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.