The Earth's shape is the result of various forces. Among these forces, the most important on the planetary scale are gravity and centrifugal force. The result of these two forces is what gives the orientation of the plumb line for any given point on Earth. That is the orientation of the local vertical. A continuous surface perpendicular to the local vertical is an equipotential surface. Among the equipotential surfaces one can define a specific one : the geoid. The geoid has a surface corresponding to mean sea level, assumed to extend at the same level under the continents. This geoid differs from the real terrestrial surface since it was modelled by different geodynamic factors.

In order to describe one's position on Earth, a person needs a simple surface which can be described mathematically. Since the 18th century, measurements have existed which prove that the Earth does not look like a perfect sphere, but rather is flattened at the poles. The appropriate mathematical description is an ellipsoid of revolution. An ellipsoid of revolution is produced by rotating an ellipse about one of its two axes. One defines the ellipsoid by describing the ellipse and the axis of rotation. The ellipsoid of the Earth has as axis of rotation the small axis of an ellipse whose precise dimensions are established at international scientific congresses. Technological progress has made it possible to define these dimensions with ever-increasing accuracy.

Depending on the country or the continent where one is working, one chooses a geodetic datum which minimises the deviation from the geoid. The geodetic data are different each time, and their centre does not coincide with the Earth's centre of mass.

As spatial measurements developed, it was necessary to define global geocentric reference systems. The most widely used system is the WGS84 (World Geodetic System 1984). This is the reference system used for the GPS.

Once the datum is defined, one can describe a point on the surface of the Earth in either of two ways : by employing geocentric Cartesian co-ordinates, or by using geographical co-ordinates.
Cartographers and navigators have long since grow accustomed to locating a point with the aid of geographical co-ordinates expressed in longitude l and latitude j.

The longitude l is the dihedral angle formed by the meridian of a point with the meridian of Greenwich and counted from this original meridian, positively towards the west and negatively towards the east.

The latitude j of a point is the angle formed by the vertical of the point with the plane of the equator. Latitudes are counted from the equator, positively towards the north and negatively towards the south.

The longitude and the latitude make it possible to define a point on the ellipsoid. To situate a given point, one must also define its position with a third parameter, called the ellipsoidal height h.

One can also describe the location of a point by its geocentric Cartesian co-ordinates. Such a positioning system locates the points in X, Y and Z with the centre of the ellipsoid as origin.

Cartography

A map is a plane representation of the terrestrial surface.
Because it is impossible to develop an ellipsoid in a plane surface, one must transpose the points on the ellipsoid onto this plane surface via a projection. The points of the ellipsoid in longitude l and latitude j are transposed into points x and y of the map's plane surface.

x = f (l,j)
y = g (l,j)

This transposition inevitably introduces distortions. Many systems of projections have already been imagined, and more are still being conceived today. In principle, a projection is either conformal, equivalent or aphylactic.

A conformal projection is one which preserves the original values of the angles. An equivalent projection is one which preserves the surface relations between the ellipsoid and the plane. The transposition cannot be simultaneously conformal and equivalent. Projections which are neither conformal nor equivalent are called aphylactic. Among the latter, one can distinguish the equidistant projections, which maintain the scale in one direction.

In some transpositions of co-ordinates of the ellipsoid towards the plane, one makes use of developable surfaces. Among these transpositions, one distinguishes among cylindrical projections, conical projections and azimuthal projections, depending on whether the involute surface is a cylinder, a cone or a plane. These involute surfaces can be tangent or secant to the ellipsoid and can have different aspects. One distinguishes between direct (or normal) aspect, transverse aspect and oblique aspect. Moreover, there are an infinite number of ways to project a point of the ellipsoid onto the involute surface.