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Points and Coordinates

The point in this blog posting has been made previously but I believe that it merits repeating, namely that points in a point space (e.g. surface of the earth, or on a model of the surface of the earth) are not the same thing as the coordinates that may be assigned to them.

A physical point in a real world space can sometimes be designated (approximately) by marking the point in question with some kind of marker. You may have seen the bronze monuments for ground control points deployed by national or regional geological survey or mapping organizations. Such an approach is, of course, not possible for points that are in space, under the earth, or the ocean. In fact, in the majority of cases, we can only specify a point by constructing a coordinate reference system and then specifying the coordinates associated to the point through that coordinate reference system. Such a point specification is really only the specification of a point in a model point space that approximates a portion of the physical world.

Let us illustrate this with a simple example. We assume a point space model which is a vector space. Each point is specified by a vector from the origin to the point in question. We then select a basis for the vector space and this, in turn, defines the coordinates to be associated to the point.

points-and-coordinates

The Point O is easily related to the vector OP (the frame F’ defines the point space of interest).  Now given the basis {e1, e2, e3} we can write the vector OP as follows:

OP = a1.e1 + a2.e2 + a3.e3

where (a1, a2, a3) are real numbers.

The numbers a1,a2,a3 are called coordinates and (a1, a2, a3) is called a coordinate tuple.  Note that the tuple is NOT the point P, even if we could estimate the coordinate values exactly (zero error).  The coordinate tuple can ONLY be associated to the point P through the coordinate system, in this case the Frame F.

To make this point even more strongly, consider a vector space which is spanned by the basis “vectors” {1, x, x2, x3}.  This is the vector space of polynomials of degree 3.  Given a vector in this point space, i.e. a polynomial of degree <=3 , we can assign coordinate using the basis.  For example, the polynomial 2 + x-x2 can be wriiten 2.1 + 1.x -1.x2 + 0.x3, hence the coordinates are (2, 1, -1, 0).  Equally clearly, the coordinate tuple (2, 1, -1, 0) is NOT a polynomial.

So… bottom line…  Points are geometric entities that live in a point space (geometric space) that can model some aspect of the real world.  Coordinates (real or complex numbers in most cases) are assigned to a point through the use of a coordinate reference system.