*4.9.2011*
To describe the figure of the earth and its gravity field is the aim of geodesy.

There exist various typs of mathematical, physical und topographical earth models
to describe the
figure, shape and size of the earth.

Three definitions of "

*figure of the earth*" [Moritz 1990, p. 1]:
A) The solid and liquid earth bounded by the *physical
earth's surface*, or *topographic surface*, which
is the surface which we see, on which we stand, walk, drive, and, occasionally, swim. It is highly
irregular, even after some obvious smoothing which is always necessary to make it a smooth
surface amenable to mathematical treatment, and also after some averaging with respect to time
since this surface undergoes temporal variations (on the order of decimeter or more) because of
tidal effects, etc.

B) The (part of the earth bounded by the) *geoid*,
which is a level surface coinciding (somewhat
loosely speaking) with the free surface of the oceans together with its continuation under the
continents. It is the geoid above which "heights above sea level" are measured. A level surface
is
everywhere horizontal, that is, perpendicular to the direction of the plumb line. Level surfaces are
surfaces of constant gravity potential *W*. , *W* = const. and the geoid is one of them, *W
= W**0*,
denoting the constant geoid potential by *W**0*. Again we are disregarding
temporal (tidal) variations.
Whereas the physical earth's surface, in its picturesque variety and beauty, is very irregular, the
geoid is smoother and subject to a mathematical equation, *W = W**0*; however, even the gravity
potential *W* is far from being a simple mathematical function. Therefor, the geoid is referred
to a
much regular, "normal", surface which approximates the geoid while being more regular in an
mathematical or physical sense. Thus we arrive at the concept of a

C) *Normal earth*, or *reference earth*, or *earth
model*. Mathematically the simplest model is an
*ellipsoid *of revolution, which therefor is practically almost exclusively used. Physically the
best
reference for describing the small, more or less elastic, temporal variations (free and forced
oscillations such as earth tides), is a hydrostatic *equilibrium figure*. Figures of hydrostatic
equilibrium for the earth are very close to ellipsoids, but do not exactly coincide with an ellipsoid
as
we shall have ample opportunity to see in this book. By the way, we are frequently not
distinguishing between a figure and the surface bounding it; this is costumary and should not
cause any confusion.