# Extensive Definition

Geodesy (), also called geodetics, a branch of
earth
sciences, is the scientific discipline that deals with the
measurement and representation of the Earth, including its
gravitational
field, in a three-dimensional time-varying space. Besides the
Earth's gravitational field, geodesists also study geodynamical phenomena such
as crustal
motion, tides, and polar
motion. For this they design global and national Control
networks, using Space
and terrestrial techniques while relying on datums
and coordinate
systems.

## Definition

Geodesy (from Greek Γεωδαισία lit. division of the Earth) is primarily concerned with positioning within the temporally varying gravity field. Somewhat obsolete nowadays, geodesy in the German speaking world is divided into "Higher Geodesy" ("Erdmessung" or "höhere Geodäsie"), which is concerned with measuring the Earth on the global scale, and "Practical Geodesy" or "Engineering Geodesy" ("Ingenieurgeodäsie"), which is concerned with measuring specific parts or regions of the Earth, and which includes surveying.The shape of the Earth is to a large extent the
result of its rotation, which causes its equatorial bulge, and the
competition of geological processes such as the collision of plates
and of vulcanism, resisted
by the Earth's gravity
field. This applies to the solid surface, the liquid surface
(dynamic
sea surface topography) and the Earth's
atmosphere. For this reason, the study of the Earth's gravity
field is called physical
geodesy by some.

## History

## Geoid and reference ellipsoid

The geoid is essentially the figure of the Earth abstracted from its topographical features. It is an idealized equilibrium surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. and continued under the continental masses. The geoid, unlike ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between it and the reference ellipsoid is called the geoidal undulation. It varies globally between ±110 m.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), is a purely geometrical one.
The mechanical ellipticity of the Earth (dynamical flattening,
symbol J2) can be determined to high precision by observation of
satellite orbit perturbations. Its relationship with the
geometrical flattening is indirect. The relationship depends on the
internal density distribution, or, in simplest terms, the degree of
central concentration of mass.

The 1980 Geodetic Reference System (GRS80) posited a
6,378,137 m semi-major axis and a 1:298.257 flattening.
This system was adopted at the XVII General Assembly of the
International Union of Geodesy and Geophysics (IUGG). It is
essentially the basis for geodetic positioning by the Global
Positioning System and is thus also in extremely widespread use
outside the geodetic community.

The numerous other systems which have been used
by diverse countries for their maps and charts are gradually
dropping out of use as more and more countries move to global,
geocentric reference systems using the GRS80 reference
ellipsoid.

## Coordinate systems in space

see also Geodetic system The locations of points in three-dimensional space are most conveniently described by three cartesian or rectangular coordinates, X, Y and Z. Since the advent of satellite positioning, such coordinate systems are typically geocentric: the Z axis is aligned with the Earth's (conventional or instantaneous) rotation axis.Prior to satellite
geodesy era, the coordinate systems associated with a geodetic
datum
attempted to be geocentric, but their origins
differed from the geocentre by hundreds of metres, due to regional
deviations in the direction of the plumbline (vertical). These
regional geodetic datums, such as ED50 (European Datum
1950) or NAD83 (North American
Datum 1983) have ellipsoids associated with them that are regional
'best fits' to the geoids
within their areas of validity, minimising the deflections of the
vertical over these areas.

It is only because GPS
satellites orbit about the geocentre, that this point becomes
naturally the origin of a coordinate system defined by satellite
geodetic means, as the satellite positions in space are themselves
computed in such a system.

Geocentric coordinate systems used in geodesy can
be divided naturally into two classes:

- Inertial reference systems, where the coordinate axes retain their orientation relative to the fixed stars, or equivalently, to the rotation axes of ideal gyroscopes; the X axis points to the vernal equinox
- Co-rotating, also ECEF ("Earth Centred, Earth Fixed"), where the axes are attached to the solid body of the Earth. The X axis lies within the Greenwich observatory's meridian plane.

The coordinate transformation between these two
systems is described to good approximation by (apparent) sidereal
time, which takes into account variations in the Earth's axial
rotation (length-of-day
variations). A more accurate description also takes polar motion
into account, a phenomenon closely monitored by geodesists.

