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.

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.

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