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caling operation in addition to a simple translation in the plane a helmert transformation has four parameters in space seven a note on terminology in the abstract a coordinate system as used in mathematics and geodesy is called a coordinate system in iso terminology whereas the international earth rotation and reference systems service iers uses the term reference system when these coordinates are realized by choosing datum points and fixing a geodetic datum iso says coordinate reference system while iers says reference frame the iso term for a datum transformation again is a coordinate transformation 5 point positioning edit geodetic control mark example of a deep benchmark point positioning is the determination of the coordinates of a point on land at sea or in space with respect to a coordinate system point position is solved by computation from measurements linking the known positions of terrestrial or extraterrestrial points with the unknown terrestrial position this may involve transformations between or among astronomical and terrestrial coordinate systems the known points used for point positioning can be triangulation points of a higher order network or gps satellites traditionally a hierarchy of networks has been built to allow point positioning within a country highest in the hierarchy were triangulation networks these were densified into networks of traverses polygons into which local mapping surveying measurements usually with measuring tape corner prism and the familiar where red and white poles are tied nowadays all but special measurements e g underground or high precision engineering measurements are performed with gps the higher order networks are measured with static gps using differential measurement to determine vectors between terrestrial points these vectors are then adjusted in traditional network fashion a global polyhedron of permanently operating gps stations under the auspices of the iers is used to define a single global geocentric reference frame which serves as the zero order global reference to which national measurements are attached for surveying mappings frequently real time kinematic gps is employed tying in the unknown points with known terrestrial points close by in real time one purpose of point positioning is the provision of known points for mapping measurements also known as horizontal and vertical control in every country thousands of such known points exist and are normally documented by national mapping agencies surveyors involved in real estate and insurance will use these to tie their local measurements geodetic problems edit main article geodesics on an ellipsoid solution of the direct and inverse problems in geometric geodesy two standard problems exist the first direct or forward and the second inverse or reverse first direct or forward geodetic problem given a point in terms of its coordinates and the direction azimuth and distance from that point to a second point determine the coordinates of that second point second inverse or reverse geodetic problem given two points determine the azimuth and length of the line straight line arc or geodesic that connects them in plane geometry valid for small areas on earth s surface the solutions to both problems reduce to simple trigonometry on a sphere however the solution is significantly more complex because in the inverse problem the azimuths will differ between the two end points of the connecting great circle arc on the ellipsoid of revolution geodesics may be written in terms of elliptic integrals which are usually evaluated in terms of a series expansion see for example vincenty s formulae in the general case the solution is called the geodesic for the surface considered the differential equations for the geodesic can be solved numerically observational concepts edit here we define some basic observational concepts like angles and coordinates defined in geodesy and astronomy as well mostly from the viewpoint of the local observer plumbline or vertical the direction of local gravity or the line that results by following it zenith the point on the celestial sphere where the direction of the gravity vector in a point extended upwards intersects it it is more correct to call it a direction rather than a point nadir the opposite point or rather direction where the direction of gravity extended downward intersects the obscured celestial sphere celestial horizon a plane perpendicular to a point s gravity vector azimuth the direction angle within the plane of the horizon typically counted clockwise from the north in geodesy and astronomy or the south in france elevation the angular height of an object above the horizon alternatively zenith distance being equal to 90 degrees minus elevation local topocentric coordinates azimuth direction angle within the plane of the horizon elevation angle or zenith angle distance north celestial pole the extension of earth s precessing and nutating instantaneous spin axis extended northward to intersect the celestial sphere similarly for the south celestial pole celestial equator the instantaneous intersection of earth s equatorial plane with the celestial sphere meridian plane any plane perpendicular to the celestial equator and containing the celestial poles local meridian the plane containing the direction to the zenith and the direction to the celestial pole measurements edit further information satellite geodesy geodetic astronomy surveying gravimetry and levelling a nasa project manager talks about his work for the space geodesy project including an overview of its four fundamental techniques gps vlbi slr and doris the level is used for determining height differences and height reference systems commonly referred to mean sea level the traditional spirit level produces these practically most useful heights above sea level directly 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 knowledge accumulates one may expect the use of gps heighting to spread the theodolite is used to measure horizontal and vertical angles to target points these angles are referred to the local vertical the tacheometer additionally determines electronically or electro optically the distance to target and is highly automated to even robotic in its operations the method of free station position is widely used for local detail surveys tacheometers are commonly employed although the old fashioned rectangular technique using angle prism and steel tape is still an inexpensive alternative real time kinematic rtk gps techniques are used as well data collected are tagged and recorded digitally for entry into a geographic information system gis database geodetic gps receivers produce directly three dimensional coordinates in a geocentric coordinate frame such a frame is e g wgs84 or the frames that are regularly produced and published by the international earth rotation and reference systems service iers gps receivers have almost completely replaced terrestrial instruments for large scale base network surveys for planet wide geodetic surveys previously impossible we can still mention satellite laser ranging slr and lunar laser ranging