The Reference Ellipsoid

A reference ellipsoid is the smallest geodetic object — a pure mathematical surface, defined by two numbers, that approximates the overall shape of Earth. Every modern coordinate reference system builds on one. The /learn/what-is-a-geodetic-datum pillar treats it as one of the four components of a datum; /learn/why-the-earth-is-not-a-sphere covers why we need a flattened ellipsoid rather than a sphere. This article goes one level deeper: the ellipsoid as a building block, its defining parameters, the historical list, and the relationship to datums.

The two defining parameters

A reference ellipsoid is fully specified by two numbers:

• Semi-major axis (a) — the equatorial radius, in metres. The longest radius of the ellipsoid.
• Flattening (f) — the ratio (a − b) / a where b is the semi-minor (polar) radius. Captures how much shorter the polar radius is than the equatorial.

For the WGS 84 reference ellipsoid (the most-used in the world):

a = 6,378,137.0 m
f = 1/298.257223563 ≈ 0.00335281066

That's the entire definition. Every other ellipsoid parameter is derived from these two.

The derived parameters

From a and f, several derived quantities are computed and used throughout geodetic formulas:

| Quantity | Formula | WGS 84 value | | ------------------------------ | ------------------------ | ------------------------ | | Semi-minor (polar) axis | b = a(1 − f) | 6,356,752.3142 m | | First eccentricity squared | e² = f(2 − f) | 0.00669437999014 | | First eccentricity | e = √(e²) | 0.0818191908426 | | Second eccentricity squared | e'² = e²/(1 − e²) | 0.00673949675659 | | Mean radius | R = (2a + b)/3 | 6,371,008.7714 m | | Authalic radius (equal-area sphere) | derived | 6,371,007.181 m | | Volumetric radius (equal-volume sphere) | derived | 6,371,000.79 m |

The eccentricity e appears throughout closed-form coordinate formulas — Vincenty's distance, the Transverse Mercator projection, ECEF (Earth-Centred Earth-Fixed) Cartesian conversion, and many others. The mean / authalic / volumetric radii are used when the workflow needs to approximate the ellipsoid as a sphere (e.g., haversine distance, web map tile arithmetic).

How the parameters define the surface

In a geocentric Cartesian coordinate system with Z along the rotation axis, the reference ellipsoid is the set of points satisfying:

(x² + y²) / a² + z² / b² = 1

This is the implicit equation of an oblate spheroid. For each geodetic latitude φ and longitude λ, the corresponding point on the ellipsoid is:

N = a / √(1 − e² sin²φ)                       (prime vertical radius of curvature)
x = N · cos(φ) · cos(λ)
y = N · cos(φ) · sin(λ)
z = N · (1 − e²) · sin(φ)

These are the ECEF coordinates (EPSG:4978 when the datum is WGS 84) — the Cartesian representation that satellite-tracking arithmetic uses internally. The same physical point can be expressed as either geographic (φ, λ, h) or geocentric Cartesian (x, y, z); the formulas above are the conversion between them.

A worked ECEF example

Convert the Empire State Building's geographic coordinates (40.7484°N, 73.9857°W, h = 0 m) to ECEF on the WGS 84 ellipsoid:

Inputs:
  φ = 40.7484° → 0.7112 rad
  λ = −73.9857° → −1.2912 rad
  a = 6,378,137.0 m
  e² = 0.00669437999014

Prime vertical radius of curvature:
  N = a / √(1 − e²·sin²φ)
  sin(0.7112) = 0.6526
  sin² = 0.4259
  e²·sin² = 0.00669438 × 0.4259 = 0.002851
  1 − 0.002851 = 0.997149
  √(0.997149) = 0.998574
  N = 6,378,137 / 0.998574 = 6,387,247 m

ECEF coordinates (with h = 0):
  X = N·cos(φ)·cos(λ)
    = 6,387,247 × cos(0.7112) × cos(−1.2912)
    = 6,387,247 × 0.7577 × 0.2774
    = 1,342,538 m

