Ellipsoid represents an approximation of the shape of the Earth that does not account for variations caused by the Earths non-uniform density. Synonymous with spheroid and geoid. The Earth has a highly irregular and constantly changing surface. Models of the surface of the Earth are used in navigation, surveying, and mapping. Topographic and sea-level models attempt to model the physical variations of the surface, while gravity models and geoids are used to represent local variations in gravity that change the local definition of a level surface.

Reference ellipsoids are usually defined by semi-major (equatorial radius) and flattening (the relationship between equatorial and polar radii). Other reference ellipsoid parameters such as semi-minor axis (polar radius) and eccentricity can computed from these terms.

Tri-axial ellipsoid with distinct semi-axes a, b and c |
Ellipsoid of revolution (spheroid) with a pair of equal semi-axes (a) and a distinct third semi-axis (c) |

Geoid

Geoid is an ellipsoid with a highly irregular surface used to describe the shape of the Earth. Ellipsoid represents an ellipse rotated about its minor axis, while this figure deviates by less than 100m from geoid and is mathematically easier to use. Geoid models attempt to represent the surface of the entire Earth over both land and ocean as though the surface resulted from gravity alone.

Image showing the ellipsoid, geoid, and topographic surface (the landmass topography as well as the ocean bathymetry) |
The elevation (H) above the geoid, the ellipsoid height (h), and the geoid height (N), or undulation, above the ellipsoid |

Topographical Surface

The topographical surface of the Earth is the actual surface of the land and sea at some moment in time. Aircraft navigators have a special interest in maintaining a positive height vector above this surface.

Sea Level

Sea level is the average (methods and temporal spans vary) surface of the oceans. Tidal forces and gravity differences from location to location cause even this smoothed surface to vary over the globe by hundreds of meters.

Gravity Models

Gravity models attempt to describe in detail the variations in the gravity field. The importance of this effort is related to the idea of leveling. Plane and geodetic surveying uses the idea of a plane perpendicular to the gravity surface of the Earth, the direction perpendicular to a plumb bob pointing toward the center of mass of the Earth. Local variations in gravity, caused by variations in the earth's core and surface materials, cause this gravity surface to be irregular.