### Coordinate systems in the plane

In surveying and mapping, important fields of application of geodesy, two general types of coordinate systems are used in the plane:- Plano-polar, in which points in a plane are defined by a distance s from a specified point along a ray having a specified direction \alpha with respect to a base line or axis;
- Rectangular, points are defined by distances from two perpendicular axes called x and y. It is geodetic practice — contrary to the mathematical convention — to let the x axis point to the North and the y axis to the East.

Rectangular coordinates in the plane can be used
intuitively with respect to one's current location, in which case
the x axis will point to the local North. More formally, such
coordinates can be obtained from three-dimensional coordinates
using the artifice of a map
projection. It is not possible to map the curved surface of the
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 are mapped as small circles and small squares as
squares.

An example of such 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 \alpha and s respectively, then we have

\begin x &=& s \cos \alpha\\ y
&=& s \sin \alpha \end

The reverse transformation is given by:

\begin s &=& \sqrt\\ \alpha &=&
\arctan. \end

## Heights

In geodesy, point or terrain heights are "above sea level", an irregular, physically defined surface. Therefore a height should ideally not be referred to as a coordinate. It is more like a physical quantity, and though it can be tempting to treat height as the vertical coordinate z, in addition to the horizontal coordinates x and y, and though this actually is a good approximation of physical reality in small areas, it quickly becomes invalid for regional considerations.Heights come in the following variants:

Each has its advantages and disadvantages. Both
orthometric and normal heights are heights in metres above sea
level, whereas geopotential numbers are measures of potential
energy (unit: m² s−2) and not metric. Orthometric and
normal heights differ in the precise way in which mean sea level is
conceptually continued under the continental masses. The reference
surface for orthometric heights is the geoid, an equipotential surface
approximating mean sea level.

None of these heights is in any way related to
geodetic or ellipsoidial heights, which express the height of a
point above the reference
ellipsoid. Satellite positioning receivers typically provide
ellipsoidal heights, unless they are fitted with special conversion
software based on a model of the geoid.

## Geodetic data

Because geodetic point coordinates (and heights) are always obtained in a system that has been constructed itself using real observations, geodesists introduce the concept of a geodetic datum: a physical realization of a coordinate system used for describing point locations. The realization is the result of choosing conventional coordinate values for one or more datum points.In the case of height datums, it suffices to
choose one datum point: the reference bench mark, typically a tide
gauge at the shore. Thus we have vertical datums like the NAP
(Normaal
Amsterdams Peil), the North American Vertical Datum 1988
(NAVD88), the Kronstadt
datum, the Trieste datum, and so on.

In case of plane or spatial coordinates, we
typically need several datum points. A regional, ellipsoidal datum
like ED50 can
be fixed by prescribing the undulation of the geoid and the deflection of the
vertical in one datum point, in this case the Helmert
Tower in Potsdam. However,
an overdetermined ensemble of datum points can also be used.

Changing the coordinates of a point set referring
to one datum, so to make them refer to another datum, is called a
datum transformation. In the case of vertical datums, this consists
of simply adding a constant shift to all height values. In the case
of plane or spatial coordinates, datum transformation takes the
form of a similarity or Helmert transformation, consisting of a
rotation and scaling 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, e.g., in ISO terminology, referred to as a coordinate system. International geodetic organizations like the IERS (International Earth Rotation and Reference Systems Service) speak of a reference system.When these coordinates are realized by choosing
datum points and fixing a geodetic datum, ISO uses the terminology
coordinate reference system, while IERS speaks of a reference
frame. A datum transformation again is referred to by ISO as a
coordinate transformation. (ISO 19111: Spatial referencing by
coordinates).

## Point positioning

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 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 the national mapping
agencies. Surveyors involved in real estate and insurance will use
these to tie their local measurements to.

## Geodetic problems

In geometric geodesy, two standard problems exist:### First 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) geodetic problem

- Given two points, determine the azimuth and length of the line (straight line, arc or geodesic) that connects them.