  Y = N·cos(φ)·sin(λ)
    = 6,387,247 × 0.7577 × sin(−1.2912)
    = 6,387,247 × 0.7577 × (−0.9608)
    = −4,650,128 m

  Z = N·(1 − e²)·sin(φ)
    = 6,387,247 × 0.99331 × 0.6526
    = 4,141,054 m

So the Empire State Building's ECEF position is approximately (1,342,538, −4,650,128, 4,141,054) m. The same physical point expressed as a geographic triple (lat, lon, height) and as a Cartesian triple (X, Y, Z) — different representations of the same location relative to the WGS 84 ellipsoid's centre. ECEF coordinates are used internally by satellite-tracking software, GNSS receivers, and any code that needs Cartesian arithmetic on points relative to Earth's centre.

A short list of historical ellipsoids

| Ellipsoid | Semi-major axis | Flattening (1/f) | Year | Notes | | ------------------ | ------------------ | -------------------- | ---- | ------------------------------ | | Airy 1830 | 6,377,563.396 m | 299.3249646 | 1830 | OSGB36 (Britain) | | Bessel 1841 | 6,377,397.155 m | 299.1528128 | 1841 | Older Asian datums | | Clarke 1866 | 6,378,206.4 m | 294.978698 | 1866 | NAD 27 | | Clarke 1880 | 6,378,249.145 m | 293.465 | 1880 | Older African and Middle Eastern datums | | International 1924 | 6,378,388 m | 297.0 | 1924 | Older European datums | | Krassovsky 1940 | 6,378,245 m | 298.3 | 1940 | Soviet Union datums | | WGS 60 | 6,378,165 m | 298.3 | 1960 | First WGS — superseded | | WGS 66 | 6,378,145 m | 298.25 | 1966 | Superseded | | WGS 72 | 6,378,135 m | 298.26 | 1972 | Superseded | | GRS 80 | 6,378,137 m | 298.257222101 | 1980 | NAD 83, ETRS89, GDA2020, ITRF | | WGS 84 | 6,378,137 m | 298.257223563 | 1984 | GPS, GeoJSON, EPSG:4326 |

The progression shows the parameters converging as measurement techniques improved. The 18th- and 19th-century ellipsoids (Airy, Bessel, Clarke) were derived from regional triangulation surveys; the 20th-century ones (International 1924 onwards) used increasingly global data. GRS 80 — the IAG's 1980 standard — is the modern reference; WGS 84 is essentially a re-realization of GRS 80 by the US DoD with a marginally different flattening (the two are interchangeable in practice).

The ellipsoid–datum relationship

An ellipsoid alone doesn't locate a point on Earth. To do that, the ellipsoid has to be tied to physical Earth: its centre placed somewhere in space, its axes oriented to the rotating planet, and its surface anchored to observed reference points. That wrapping turns an ellipsoid into a datum.

The relationship is many-to-many in principle but mostly one-to-many in practice:

• GRS80 is the ellipsoid of NAD 83 (1986 origin), NAD 83(2011), ETRS89 (1989 origin, plate-fixed to Eurasia), GDA2020 (plate- fixed to Australia), ITRF (global, IERS).
• WGS 84 is the ellipsoid of the WGS 84 datum (G2139 current realization).
• Clarke 1866 is the ellipsoid of NAD 27 — and is otherwise retired.
• Airy 1830 is the ellipsoid of OSGB36 (Britain) — used only there.

The EPSG registry maintains separate codes for ellipsoids (e.g., 7030 for WGS 84 ellipsoid, 7019 for GRS80) and for the datums that use them. A coordinate is unambiguously specified only with the full CRS code (e.g., EPSG:4326 for WGS 84 geographic), not just the ellipsoid.

