## Overview

A great deal of discussion has taken place in the OGC and ISO TC/211, over the past several years, with respect to coordinate reference systems. Much of this discussion has been confusing, and many of the ideas discussed are subtle and hard to express. This has resulted in less generality than one might have liked for abstract specifications such as ISO 19111. Attempts to apply encodings and models, which depending on these abstract specifications, have subsequently run into trouble as a result.

One of the most poorly understood concepts has been that of datum. In common parlance, and in land surveying, a datum is a reference point or reference surface. Sometimes this is also interpreted as the zero point for a measurement, such as the freezing point of water being a datum for a Celsius thermometer. These definitions, while correct in themselves, provide a poor basis for generalization, and as a result progress on a general treatment for coordinates and coordinate reference systems in OGC and ISO has suffered.

We propose here a notion of datum which we believe provides the required degree of generality and allows the transportation, engineering, physics, and geodesy use cases to be integrated together.

## What are coordinates?

The most general way to think of coordinates is to think of a Coordinate Vector Space. This is set of n-tuples of real numbers (other fields could be used) in which addition of these tuples is defined by component wise addition, and in which multiplication by scalars is defined by component wise multiplication. The identity element in such a set is (0, 0, … 0) and it is clear that any element (a1, a2, … an) has an additive inverse, namely (-a1, -a2, … -an). Hence the set of n-tuples forms a vector space which we call the Coordinate Vector Space. The numbers a1, a2, etc., appearing in these tuples, are coordinates.

## Assigning coordinates to points

From the point of view of geometry, points are more primitive than coordinates, and one can talk about points in a geometric set without having any notion of coordinates. In fact, a point could have multiple coordinate representations. Under certain assumptions about the geometric set, it does make sense to talk of coordinates, and we can think of coordinate reference systems as enabling the assignment of coordinates to a point. Note that the geometric set can be quite abstract and need not represent physical space; we could have a set of points where each point is a branch voltage or the concentration of a chemical species.

If we are to talk about assigning coordinates to points, we first have to be clear what set of points we are talking about. This means making a model of the real world “region” of interest, and then using mathematics to construct coordinates from that model.

If we model our physical “region” as a vector space, meaning the points of the “process” are vectors, then we can assign coordinates to these points simply by choosing a basis {**e _{1}**

_{,}

**e**

_{2}_{,}…

**e**

_{n}_{,}} for the vector space, as we know, any vector

**v**can be uniquely expressed as

**v**= a

_{1}.

**e**+ a

_{1 }_{2}.

**e**

_{2}_{ }+ … + a

_{n}.

**e**

_{n}_{, }hence the coordinates of the point P (corresponding to the endpoint of the vector

**v**=

**OP**) are (a

_{1,}a

_{2, }… a

_{n}).

## We can’t model everything as a vector space

It should be clear that not everything of interest can be modeled as a vector space. Consider, for example, the set of possible orientations of a rigid body. Each orientation of the body can be seen as the result of a rotation, so we can assign rotations to orientations. Think of a partly symmetric body like a fixed wing aircraft. If we perform two rotations in succession (e.g. 1^{st} about the longitudinal axis of symmetry and the 2^{nd} about a normal to this axis) we can easily see that the result of these two rotations can be different (http://farside.ph.utexas.edu/teaching/301/lectures/node100.html ^{[2]}). If the two rotations were then thought of as quantities **a** and **b** then this says that **a+b** is not the same as **b+a.** This means that the set of rotations (orientations) does NOT form a vector space, and rotations are NOT vectors, even though such a quantity can be expressed in terms of a well defined magnitude and direction.

If we think about points on the surface of the Earth (which is more or less a ball), we can come to a similar conclusion. Consider any such point and multiply it by a scale factor other than 1. One can readily see that the new point will NOT be on the surface of the Earth. Vector spaces require that the set of vectors be closed under multiplication by scalars and clearly this is not the case.

## What to do? What about locally?

Without getting too technical, in many cases it is possible to model a “region” locally as a vector space. This is the case for orientations of a rigid body, and for positions on the Earth. When we say that the Earth is not “flat” – and that one cannot flatten the earth without tearing it (you may remember this from grade school) – what we are saying is that a model of the Earth is fundamentally different, as a surface, than a plane (which can be modeled as a vector space). To address this difference, we use projections to locally map the Earth (model) to a plane. For example, we may model the Earth as a sphere, put that sphere inside a cylinder, and project points on the sphere to points on the cylinder (e.g. draw rays from the center of the sphere through a point on the Earth to intersect a point on the cylinder). When we unroll the cylinder, we get a flat representation of the Earth. It is clear, however, that this model does not include: (1) the poles (these are mapped to the bounding lines), and (2) the points on the line where we broke the cylinder. The target of the projection is a 2D plane, which CAN be modeled as a vector space and thus allow us to use our scheme above to construct coordinates, but we need more than ONE of these projections to cover the entire Earth – it CANNOT be done with ONE only. It does, however, mean that the Earth (model) can be modeled locally as a vector space and that, locally, we can assign coordinates (given a projection), and that these coordinates are relative to a choice of basis – e.g. latitude and longitude. So in general we must think of coordinates in terms of vector spaces in ALL cases.

## So what is a datum?

At the outset of this discussion, we said that a datum in common parlance was a reference surface or zero point. Now we will introduce a different idea, namely, that the datum is the “space” of points in our model of the real world process. It is the “space” of points to which we then assign coordinates through the selection of a coordinate reference system.

If you think of the reference surface viewpoint on datums for a moment you can see that it is not really inconsistent with the view presented here. If we state that a datum for temperature measurement is the freezing point of water, implicit in this statement is the assumption that other temperature values can be represented as points on a line, in effect mapping temperature values into the Real number line which is, of course, a 1D vector space.

Considered in this way, a datum is the “surface” model for the application “region” to which we assign coordinates through the selection of a coordinate reference system.

We can then see that a datum for the positions and orientations of a rigid body is specified by a product of a geometric (affine) space of positions and a space of 3-frames attached at every point of this affine space. A point in this space consists then of a position point P, and a frame F_{P} attached at P.

^{[4]}

The Geodetic datum case also falls under the same concept. A geodetic datum is, in effect, an earth model which has been “fit” to the earth by the selection of a model surface (e.g. ellipsoid), and the specification of the model surface parameters (e.g. sphere radius), and sufficient parameters (usually selection of real world points) to enable the 1:1 identification of points on the earth with points in the model. This “fitted model” is a 2-dimensional manifold, and using this manifold and suitable “projections” we can construct a variety of coordinate systems.

The notion of datum as the point space for our models of the physical world thus serves to unify the treatment of coordinates and coordinate reference systems over a multiplicity of domains.