipsoids associated with them that are regional "best fits" to

1632-471: The figure of the geoid over these areas. The most accurate relative gravimeters are called superconducting gravimeter s, which are sensitive to one-thousandth of one-billionth of Earth-surface gravity. Twenty-some superconducting gravimeters are used worldwide in studying Earth's tides , rotation , interior, oceanic and atmospheric loading, as well as in verifying the Newtonian constant of gravitation . In

1680-401: The future, gravity and altitude might become measurable using the special-relativistic concept of time dilation as gauged by optical clocks . Geographical latitude and longitude are stated in the units degree, minute of arc, and second of arc. They are angles , not metric measures, and describe the direction of the local normal to the reference ellipsoid of revolution. This direction

1728-412: The geoid and a reference ellipsoid is called geoidal undulation , and it varies globally between ±110 m based on the GRS 80 ellipsoid. A reference ellipsoid, customarily chosen to be the same size (volume) as the geoid, is described by its semi-major axis (equatorial radius) a and flattening f . The quantity f = ⁠ a − b / a ⁠ , where b is the semi-minor axis (polar radius),

1776-399: The global scale, or engineering geodesy ( Ingenieurgeodäsie ) that includes surveying — measuring parts or regions of Earth. For the longest time, geodesy was the science of measuring and understanding Earth's geometric shape, orientation in space, and gravitational field; however, geodetic science and operations are applied to other astronomical bodies in our Solar System also. To

1824-430: The height of a point above the reference ellipsoid . Satellite positioning receivers typically provide ellipsoidal heights unless fitted with special conversion software based on a model of the geoid. Because coordinates and heights of geodetic points always get obtained within a system that itself was constructed based on real-world observations, geodesists introduced the concept of a "geodetic datum" (plural datums ):

1872-483: The height systems of the European countries. Heights in this system are given in meters above NHN or m (NHN) . The NHN was introduced because for heights above Normalnull the actual gravitational field of the Earth was not taken into account. As a result, there were changes in both the old West German normal orthometric heights (new methods of calculation) and the normal heights of East Germany (with respect to

1920-527: The highest point on the ridge between Schauinsland and Feldberg and is only separated from the latter by a wide saddle. To the north is the St. Wilhelm Valley , to the south the Wiesental with Todtnau and Todtnauberg . The summit of the Stübenwasen is not wooded. The sudden transitions to forest show, however, that this is not a natural treeline . The Stübenwasen would not be treeless just on account of its height;

1968-405: The inverse problem, the azimuths differ going between the two end points along the arc of the connecting great circle . The general solution is called the geodesic for the surface considered, and the differential equations for the geodesic are solvable numerically. On the ellipsoid of revolution, geodesics are expressible in terms of elliptic integrals, which are usually evaluated in terms of

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2016-496: The more economical use of GPS instruments for height determination requires precise knowledge of the figure of the geoid , as GPS only gives heights above the GRS80 reference ellipsoid. As geoid determination improves, one may expect that the use of GPS in height determination shall increase, too. The theodolite is an instrument used to measure horizontal and vertical (relative to the local vertical) angles to target points. In addition,

2064-532: The new digital topographic mapping at 1:25,000 scale (DTK). Not all the maps have appeared in print yet. On the DTK25 maps, NHN is used for elevations; however, on the DTK25-V scanned topographic maps Höhennull (HN) and Normalnull (NN) are still being used. In East Germany normal heights used to be referred to as heights above Höhennormal or HN . The 1958 Kronstadt Tide Gauge ( Kronstädter Pegel )

2112-507: The plane for one's current location, in which case the x -axis will point to the local north. More formally, such coordinates can be obtained from 3D coordinates using the artifice of a map projection . It is impossible to map the curved surface of Earth onto a flat map surface without deformation. The compromise most often chosen — called a conformal projection — preserves angles and length ratios so that small circles get mapped as small circles and small squares as squares. An example of such

2160-422: The sky to a traveler headed South. In English , geodesy refers to the science of measuring and representing geospatial information , while geomatics encompasses practical applications of geodesy on local and regional scales, including surveying . In German , geodesy can refer to either higher geodesy ( höhere Geodäsie or Erdmessung , literally "geomensuration") — concerned with measuring Earth on

2208-563: The treeline here, about 100 kilometres north of the nearest north Alpine peak could be expected to lie at about 1,650– 1,700 m . The highlands are used in summer as cattle pasture, in winter as a ski area. The Stübenwasen is thus well developed for tourism, although it is not accessible by car, like for example the Feldberg or the Kandel . West of the summit is an inn, the Berggasthaus Stübenwasen (1,270 m), built in 1935. It

2256-455: The very word geodesy comes from the Ancient Greek word γεωδαισία or geodaisia (literally, "division of Earth"). Early ideas about the figure of the Earth held the Earth to be flat and the heavens a physical dome spanning over it. Two early arguments for a spherical Earth were that lunar eclipses appear to an observer as circular shadows and that Polaris appears lower and lower in

2304-578: Was used as the datum. The new NHN heights are typically 12–15 cm higher. The maximum deviations in the spirit level points of first order are between 7 and 16 cm. Geodesy Geodynamical phenomena, including crustal motion, tides , and polar motion , can be studied by designing global and national control networks , applying space geodesy and terrestrial geodetic techniques, and relying on datums and coordinate systems . Geodetic job titles include geodesist and geodetic surveyor . Geodesy began in pre-scientific antiquity , so

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