The ellipsoid provides a mathematical surface which closely resembles the earth's shape, and is used in surveying with plane coordinate systems and in geodesy. For a point, it also provides two dimensions, latitude and longitude, of the point's three-dimensional position. The third dimension, gravity, can be approximated by assigning an approximate gravitational field to the ellipsoid. This field should closely approximate the actual gravity field of the earth. In so defining this equipotential surface, the direction of a plumb line becomes perpendicular to the ellipsoid and thus is known as the normal gravity field. This surface is sometimes known as the level ellipsoid.
The normal figure of the earth should guarantee a
Theorem of Stokes: If a body of total mass M rotates with constant angular velocity, w (omega), about a fixed axis, and if S is a level surface of its gravity field enclosing the entire mass, then the gravity potential in the exterior space of S is uniquely determined by M, w, S, a, f.
The figure to the right shows a set of confocal ellipsoids with constant linear eccentricity
e.
The level surfaces coincide with surfaces of equal density and equal pressure.
The hydrostatic equilibrium of the normal figure is created by redistribution of the actual masses of the Earth.
According to theory, the closed representation of the normal gravity potential may be achieved in a system of ellipsoid coordinates.
The point P is then specified by ellipsoidal coordinates u (the
semi-minor axis), b (the reduced latitude), and l
(the geographic
longitude). From these terms, we can derive that the semi-major axis is
The transformation into Cartesian coordinates is
U is composed of gravitational potential and the potential of centrifugal acceleration and is computed as
U = V + F
The gravitational ellipsoid satisfies DV = 0 in the space exterior to the ellipsoid containing mass M. Imposing rotational symmetry on the normal gravity field, the nonzonal terms in the spherical harmonic expansion disappear. In exterior space, closed expression for the potential of normal gravity is
where q is
where a and b are the semi-major and semi-minor axes of the ellipsoid, respectively,
are the gravity at the equator and poles, respectively.
The equation for the normal gravity field above the ellipsoid is given by Heiskanen and Moritz (1967) as
where h denotes the height above the ellipsoid in meters.
where b is the gravity flattening,
, and e' is the second eccentricity. From the above two theorems, it can be seen that there are only four
independent quantities.
The potential of normal gravity is
For n=2 with the substitution of 
, we get
Solving for r and setting U = U0 gives the radius vector to the level ellipsoid where r = a, or
and the normal gravity with respect to r is
Substituting in a latitude of 0 and 90 degrees, the above equations yield a semi-major axis of a and equatorial gravity of ga. Also from these equations we see that the geometric flattening f is
is the ratio of centrifugal acceleration to the normal gravity at the equator. Using the above equations, an approximation for the theorem of Pizzetti is
and to Clairaut's theorem is
From these formulas we obtain Newton's gravity formula
Thus if two gravity values are known on the ellipsoid and at different geodetic latitudes, ga and b can be computed. Similarly with known values for a and angular velocity, the quantity m can be computed. The following series expansions are also useful
Notice that in the above equations, the definition for the normal gravity potential is defined by a function of a (the earth's semi-major axis), J2 (dynamic form factor), GM (the product of Newton's gravitational constant and the earth's mass), and the earth's angular velocity w. For GRS80 these constants are
The derived constants for GRS80 are
where h denotes the height above the ellipsoid.