||It might help to have a more general understanding of coordinate systems and projections, because that will put the earth-centric situation into a larger perspective. I would like to share some conventional mathematical definitions, because they offer the potential to clarify things. The following discussion applies to arbitrary surfaces as well as higher-dimensional spaces, but I will use language specific to the problem of mapping the earth.
There are three mathematical elements to a system of geographic reference: a projection and *two* coordinate systems. After defining and describing these, I will provide a few simple examples.
(i) There is a mathematical transformation, the “projection,” between an idealized surface representing the earth (the ellipsoid) and the Cartesian plane. This transformation does not require either surface to have coordinates. We are all aware that some transformations can be defined geometrically, for instance (stereographic, equal-area cylindrical, and conic projections come to mind); these provide examples where coordinates are not needed. In most situations, though, a simple geometric definition is not possible (or has not been discovered).
(ii) There are coordinates for the earth's surface. There are two fundamentally different methods used. One is _intrinsic_: a unique point on the earth’s surface is associated to each pair of real numbers that are “valid” coordinates. Key things to note about this definition include
(a) The *inverses* of the coordinate functions are a pair of real-valued functions defined on the surface. They have to satisfy certain topological and differential properties that I won't go into here; suffice it to say that they must be "well behaved."
(b) The coordinate functions do not have to cover the entire surface. This implies any coordinate system might be limited in coverage. The most common example is lat-lon. Longitude and latitude are the *inverses* of the coordinates, because they are functions assigning coordinates to points on the ellipsoid, rather than functions assigning points to coordinates. Longitude is *undefined* at the poles. This causes problems when people forget this and try to use lat-lon for calculations near the poles. Another common example is a “local” or “site-specific” coordinate system set up in some small region; there never is any intention to use such systems to reference points over the entire earth.
The usual intrinsic coordinates for an ellipsoid are based on lat-lon, but many others are also in common use. For these functions to be useful, you need a _datum_. This is the information needed to match a physical location to any pair of coordinates. It usually includes an explicit model of the ellipsoid, an origin (a distinguished point on the ellipsoid), and a distinguished direction at the origin. The need for a datum is clear mathematically: the coordinates give us a point on an abstract ellipsoid, but how exactly does that ellipsoid correspond to the earth itself? That is a matter for a surveyor to determine.
The other kind of coordinates are _extrinsic_: they are coordinates for a region of space surrounding the earth. Unlike property (b) of intrinsic coordinate, extrinsic coordinates usually cover every point of the earth. Again a datum--albeit of a different sort--is needed to associate coordinates with physical points. The datum consists of an origin, often taken to be the center of the earth, and three vectors at that origin defining three distinct directions (and distances), often taken to be rotating with the earth. What is interesting about these is that most coordinate combinations do not correspond to points on the earth’s surface, but rather to points above or below it. Given specific coordinate values (x,y,z), it is not usually easy to tell whether they correspond to a point on the earth’s surface or not: you have to do a calculation.
(iii) There are coordinates in the Cartesian plane. Many types of system exist: rectangular, polar, confocal elliptical, etc. In addition to the type of coordinates, these come with a datum of their own: the origin, orientation, and scale. Think of the plane as a blank sheet of paper; the datum is established when you draw two coordinate axes and mark off scales on them (for rectangular coordinates) or when you draw an origin and a direction at that origin denoting the zero angle (for polar coordinates) and mark off a scale along that direction.
There are interactions among these two coordinate systems and the projection. For example, rescaling the coordinates on the earth and rescaling the coordinates on the plane can accomplish the same thing, mathematically. This can cause some confusion.
As another example, the "false easting" or "false northing" sometimes applied to location data are changes of datum *in the plane*, not on the ellipsoid. This shows as clearly as anything that there really are two coordinate systems involved in any useful map of the earth.
Coordinates (ii) and (iii) are needed for several reasons. In addition to providing names for points on the earth and the map, respectively, they also provide a mechanism to write down projections in terms of formulas. Obviously we cannot write a formula for a projection until we have numbers for referring both to points on the earth's surface and points on the map.
Here is an example. The ESRI 'geographic' coordinate system (i) is a Plate Carree projection, (ii) uses lat-lon in decimal degrees [the coordinates] relative to the Greenwich prime meridian on a spherical earth model [the datum], and (iii) uses rectangular coordinates in the Cartesian plane. If we let (lambda, phi) be the lon-lat coordinates on the earth and (x, y) be the planar coordinates, then the projection has an extremely simple definition: y = phi and x = lambda. By equating the coordinates in this way we can be misled into thinking there is 'no projection,' but that evidently is not true.
As another example, let's track the process of locating a physical point on the earth based on map coordinates. Beginning with coordinates (x, y), one uses the map's datum and coordinate system to find a point on the map whose coordinates are (x, y). The projection formula (or rather its inverse) associates that point with a unique set of earth coordinates. The earth's datum and coordinate system tell us how to use those coordinates to go to a definite point on the earth's surface.
The datums associate the abstract mathematical entities--an ellipsoid and a plane--with physical points on the earth and the map, respectively. Everything else--the two coordinate systems and the projection--is purely mathematical.
I think it is helpful to bear in mind that every system of geographic reference necessarily involves a projection, *two* coordinate systems, and *two* datums. All these ingredients are always present, whether they are made explicit or not, and regardless of what terminology your GIS software or data vendor decides to use at the moment.