Reference ellipsoid in the context of Normal height

An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's shape and size, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different reference ellipsoids have been used as approximations.

It is an oblate spheroid (an ellipsoid of revolution) whose minor axis (polar diameter), connecting the geographical poles, is approximately aligned with the Earth's axis of rotation. The ellipsoid is also defined by the major axis (equatorial axis); the difference between the two axes is slightly more than 21 km or 0.335%.

The geoid (/ˈdʒiː.ɔɪd/ JEE-oyd) is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended through the continents (such as might be approximated with very narrow hypothetical canals). According to Carl Friedrich Gauss, who first described it, it is the "mathematical figure of the Earth", a smooth but irregular surface whose shape results from the uneven distribution of mass within and on the surface of Earth. It can be known only through extensive gravitational measurements and calculations. Despite being an important concept for almost 200 years in the history of geodesy and geophysics, it has been defined to high precision only since advances in satellite geodesy in the late 20th century.

The geoid is often expressed as a geoid undulation or geoidal height above a given reference ellipsoid, which is a slightly flattened sphere whose equatorial bulge is caused by the planet's rotation. Generally the geoidal height rises where the Earth's material is locally more dense and exerts greater gravitational force than the surrounding areas. The geoid in turn serves as a reference coordinate surface for various vertical coordinates, such as orthometric heights, geopotential heights, and dynamic heights (see Geodesy).

The Earth-centered, Earth-fixed coordinate system (acronym ECEF), also known as the geocentric coordinate system, is a cartesian spatial reference system that represents locations in the vicinity of the Earth (including its surface, interior, atmosphere, and surrounding outer space) as X, Y, and Z measurements from its center of mass. Its most common use is in tracking the orbits of satellites and in satellite navigation systems for measuring locations on the surface of the Earth, but it is also used in applications such as tracking crustal motion.

The distance from a given point of interest to the center of Earth is called the geocentric distance, R = (X + Y + Z), which is a generalization of the geocentric radius, R₀, not restricted to points on the reference ellipsoid surface.The geocentric altitude is a type of altitude defined as the difference between the two aforementioned quantities: h′ = R − R₀; it is not to be confused for the geodetic altitude.

Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a reference ellipsoid.They include geodetic latitude (north/south) ϕ, longitude (east/west) λ, and ellipsoidal height h (also known as geodetic height). The triad is also known as Earth ellipsoidal coordinates (not to be confused with ellipsoidal-harmonic coordinates).

The vertical deflection (VD) or deflection of the vertical (DoV), also known as deflection of the plumb line and astro-geodetic deflection, is a measure of how far the gravity direction at a given point of interest is rotated by local mass anomalies such as nearby mountains. They are widely used in geodesy, for surveying networks and for geophysical purposes.

The vertical deflection are the angular components between the true zenith–nadir curve (plumb line) tangent line and the normal vector to the surface of the reference ellipsoid (chosen to approximate the Earth's sea-level surface). VDs are caused by mountains and by underground geological irregularities. Typically angle values amount to less than 10 arc-seconds in flat areas or up to 1 arc-minute in mountainous terrain.

The Struve Geodetic Arc is a chain of survey triangulations stretching from Hammerfest in Norway to the Black Sea, through ten countries and over 2,820 kilometres (1,750 mi), which yielded the first accurate measurement of a meridian arc.

The chain was established and used by the German-born Russian scientist Friedrich Georg Wilhelm von Struve in the years 1816 to 1855 to establish the exact size and shape of the earth. At that time, the chain passed merely through three countries: Norway, Sweden and the Russian Empire. The Arc's first point is located in Tartu Observatory in Estonia, where Struve conducted much of his research. Measurement of the triangulation chain comprises 258 main triangles and 265 geodetic vertices. The northernmost point is located near Hammerfest in Norway and the southernmost point near the Black Sea in Ukraine.