## 3.2 The geoid and the ellipsoid of revolution

The mathematical description of the geoid is possible in several ways. It is possible to give the radius lengths from the geometric center to the geoid surface at crosshairs of the parallels and meridians (latitude-longitude grid). We can also give just the vertical difference of the geoid and an ellipsoid (the geoid undulations) in the same system. The geoid can be also described by the form of the spherical harmonics. Describing a local or regional geoid part, a grid in map projection can be also used.

Selecting any form from these possibilities, it is obvious that the geoid is a very complex surface. If we are about to make a map, we have to chose a projection. The projections, that are quite easy if we suppose the Earth as a sphere, become complicated in ellipsoidal case, while they cannot be handled at all, if the original surface is the real geoid. It was even more impossible to use in the pre-computer age, while the mathematics of the map projections was invented. So, in the geodetic and cartographic applications, the true shape of our planet, the geoid, is substituted by the ellipsoid of revolution.

The ellipsoid for this approximation is generally a well known surface with pre-set semi-major axis and flattening/eccentricity. We can note, that in case of some ellipsoids, characterized by the same name and year, it is possible to find different semi-major axis lengths (such as at the Everest ellipsoid, or the, aforementioned Bessel-Namibia, see Table 3). The cause of this is according to the original definition of these ellipsoids, the semi-major axis was not given in meters but in other units, e.g. in yards of feet. Converting to meters, it is important to give enough decimal figures in the conversion factor. Omitting the ten thousandth parts in this factor (the fourth decimal digit after the point) won’t cause much difference in the everyday life, however if we have millions of feet (such in case of the Earth’s radius we do) the difference is up to several hundred meters.

 name a b 1/f f e Laplace 1802 6376615 6355776.4 306.0058 0.003268 0.08078 Bohnenberger 1809 6376480 6356799.51 324 0.003086 0.07851 Zach 1809 6376480 6355910.71 310 0.003226 0.08026 Zach-Oriani 1810 6376130 6355562.26 310 0.003226 0.08026 Walbeck 1820 6376896 6355834.85 302.78 0.003303 0.08121 Everest 1830 6377276 6356075.4 300.8 0.003324 0.08147 Bessel 1841 6377397 6356078.96 299.1528 0.003343 0.08170 Struve 1860 6378298 6356657.14 294.73 0.003393 0.08231 Clarke 1866 6378206 6356583.8 294.98 0.00339 0.08227 Clarke 1880 6378249 6356514.87 293.465 0.003408 0.08248 Hayford (Int'l) 1924 6378388 6356911.95 297 0.003367 0.08199 Krassovsky 1940 6378245 6356863.02 298.3 0.003352 0.08181 GRS67 6378160 6356774.52 298.2472 0.003353 0.08182 GRS80 6378137 6356752.31 298.2572 0.003353 0.08182 WGS84 6378137 6356752.31 298.2572 0.003353 0.08182 Mars (MOLA) 3396200 3376200 169.81 0.005889 0.10837

Table 3. Data of some ellipsoids used in cartography. a: semi-major axis; b: semi-minor axis; 1/f: inverse flattening; f: flattening; e: eccentricity.

The fitting of the ellipsoid to the geoid is an important exercise of the physical geodesy. Prior to the usage of cosmic geodesy, this task could be accomplished by creating of geodetic or triangulation networks and (later) by their adjustment.