In geodesy, precise computations are made
an ellipsoid. Unfortunately,
measurements made on the earth's surface are not made on a mathematical ellipsoid. The surface is
called a geoid.
a. General. The geoid is the surface which the ocean waters of the earth would conform to if they
were free to adjust to the forces acting on them. The ocean waters would conform to the surface under
the continents if allowed to flow freely through sea-level canals. The forces acting on the oceans
include the actual attraction of the earth's mass, attractions due to density differences in the earth's crust,
and centrifugal force due to the earth's rotation. The component of centrifugal force opposing the
attraction of gravity is greater at the equator than near the poles. Since terrain features such as
mountains, valleys, and ocean islands exert gravity forces, they also affect the shape of the geoid. The
geoid can also be defined as the actual shape of a surface at which the gravity potential is the same.
While this surface is smoother than the topographic surface, the geoid still has bumps and hollows.
b. Characteristics. There are two very important characteristics of the geoid. First, the gravity
potential in the geoid is the same everywhere, and the direction of gravity is perpendicular to the geoid.
Second, whenever you use an optical instrument with level bubbles, properly adjusted, the vertical axis
of the instrument should coincide with the direction of gravity and is, therefore, perpendicular to the
geoid. The second factor is very important because the attraction of gravity is shown by the direction of
the plumb lines.
c. Deflection of the Vertical. Since the ellipsoid is a regular surface and the geoid is irregular, the
two surfaces do not coincide. However, they do intersect, forming an angle between the two surfaces.
Geometry has taught us that the angle between the two surfaces is also the angle formed between the
perpendicular to the ellipsoid and the geoid plumb line. This angle is called the deflection of the
vertical. The word normal is sometimes used to describe the perpendicular to the ellipsoid and the geoid
since a normal is a line perpendicular to the tangent at a curve. In less precise language, this is known as
perpendicular to a curve (Figure 1-4).
d. Separations. The separations between the geoid and the ellipsoid are called undulations of the
geoid, geoid separations, or geoid heights. The geoid height reveals the extent to which an ellipsoid fits
the geoid and thus helps to determine the bestfitting ellipsoid. For purposes of illustration, the
undulations of the geoid in Figure 1-4 and other figures are highly exaggerated.