Geoid

A geoid is a close representation, physical model, of the figure of the Earth. According to C.F. Gauss, it is the "mathematical figure of the Earth", in fact, of her gravity field. It is that equipotential surface (surface of fixed potential value) which coincides on average with mean sea level.

The geoid surface is more irregular that the ellipsoid of revolution often used to approximate the shape of the physical Earth, but considerably more smooth than the Earth's physical surface. While the latter has excursions of +8,000 m (Mount Everest) and -11,000 m (Marianas Trench), the geoid varies by only approx. ±100 m about the reference ellipsoid of revolution.

Because the force of gravity is everywhere perpendicular to the geoid (being an equipotential surface), sea water if left to itself would assume a surface equal to it. Similarly if sea water would be allowed to freely penetrate the continental masses, e.g., through tunnels. In reality it is not, of course; still, geodesists are able to derive the heights of continental points above this imaginary, but physically defined, surface by a technique called spirit levelling.

When travelling by ship, one does not notice the undulations of the geoid; the local vertical is always perpendicular to it, and the local horizon tangential to it. A GPS receiver on board may show the height variations relative to the (mathematically defined) reference ellipsoid, the centre of which coincides with the Earth's centre of mass, the centre of orbital motion of the GPS satellites.

Spherical harmonics are often used to approximate the shape of the geoid.