How ellipsoid parameters are determined

Modern reference ellipsoids are derived from large-scale measurement campaigns. The two defining parameters (a and f) are chosen to minimise the misfit between the ellipsoid and the observed Earth surface — typically the geoid.

Historically, this was done by measuring the lengths of meridian arcs at multiple latitudes (the Cassini-vs-Newton expeditions of the 1730s, then subsequent triangulation networks across continents in the 19th century). Comparing the observed lengths of degrees of latitude reveals the ellipsoid's curvature parameters; least- squares adjustment of many measurements gives the best-fit a and f.

In the satellite era, the parameter determination shifted to global gravity observations. NASA's GRACE / GRACE-FO and ESA's GOCE measure Earth's gravity field continuously; the equipotential surface that best fits Earth's mean shape defines the modern reference. The IAG's GRS80 ellipsoid (and the closely-aligned WGS 84 ellipsoid) was determined this way: satellite-tracking data, terrestrial gravity, and altimetry all combined into a global least-squares fit.

The result is that the modern a and f values are not arbitrary choices — they are the empirical best fit to Earth's mean shape, refined every few decades as measurement precision improves.

Choosing an ellipsoid for a project

In practice, the choice is usually made by the choice of datum, which is in turn made by the choice of region or application:

• Global / GPS / web mapping → WGS 84 datum on WGS 84 ellipsoid (EPSG:4326).
• US federal / state mapping → NAD 83 datum on GRS80 ellipsoid (EPSG:4269 for current epoch).
• European INSPIRE-compliant data → ETRS89 datum on GRS80 ellipsoid (EPSG:4258).
• Australian national data → GDA2020 datum on GRS80 ellipsoid (EPSG:7844).
• British Ordnance Survey → OSGB36 datum on Airy 1830 ellipsoid (EPSG:4277).
• Legacy US data (pre-1983) → NAD 27 datum on Clarke 1866 ellipsoid (EPSG:4267).

Storing the ellipsoid as part of the data's metadata is essential when integrating across systems. Two coordinates on the same physical point can have different numbers under different ellipsoid-datum combinations; without metadata, you cannot reliably integrate.

Common misconceptions

“The ellipsoid is the same as the datum.” The ellipsoid is the shape. The datum is the ellipsoid plus origin, orientation, and gravity model. ISO 19111 keeps them separate concepts. Multiple datums can share an ellipsoid (GRS80 in NAD 83, ETRS89, GDA2020, ITRF); each is a distinct datum because the origin and realization differ.

“An ellipsoid is fixed forever.” New ellipsoids are defined occasionally as measurements improve (WGS 60 → 66 → 72 → 84 is one such sequence). Once defined, an ellipsoid's parameters don't change; what changes is which ellipsoid the contemporary standards use.

“WGS 84 and GRS80 are different.” Technically yes (in the eighth decimal of flattening); practically no (the coordinate offset is below 0.1 mm anywhere on Earth). For sub-millimetre work, the distinction matters; for everything else, the two ellipsoids are interchangeable. Most modern software treats them as equivalent and rounds to whichever is needed at output.

“A spherical Earth model is good enough.” For visualisation, yes. For coordinate computation, no — the 1 % spherical-vs-ellipsoidal accuracy gap exceeds the requirements of surveying, navigation, and engineering. Modern systems use the ellipsoid model.

“The ellipsoid's centre is the centre of Earth.” For modern global datums (WGS 84, GRS80-based geocentric datums) yes — the ellipsoid centre is placed at the Earth's geocentre. For historical topocentric datums (NAD 27, Tokyo Datum 1918) the ellipsoid centre was offset to make the ellipsoid fit a regional surface; geocentric placement only became standard in the satellite era.

“You can pick any ellipsoid for any coordinate.” The coordinate's numerical value depends on the ellipsoid; reading a coordinate from one ellipsoid as if it were on another produces a metre-scale error. Always know which ellipsoid (or full datum) the coordinate is referenced to.