notes on space

representations of physical space

The fact that we are dealing with electronic spaces (avoiding the word virtual as we are using that for something else) that are representations of some part of physical space is really important. It constrains the sorts of locational systems and relationships that are meaningful. However, it there is not a simple relationship between the real and the represented. This is because the relationship between real and represented is an existential one not always a realised one. We know that the interior of the car has a physical location in space and as such has constraints of physicality in its coordinate system and objects. However, we may have no idea exactly where in physical space it is!

This relationship to real space actually shows up in the limits on the interrelationship of measured space as in some way the real space is as virtual to the electronic space as the electronic is to the physical.

three kinds of location system for measured space

Although we talked about the measured spaces having their own coordinate system, we really mean something more like location system as not all are based on coordinates as such.

There seam to be three main classes of locational information within spaces:

coordinates: some sort of explicit dimensional representation, not necessarily orthogonal and not necessarily Euclidean (e.g. polar, spherical, OS grid). Note that these coordinate systems will typically have only validity over a finite space (OS grid meaningless over North America) and have an even smaller space over which they are expected to have values (coordinate system based on north-east corner of my room would have validity over most of campus, but would only be expected to be used in and near the room - and the latter is very important as we do need to use it in the doorway area).
zonal: objects are located within some zone - IR sensor in a room, mobile phone cells, wavelan areas in Guide. The simplest form of zonal space is a single proximity sensor (which is also the simplest form of relational space below)
relational: where objects report some form of relative location information - device A is close to device B, or device A is about 3 metres northeast of device B. This leads to a space which can be regarded as a graph of all the possible sensors/devices with knowledge of some arcs (approximate distance, possibly direction) and sometimes knowledge between arcs (angle between devices). It is interesting to note that most traditional cartography starts with precisely this information.

Note that these relate principally to the way in which objects are represented within the space. So that even in a zonal space, the zones may have known physical extent within some coordinate system.

Just to confuse things still further. Sensors may not return location, but may still fit within a coordinate space. A compass clearly is related to a magnetic map of the earth (in the small Cartesian space), but delivers orientation only.

subspaces

Following the mathematician's habit of always looking for structure-substructure relationships it is clear that there are some here. In coordinate spaces virtually any region may (with some transformation of coordinates) be regarded as a subspace. Similarly the local coordinate space of an individual device in the 'relational' system may be regarded as either a zonal or coordinate space in its own right.

There may be some advantage in being clear about this as it stops angst about whether the 'space' is one thing or another. We just say both!

Operationally, in any application we need to decide what we represent as spaces (although even then things may be used in different ways at different times), and also what are 'heavy weight' spaces with permanent representations and what are light weight ones used when appropriate. It may be interesting to ask this question of the spaces used in the different real Equator systems.

The subspace relationship is complicated by various factors including the dynamic relationship between spaces and location systems.

In fact, as in other areas of mathematics, the subspace relationship is a special case of general relationships between spaces, so we can worry about it after doing the latter!

relationships

Again these seem to be of three classes (is it just the mystical nature of threes?):

topological: separate, overlaps, touches, contained. As we've said there are a standard set of these and relationships between lines and regions that are used for qualitative reasoning in AI. One paper that seems to go into these issues in some detail (Grigni 1995) and also one by one of the authors that discusses the issues of qualitative direction (Papadias 1996). There is a similar set of temporal relationships (before, overlaps end, etc.) defined by Allen (1991). Note that spatial overlaps and touches have subcases depending on whether there is a single overlap or lots of parts. The latter is not possible if both regions are convex.
boundary: used in the sense in Koliva (1999) - how are the spaces linked and how do trajectories or boundary points map onto one another.
mapping: where there is a more significant overlap, how do the location systems map onto one another

For the topological relationships the issue of the extent of the 'normal' area of the space becomes important. We are not talking about the coordinate system intersecting but the natural regions that the spaces represent. That is we have two very distinct ontological categories:

space as extent within another (perhaps not measured)
space as potential to measure locations

Topological and existence of boundaries relates primarily to the former. Mapping and mapping at boundaries is largely about the latter.