Geodesy

Geodesy is the scientific discipline concerned with measuring and representing the Earth, including its geometric shape, gravitational field, and orientation in three-dimensional, time-varying space. Modern definitions extend this scope to include other planetary bodies and natural satellites, forming the broader field of planetary geodesy. Working across both global and regional scales, geodesy underpins surveying, navigation, mapping, satellite positioning, and studies of dynamic Earth processes such as crustal deformation, tides, and polar motion.
Geodetic science relies on terrestrial and space-based techniques, together with global and national geodetic control networks, to determine precise positions and monitor temporal changes. These measurements support navigation systems such as GNSS, contribute to geodynamic research, and inform engineering, environmental monitoring, and Earth system science.

Historical Background

Geodesy has its roots in ancient attempts to determine the Earth’s size and shape. The term derives from the Ancient Greek word geodaisia, meaning “division of the Earth.” Early cosmological views often envisaged a flat Earth beneath a firmament, but empirical observations gradually pointed to a spherical shape. Notable early arguments included the circular shadow observed during lunar eclipses and the apparent lowering of Polaris as travellers moved southward.
Throughout history, surveying and cartography drove the development of geodetic ideas. Over time, geodesy evolved from a geometry-based discipline concerned with Earth’s surface to a comprehensive science addressing Earth’s gravity, rotational behaviour, and internal mass distribution. By the twentieth century, geodesy incorporated principles from astronomy, physics, and mathematics, and with the introduction of satellite technology, it developed into a precise, globally integrated discipline.

Definition and Scope

In English usage, geodesy denotes the science of measuring and representing geospatial information, while geomatics refers to practical applications such as surveying, mapping, and GIS. German usage distinguishes between higher geodesy, focused on global measurements, and engineering geodesy, which concerns regional or local surveying tasks.
The scope of geodesy encompasses:

• Determining Earth’s geometric shape and size.
• Measuring the gravity field and associated potential surfaces.
• Establishing Earth’s orientation and rotational behaviour in space.
• Monitoring changes over time resulting from tectonics, tides, and other geodynamic phenomena.
• Applying its methods to other planets and natural satellites.

The Earth’s shape results from several competing influences: rotation causes equatorial bulging, while geological processes such as tectonic collision and volcanism deform the crust. The gravity field counterbalances these forces, shaping both solid surfaces and dynamic features such as sea-surface topography and atmospheric layers.

The Geoid and the Reference Ellipsoid

Three primary surfaces are fundamental to geodetic measurement: the geoid, the reference ellipsoid, and the terrain.

• The geoid is an irregular but physically meaningful surface that approximates mean sea level in the absence of currents and atmospheric pressure variations. It extends beneath continents and can be located using physical measurements such as tide gauges. It represents an equipotential surface of the Earth’s gravity field.
• The reference ellipsoid is a mathematically defined, smooth surface used to approximate the geoid for computational purposes. It is described by its semimajor axis and flattening. The reference ellipsoid is abstract and can take various instantiations depending on intended applications.
• Geoidal undulation is the separation between the geoid and the reference ellipsoid. It varies globally, typically within about ±100 metres relative to the GRS 80 ellipsoid.

The geometrical flattening of the reference ellipsoid differs from the dynamical flattening (J₂), which reflects internal mass distribution and is determined through satellite orbital perturbations. The Geodetic Reference System 1980 (GRS 80) specifies widely adopted ellipsoid parameters, including a semimajor axis of 6,378,137 metres and a flattening of 1/298.257. These parameters underpin GNSS systems, including GPS, and provide a common global standard for geodetic positioning.

Coordinate Systems in Three Dimensions

Geodesy employs three-dimensional coordinate systems to specify positions in space. Modern systems are typically geocentric, with:

• X-axis lying in the Greenwich meridian plane.
• Y-axis orthogonal to the X-axis, forming a right-handed system.
• Z-axis aligned with Earth’s rotational axis.

Before satellite geodesy, regional geodetic datums attempted to approximate geocentric coordinate systems but were offset from the true geocentre due to local vertical deviations. Examples include the European Datum 1950 and the North American Datum 1927.
Geocentric systems fall into two main categories:

• Inertial reference systems, whose axes remain fixed with respect to distant celestial objects. The X-axis points toward the equinox.
• Earth-centred Earth-fixed (ECEF) systems, whose axes rotate with the Earth. These are used in satellite positioning and mapping.

Transformations between inertial and rotating systems involve apparent sidereal time and, at higher precision, corrections for polar motion monitored by geodesists.

Coordinate Systems in the Plane

Two coordinate types dominate plane mapping:

• Polar (planopolar) coordinates, specifying a point by distance from a baseline and an angle from a reference direction.
• Rectangular coordinates, specifying distances along perpendicular axes (x and y). In geodetic practice, x denotes northing and y denotes easting.

Rectangular coordinates in mapping arise from map projections, since a curved Earth surface cannot be represented on a flat plane without distortion. Conformal projections, such as the Universal Transverse Mercator (UTM), preserve local shape and angle properties, making them popular for large-scale mapping. In such projections, the reference north direction is map north, which may differ from true north; this angular difference is known as convergence.
Transformations between polar and rectangular coordinates in the plane are given by:

• x=scos⁡αx = s \cos \alphax=scosα
• y=ssin⁡αy = s \sin \alphay=ssinα

with the reverse transformations:

• s=x2+y2s = \sqrt{x^2 + y^2}s=x2+y2​
• α=arctan⁡(y/x)\alpha = \arctan(y/x)α=arctan(y/x)

Heights and Vertical Datums

Geodetic height measurement refers to the elevation of points relative to the geoid, the physically defined mean sea level surface. Several height systems are in use:

• Orthometric heights, representing elevation above sea level along the plumb-line.
• Normal heights, referencing a theoretical surface closely approximating the geoid.
• Dynamic heights, used in hydrological and large-scale geophysical applications.
• Geopotential heights, expressing height in units of gravitational potential rather than distance.

Orthometric and normal heights are expressed in metres, whereas geopotential numbers carry energy-based units. The geoid serves as the reference surface for most vertical datums, ensuring consistency across regional and global applications.

Geodetic Applications and Geodynamics

Geodesy supports numerous scientific and practical fields:

• Geodynamics, studying Earth’s crustal movements, tides, sea-level change, and polar motion.
• Navigation and positioning, especially through GNSS, which relies on precise geodetic reference frames such as those based on GRS 80.
• Surveying and mapping, underpinning national cadastral systems and engineering projects.
• Earth observation, monitoring environmental and climate-related changes.
• Planetary studies, applying geodetic principles to other celestial bodies.