llr and very long baseline interferometry vlbi techniques all these techniques also serve to monitor irregularities in earth s rotation as well as plate tectonic motions gravity is measured using gravimeters of which there are two kinds first absolute gravimeters are based on measuring the acceleration of free fall e g of a reflecting prism in a vacuum tube they are used to establish the vertical geospatial control and can be used in the field second relative gravimeters are spring based and are more common they are used in gravity surveys over large areas for establishing the figure of the geoid over these areas the most accurate relative gravimeters are called superconducting gravimeters which are sensitive to one thousandth of one billionth of earth surface gravity twenty some superconducting gravimeters are used worldwide for studying earth s tides rotation interior and ocean and atmospheric loading as well as for verifying the newtonian constant of gravitation in the future gravity and altitude will be measured by relativistic time dilation measured by optical clocks units and measures on the ellipsoid edit further information geodetic coordinates 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 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 fairly well provided an ellipsoidal model of the figure of earth is used one geographical mile defined as one minute of arc on the equator equals 1 855 32571922 m one nautical mile 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 as is the nautical mile a metre was originally defined as the 10 millionth part of the length from equator to north pole along the meridian through paris the target was not quite reached in actual implementation so that is off by 200 ppm in the current definitions this means that one kilometre is roughly equal to 1 40 000 360 60 meridional minutes of arc which equals 0 54 nautical mile though this is not exact because the two units are defined on different bases the international nautical mile is defined as exactly 1 852 m corresponding to a rounding of 1 000 0 54 m to four digits temporal change edit see also geoid temporal change in geodesy temporal change can be studied by a variety of techniques points on earth s surface change their location due to a variety of mechanisms continental plate motion plate tectonics 6 episodic motion of tectonic origin especially close to fault lines periodic effects due to tides and tidal loading 7 postglacial land uplift due to isostatic adjustment mass variations due to hydrological changes including the atmosphere cryosphere land hydrology and oceans sub daily polar motion 8 length of day variability 9 earth s center of mass geocenter variations 10 anthropogenic movements such as reservoir construction or petroleum or water extraction the science of studying deformations and motions of earth s crust and its solidity as a whole is called geodynamics often study of earth s irregular rotation is also included in its definition the geodynamics studies require terrestrial reference frames 11 that are realized by the stations belonging to the global geodedetic observing system ggos 12 techniques for studying geodynamic phenomena on the global scale include satellite positioning by gps glonass galileo and beidou very long baseline interferometry vlbi satellite laser ranging slr 13 and lunar laser ranging llr doris regionally and locally precise levelling precise tacheometers monitoring of gravity change using land airborne shipborne and spaceborne gravimetry satellite altimetry based on microwave and laser observations for studying the ocean surface sea level rise and ice cover monitoring interferometric synthetic aperture radar insar using satellite images notable geodesists edit main category geodesists it has been suggested that this section be split out into another article titled list of geodesists discuss november 2020 geodesists before 1900 arranged by date edit pythagoras 580 490 bc ancient greece 14 eratosthenes 276 194 bc ancient greece hipparchus c 190 120 bc ancient greece posidonius c 135 51 bc ancient greece claudius ptolemy c ad 83 168 roman empire roman egypt al ma mun 786 833 baghdad iraq mesopotamia abu rayhan biruni 973 1048 khorasan iran samanid dynasty muhammad al idrisi 1100 1166 arabia sicily regiomontanus 1436 1476 germany austria abel foullon 1513 1563 or 1565 france pedro nunes 1502 1578 portugal gerhard mercator 1512 1594 belgium germany snellius willebrord snel van royen 1580 1626 leiden netherlands christiaan huygens 1629 1695 netherlands pierre bouguer 1698 1758 france peru pierre de maupertuis 1698 1759 france alexis clairaut 1713 1765 france johann heinrich lambert 1728 1777 france roger joseph boscovich 1711 1787 rome berlin paris ino tadataka 1745 1818 tokyo georg von reichenbach 1771 1826 bavaria germany pierre simon laplace 1749 1827 paris france adrien marie legendre 1752 1833 paris france johann georg von soldner 1776 1833 munich germany george everest 1790 1866 england and india friedrich wilhelm bessel 1784 1846 königsberg germany heinrich christian schumacher 1780 1850 germany russian empire carl friedrich gauss 1777 1855 göttingen germany friedrich georg wilhelm struve 1793 1864 dorpat and pulkovo russian empire johann jacob baeyer 1794 1885 berlin germany george biddell airy 1801 1892 cambridge london carl christopher georg andræ 1812 1893 copenhagen denmark karl maximilian von bauernfeind 1818 1894 munich germany wilhelm jordan 1842 1899 germany hervé faye 1814 1902 france george gabriel stokes 1819 1903 england carlos ibáñez e ibáñez de ibero 1825 1891 barcelona spain henri poincaré 1854 1912 paris france alexander ross clarke 1828 1914 london england charles sanders peirce 1839 1914 united states friedrich robert helmert 1843 1917 potsdam germany heinrich bruns 1848 1919 berlin germany loránd eötvös 1848 1919 hungary 20th century geodesists alphabetically arranged edit tadeusz banachiewicz 1882 1954 poland arne bjerhammar 1917 2011 sweden guy bomford 1899 1996 india william bowie 1872 1940 us irene kaminka fischer 1907 2009 us buckminster fuller 1895 1983 united states john fillmore hayford 1868 1925 us veikko aleksanteri heiskanen 1895 1971 finland and us reino antero hirvonen 1908 1989 finland friedrich hopfner 1881 1949 vienna austria martin hotine 1898 1968 england harold jeffreys 1891 1989 london england william m kaula 1926 2000 los angeles us karl rudolf koch 1935 bonn germany feodosy nikolaevich krasovsky 1878 1948 russian empire ussr mikhail sergeevich molodenskii 1909 1991 russia john a o keefe 1916 2000 us karl ramsayer 1911 1982 stuttgart germany hellmut schmid 1914 1998 switzerland yrjö väisälä 1889 1971 finland petr vaníček 1935 fredericton canada felix andries vening meinesz 1887 1966 netherlands thaddeus vincenty 1920 2002 poland alfred wegener 1880 1930 germany and greenland hans georg wenzel 1949 1999 germany see also edit earth sciences portal geophysics portal main category geodesy fundamentals concepts and techniques in modern geography geodesics on an ellipsoid history of geodesy physical geodesy earth s circumference governmental agencies national mapping agency u s national geodetic survey national...