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*

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.