In the case of plane geometry (valid for small
areas on the Earth's surface) the solutions to both problems reduce
to simple trigonometry. On the
sphere, the solution is significantly more complex, e.g., in the
inverse problem the azimuths will differ between the two end points
of the connecting great
circle, arc, i.e. the geodesic.

On the ellipsoid of revolution, solutions in
closed form do not exist, so rapidly converging series expansions
have traditionally been used, such as Vincenty's
formulae.

In the general case, the solution is called the
geodesic for the
surface considered. It may be nonexistent or non-unique. The
differential
equations for the geodesic can be solved
numerically, e.g., in MATLAB.

## Geodetic observational concepts

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.- The plumbline or vertical is the direction of local gravity, or the line that results by following it. It is slightly curved.

- The zenith is the point on the celestial sphere where the direction of the gravity vector in a point, extended upwards, intersects it. More correct is to call it a rather than a point.

- The nadir is the opposite point (or rather, direction), where the direction of gravity extended downward intersects the (invisible) celestial sphere.

- The celestial horizon is a plane perpendicular to a point's gravity vector.

- Azimuth is the direction angle within the plane of the horizon, typically counted clockwise from the North (in geodesy and astronomy) or South (in France).

- Elevation is the angular height of an object above the horizon, Alternatively zenith distance, being equal to 90 degrees minus elevation.

- Local topocentric coordinates are azimuth (direction angle within the plane of the horizon) and elevation angle (or zenith angle) and distance.

- The North celestial pole is the extension of the Earth's (precessing and nutating) instantaneous spin axis extended Northward to intersect the celestial sphere. (Similarly for the South celestial pole.)

- The celestial equator is the intersection of the (instantaneous) Earth equatorial plane with the celestial sphere.

- A meridian plane is any plane perpendicular to the celestial equator and containing the celestial poles.

- The local meridian is the plane containing the direction to the zenith and the direction to the celestial pole.

## Geodetic measurements

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 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 Earth rotation irregularities as
well as plate tectonic motions.

Gravity is measured
using gravimeters.
Basically, there are two kinds of gravimeters. Absolute
gravimeters, which nowadays can also be used in the field, are
based directly on measuring the acceleration of free fall (for
example, of a reflecting prism in a vacuum tube). They are used for
establishing the vertical geospatial control. Most common relative
gravimeters are spring based. They are used in gravity surveys over
large areas for establishing the figure of the geoid over these
areas. Most accurate relative gravimeters are superconducting
gravimeters, and these are sensitive to one thousandth of one
billionth of the Earth surface gravity. Twenty-some superconducting
gravimeters are used worldwide for studying Earth tides, rotation, interior, and
ocean and atmospheric
loading, as well as for verifying the Newtonian constant of
gravitation.

## Units and measures on the ellipsoid

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 the 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
40-millionth part of the length of a meridian (the target wasn't
quite reached in actual implementation, so that is off by 0.02% 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
1000/0.54 m to four digits).

## Temporal change

In geodesy, temporal change can be studied by a variety of techniques. Points on the Earth's surface change their location due to a variety of mechanisms:- Continental plate motion, plate tectonics
- Episodic motion of tectonic origin, esp. close to fault lines
- Periodic effects due to Earth tides
- Postglacial land uplift due to isostatic adjustment
- Various anthropogenic movements due to, for instance, petroleum or water extraction or reservoir construction.

The science of studying deformations and motions
of the Earth's crust and the solid Earth as a whole is called
geodynamics. Often,
study of the Earth's irregular rotation is also included in its
definition.

Techniques for studying geodynamic phenomena on
the global scale include:

- satellite positioning by GPS and other such systems,
- Very Long Baseline Interferometry (VLBI)
- satellite and lunar laser ranging
- Regionally and locally, precise levelling,
- precise tacheometers,
- monitoring of gravity change,
- Interferometric synthetic aperture radar (InSAR) using satellite images, etc.

