PART A - FIGURES OF THE EARTH
1-1. Three Figures of the Earth. Before any type of measurement can take place, the surface on
which we measure must be defined. Generally, we can assume the following three figures of the earth:
topographic, mathematic, and geoidal.
a. Topographic. The surface most apparent is the actual topographic surface of the earth. This
includes the mountains, valleys, and other continental and oceanic forms. The surveyor makes the
actual measurements on these surfaces, but because of the irregularities of the land, this figure is not
suitable for mathematical computations. This surface generally concerns the topographer and the
hydrographer but interests the geodesist only with regard to the effect of the terrain features on gravity.
b. Mathematic. It is convenient to adopt a simple mathematical surface, resembling the actual
earth, to permit simplified computations of positions on the earth's surface. We might select a simple
sphere; however, the sphere is only a rough approximation of the true figure of the earth. We can and do
use a spherical form to solve most astronomical problems and for navigation. The sphere is used to
represent the earth because it is a simple surface that is easy to deal with mathematically.
c. Geoidal. The geoid is the equipotential surface within or around the earth where the plumb line
is perpendicular to each point on the surface. The geoid is considered a mean-sea-level (MSL) surface
that is extended continuously through the continents. The geoidal surface is irregular due to mass
excesses and deficiencies with the earth. The figure of the earth is considered a sea-level surface that
extends continuously through the continents. The geoid (which is obtained from observed deflections of
the vertical) is the reference surface for astronomical observations and geodetic leveling. The geoidal
surface is the reference system for orthometric heights.
1-2. Ellipsoid. Refer to Table 1-1 for ellipsoid data. The shape of the earth is more precisely
represented mathematically by an ellipsoid of revolution, which is made by rotating an ellipse around its
minor axis. The radius of the equator usually designates the size of an ellipsoid. The radius is called the
semimajor axis (Figure 1-1). The shape of the ellipsoid is given by a flattening, which indicates how
well an ellipsoid approaches the shape of a sphere. Figure 1-2, page 1-3, shows the flattening of various
figures. The ellipsoid, which represents the earth very closely, approaches a sphere since it has a
flattening of 1/300. An ellipse with such a small flattening is almost a perfect circle. Spheroid and
ellipsoid of revolution are accurate terms; however, ellipsoid has become the more accepted term.