# Reference ellipsoid

## Definition

In geodesy, a reference ellipsoid is a mathematically defined surface that approximates the true figure of the Earth or geoid. It is used as the surface on which geodetic network computations are performed and point co-ordinates calculated.

Mathematically the reference ellipsoid is an oblate (flattened) ellipsoid of revolution with two different axes, an equatorial semi-major axis [itex]a[itex] and a polar semi-minor axis [itex]b[itex]. The flattening [itex]f[itex] is defined as

[itex]

f = \frac{a-b}{a} [itex]

and the first eccentricity [itex]e[itex] by

[itex]

e^2=\frac{a^2-b^2}{a^2} = f(2-f). [itex]

For the Earth, [itex]f[itex] is around 1/300, caused by the Earth's rotation.

## Co-ordinates

The co-ordinates of a geodetic point are customarily stated as geodetic latitude and longitude, i.e., the direction in space of the geodetic normal containing the point, and the height h of the point over the reference ellipsoid. If these co-ordinates, i.e., latitude [itex]\varphi[itex], longitude [itex]\lambda[itex] and height h, are given, one can compute the geocentric rectangular co-ordinates of the point as follows:

[itex]

\begin{matrix} x &=& (N + h)\cos \varphi \cos \lambda \\ y &=& (N + h) \cos \varphi \sin \lambda \\ z &=& (N (1-e^2) + h) \sin \varphi \end{matrix} [itex]

where

[itex]

N(\varphi) = \frac{a}{\sqrt{1-e^2\sin^2\varphi}} [itex]

is the meridional radius of curvature of the ellipsoid. [itex]a[itex] is the equatorial radius and [itex]e[itex] the first eccentricity.

The radius of curvature on the transverse, i.e., East-West, direction, of the ellipsoid is given by the formula

[itex]

M(\varphi) = \frac{a(1-e^2)}{(1-e^2\sin^2\varphi)^\frac{3}{2}}. [itex]

Currently the most common reference ellipsoid used, and that used in the context of the Global Positioning System, is GRS80.

Traditional reference ellipsoids or geodetic datums are defined regionally and therefore non-geocentric, e.g., ED50. Modern geodetic datums are established with the aid of GPS and will therefore be geocentric, e.g., WGS 84.

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