## Famous geodesists

### Mathematical Geodesists before 1900

- Abu Rayhan Biruni 973-1048, Khwarezm (Iran/Persia)
- Sir George Biddell Airy 1801-1892, Cambridge & London
- Muhammad al-Idrisi 1100-1166, (Arabia & Sicily)
- Al-Ma'mun 786-833, Baghdad (Iraq/Mesopotamia)
- Johann Jacob Baeyer 1794-1885, Berlin (Germany)
- Karl Maximilian von Bauernfeind, Munich (Germany)
- Friedrich Wilhelm Bessel, Königsberg (Germany)
- Roger Joseph Boscovich, Rome/ Berlin/ Paris
- Pierre Bouguer 1698-1758, (France & Peru)
- Heinrich Bruns 1848-1919, Berlin (Germany)
- Alexis Claude Clairaut 1713-1765 (France)
- Alexander Ross Clarke, London (England)
- Loránd Eötvös 1848-1919 (Hungary)
- Eratosthenes, Alexandria (Greece & Egypt)
- Sir George Everest 1830-1843 (England & India)
- Hervé Faye 1814-1902 (France)
- Abel Foullon (France)
- Carl Friedrich Gauß 1777-1855, Göttingen (Germany)
- Friedrich Robert Helmert, Potsdam (Germany)
- Hipparchos, Nicosia (Greece)
- Christiaan Huygens 1629-1695 (Netherlands)
- Jean Henri Lambert 1728-1777 (France)
- Pierre-Simon Laplace 1749-1827, Paris (France)
- Adrien Marie Legendre 1752-1833, Paris (France)
- Johann Benedikt Listing 1808-1882 (Germany)
- Pierre de Maupertuis 1698-1759 (France)
- Gerhard Mercator 1512-1594 (Belgium & Germany)
- Friedrich H. C. Paschen, Schwerin (Germany)
- Charles S. Peirce 1839-1914 (United States)
- Henri Poincaré, Paris (France)
- J. H. Pratt 1809-1871, London (England)
- Posidonius, Alexandria (Greece & Egypt)
- Ptolemäus, Alexandria (Greece & Egypt)
- Regiomontanus (Germany/Austria)
- Georg von Reichenbach 1771-1826, Bavaria (Germany)
- Heinrich Christian Schumacher 1780-1850 (Germany & Estonia)
- Snellius (Willebrord Snel van Royen) 1580-1626, Leiden (Netherlands)
- Johann Georg von Soldner 1776-1833, Munich (Germany)
- George Gabriel Stokes (England)
- Friedrich Georg Wilhelm Struve 1793-1864, Dorpat and Pulkowa/St.-Petersburg (Russia)

### 20th century

- Arne Bjerhammar,KTH, Stockholm (Sweden)
- W. Bowie 1872-1940 (USA)
- John Fillmore Hayford (USA)
- Friedrich Hopfner, Vienna (Austria)
- Harold Jeffreys, London (England)
- John A. O'Keefe 1916-2000 (USA)
- Karl-Rudolf Koch, Bonn (Germany)
- Mikhail Sergeevich Molodenskii 1909-1991 (Russia)
- Hellmut Schmid, (Switzerland)
- Petr Vaníček, Fredericton (Canada)
- Yrjö Väisälä 1889-1971, (Finland)
- Felix Andries Vening-Meinesz 1887-1966 (Netherlands)
- Thaddeus Vincenty, (Poland)
- Alfred Wegener 1880-1930, (Germany & Greenland)

## International organizations

- International Association of Geodesy (IAG)
- International Union of Geodesy and Geophysics (IUGG)
- Fédération Internationale des Géomètres (FIG)
- European Petroleum Survey Group (EPSG) (which despite being officially disbanded in 2005 continues to refine a well tested set of Geodetic Parameters)

