28 Feb 2008 - What are coordinates?

When we see the location of a point expressed as (46.76, 120) what does that mean?  How do we know how to interpret those numbers?

Coordinates are a well defined notion from Mathematics that effectively derives from theory of vector spaces, and which is then extended to more general spaces including manifolds (like the surface of the earth), vector and fibre bundles (like linear base referencing systems).  Fortunately we can understand most of what we need simply by dealing with the real vector space case, as the other cases all reduce to this model locally.

Consider the assignment of coordinates to points in a room as shown in the figure below.

Figure 1.  Points in a Room modeled as a real vector space

In the vector space approach we model a point as the end point of a vector from some selected origin, perhaps the point O1 as in the figure.  We can then define a basis for the vector space model of the room, by assigning basis vectors.  We might do this for example by having an e1 basis vector along the intersection of the wall and the floor from O1, and e2 in a similar manner.  e3 is then defined as a vector pointing away from the floor and collinear with the intersection of the walls and pointing away from O1.  The three basis vectors are shown in Figure 2.

Figure 2.  Selecting a basis for the vector space

If we take any point P in the room this defines a vector from O1 to P (O1P).  We can then write this vector as a linear combination:

O1P = a.e1 + b.e2 + c.e3

Where a, b, and c are real numbers.

The ordered tuple of numbers (a,b,c)  are called the coordinates of the point P with respect to the basis e1, e2 and e3 as shown in Figure 2.

Now without the definition of the basis NO meaning can be assigned to the tuple (a,b,c).  Note that we can easily change the basis vectors as the selection in Figure 2. Has no magic importance.  For example we choose these vectors as in Figure 3.

Figure 3.  Selecting another basis for the vector space

Note that the expression for O1P will not change and it is only by reference to the figure – i.e. the mapping to the real world (our room) that we can differentiate one basis from another.

Each basis defines a coordinate reference system.

Now to generalize this to more complex surfaces (like the surface of the earth) goes beyond this simple discussion, however, the model is exactly the same.

Given a coordinate tuple we cannot interpret it without reference to a coordinate reference system.  Clearly, if two different people or organizations interpret (a,b,c)  using different coordinate reference systems they will understand things very differently and this can lead to expensive if not tragic results.  It is thus essential that the coordinate reference system used to interpret a coordinate tuple be completely unambiguous.  It is for this reason that the GML encoding for point contains the reference srsName attribute.
    <gml:Point srsName = “urn:ogc:def:crs:EPSG:4326”>
        <gml:pos>-45.23  110.7</gml:pos>
    </gml:Point>

There can be no ambiguity in this case.  To resolve these URN references check out the CRS registry of the OGP at http://www.epsg-registry.org.

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