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An old geodetic pillar (1855) at Ostend, Belgium







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 the geoid 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 = (ab)/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.



Coordinate systems in space








Coordinate systems in the plane


  1. 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;


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{matrix} x &=& s \cos \alpha\\ y &=& s \sin \alpha \end{matrix}

The reverse transformation is given by:

\begin{matrix} s &=& \sqrt{x^2 + y^2}\\ \alpha &=& \arctan{(y/x)}. \end{matrix}


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.[specify]

Heights come in the following variants:

  1. Orthometric heights
  2. Normal heights
  3. Geopotential heights



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

File:Geodetic Control
Geodetic Control Mark (example of a deep benchmark)






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


Second (inverse) geodetic problem


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, geodesics may be written in terms of elliptic integrals, which are usually evaluated in terms of a series expansion; for example, see 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.

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






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.



Temporal change


  • 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


Techniques for studying geodynamic phenomena on the global scale include:

Famous geodesists

Mathematical Geodesists before 1900

20th century

International organizations

Governmental agencies

Note: This list is still largely incomplete.

See also


  1. ^ OED


  • 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.)

External links


Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary



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  • IPA: /dʒiˈɑdɪsi/




geodesy (uncountable)

  1. The scientific discipline that deals with the measurement and representation of the earth, its gravitational field and geodynamic phenomena (polar motion, earth tides, and crustal motion) in three-dimensional, time-varying space.

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