## Governmental agencies

- National Geodetic Survey (NGS), Silver Spring MD, USA
- National Geospatial-Intelligence Agency (NGA), Bethesda MD, USA (Previously National Imagery and Mapping Agency NIMA, previously Defense Mapping Agency DMA)
- U.S. Geological Survey (USGS), Reston VA, USA
- Institut Géographique National (IGN), Saint-Mandé, France
- Bundesamt für Kartographie und Geodäsie (BKG), Frankfurt a. M., Germany (Previously Institut für Angewandte Geodäsie, IfAG)
- Central Research Institute for Geodesy, Remote Sensing and Cartography (CNIIGAIK), Moscow, Russia
- Geodetic Survey Division, Natural Resources Canada, Ottawa, Canada
- Geoscience Australia, Australian Federal Agency
- Finnish Geodetic Institute (FGI), Masala, Finland
- Portuguese Geographic Institute (IGEO), Lisbon, Portugal
- Brazilian Institute for Geography and Statistics - IBGE
- Spanish National Geographic Institute (IGN), Madrid, Spain
- Land Information New Zealand.

Note: This list is still largely
incomplete.

## See also

## Notes

## References

- B. Hofmann-Wellenhof and H. Moritz, Physical Geodesy, Springer-Verlag Wien, 2005. (This text is an updated edition of the 1967 classic by W.A. Heiskanen and H. Moritz).
- Vaníček P. and E.J. Krakiwsky, Geodesy: the Concepts, pp.714, Elsevier, 1986.
- Thomas H. Meyer, Daniel R. Roman, and David B. Zilkoski. "What
does height really mean?" (This is a series of four articles
published in Surveying and Land Information Science, SaLIS.)
- "Part I: Introduction" SaLIS Vol. 64, No. 4, pages 223-233, December 2004.
- "Part II: Physics and gravity" SaLIS Vol. 65, No. 1, pages 5-15, March 2005.
- "Part III: Height systems" SaLIS Vol. 66, No. 2, pages 149-160, June 2006.
- "Part IV: GPS heighting" SaLIS Vol. 66, No. 3, pages 165-183, September 2006.

## External links

- International Association of Geodesy (IAG).
- The Geodesy Page.
- Geodesy and Geomatics Home Page
- Welcome to Geodesy
- MapRef.org: The Collection of Map Projections and Reference Systems for Europe
- Geodesy on the World Wide Web
- Pennsylvania Geospatial Data Sharing Standard - Geodesy and Geodetic Monumentation
- References on Absolute Gravimeters
- Geodesy tutorial at University of New Brunswick
- Vincenty's Direct and Inverse Solutions of Geodesics on the Ellipsoid, in JavaScript
- Vincenty's Solution of Geodesics on the Ellipsoid, in C#
- Vincenty's Solution of Geodesics on the Ellipsoid, in Java
- EarthScope Project
- UNAVCO - EarthScope - Plate Boundary Observatory
- Polish Internet Informant of Geodesy

geodesist in Arabic: جيوديسيا

geodesist in Asturian: Xeodesia

geodesist in Bulgarian: Геодезия

geodesist in Catalan: Geodèsia

geodesist in Czech: Geodézie

geodesist in Danish: Geodæsi

geodesist in German: Geodäsie

geodesist in Modern Greek (1453-):
Γεωδαισία

geodesist in Spanish: Geodesia

geodesist in Esperanto: Geodezio

geodesist in Persian: ژئودزی

geodesist in French: Géodésie

geodesist in Galician: Xeodesia

geodesist in Croatian: Geodezija

geodesist in Italian: Geodesia

geodesist in Hebrew: גאודזיה

geodesist in Latvian: Ģeodēzija

geodesist in Hungarian: Geodézia

geodesist in Malay (macrolanguage):
Geodesi

geodesist in Dutch: Geodesie

geodesist in Japanese: 測地学

geodesist in Norwegian: Geodesi

geodesist in Polish: Geodezja

geodesist in Portuguese: Geodésia

geodesist in Romanian: Geodezie

geodesist in Russian: Геодезия

geodesist in Slovak: Geodézia

geodesist in Slovenian: Geodezija

geodesist in Serbian: Геодезија

geodesist in Serbo-Croatian: Geodezija

geodesist in Finnish: Geodesia

geodesist in Swedish: Geodesi

geodesist in Turkish: Jeodezi

geodesist in Ukrainian: Геодезія

geodesist in Venetian: Giodexía

geodesist in Chinese: 大地测量学