3 Location models

Various models of space do always implicitly underlie location infrastructures , [2], [9], yet they are rarely set in a proper theoretical framework by going back to the basics of what a space can be in pure mathematics. Though these models are purely abstract, they will be used in association with a particular location-sensing or ranging technology, from which they retain only relevant characteristics that can be mapped to a corresponding notion of space.

They will serve to characterise both loci and locants.

3.1 Set- theoretic

In this strictly minimal model, no metric information whatever about the shape of a locus or the precise position of a locant is assumed to be available. All that can be known is the presence/absence of a locant in a locus, modelled as an element belonging or not to a subset of space.

3.2 Topological

In these models, location properties are defined on a point-based abstraction of a locant, a locus corresponding (loosely) to the mathematical concept of neighbourhood in a topological space. Location is defined relative to a point, with special properties attributed to a given neighbourhood of this point that may not be fully characterisable in metric terms. The spatial continuity property inherent in this topological notion of neighbourhood may itself be useful in some cases, yet places strong constraints on the underlying physical location technology if taken absolutely . Other purely topological notions such as simple connectedness (absence of "holes" in an open set) may also be relevant.

Quite different in mathematical terms, yet related for being an alternative potential abstraction of RF location technologies, would be a location model based on fuzzy set theory, on which we will not elaborate.

3.3 Metric

These models assume the minimal possibility to quantify a distance between points. The physical size of a locant itself may also be metrically bounded, making it possible to go beyond the point-based abstraction of a locant. A locus may also be a similarly bounded region.

3.4 Affine and affine-euclidean

Affine and affine-euclidean spaces are a richer and practically more significant case of metric space, where it is assumed that absolute location information of a locant may be defined with respect to a suitable coordinate system. Geodetic coordinates are practically the most important example, with three main classes of coordinate systems may be used:

Geocentric cartesian coordinates.
Polar geographic coordinates (latitude, longitude, elevation).
Planar projection coordinates.

A locus may be an arbitrary region of space defined with respect to such a coordinate reference system, while richer properties may also be attached to a locant, such as its precise geometrical shape or its orientation.

3.5 Locant graphs

A completely different mathematical species, where the vertices (nodes) of the graph are locants, and loci may correspond to various subsets of this graph (e.g. paths, walks, cycles, or arbitrary subgraphs). This may be used in conjunction with a metric model (yielding a valuated graph) with a location technology that is purely relative and bilateral between objects themselves , rather than related to more or less fixed loci. Such a model is also compatible with a purely distributed management of location, eschewing any fixed infrastructure for both the location devices and the software location infrastructure: each locant may manage its own neighbourhood of objects, actually those objects for which it has a bilateral relative location information.

General graphs may be used to model all kinds of bilateral or multilateral relationships between their nodes, besides relative location as put forward here. Of course, all purely network-based models of communication already use such models, and this is not what we are attempting to reinvent Other relationships may be described for which location may still be used as a metaphor, by extracting topological properties from the graph itself. Semantic relationships may, for example, be described in a structural way, enabling inverse location queries similar to those that may be made in a physical location model.

3.6 Loci graphs

By contrast to the previous case, vertices of these graphs are the loci and not the locants. These models may be seen as enrichments of a set-based topological model, modelling not only loci as subsets or neighbourhoods of space but also their structural relationships. This is implicitly the kind of model underlying the cell pavings used in cellular networks, where adjacency relationships between cells are used for the handover of a locatable entity from one cell to another. Adjacency is but one particular case of relationship, and many other kinds of region structurations could be modelled: a hierarchical model of space (loosely underlying most of the semantic models used in directories) is another obvious case.

3.7 Semantic/symbolic models

In these models, spatial location may be defined implicitly rather than explicitly, by reference to more or less abstract concepts relevant to a given universe of discourse , i.e. a semantic frame of reference. Loci may correspond to such divisions as streets , precincts, municipalities, regions , states, as used in regular directories. At a lower level and a smaller scale, buildings , floors, rooms, or even shelves in a cupboard, cells on a given shelf, etc could be used as loci providing a spatial reference for all kinds or locants, which will themselves be defined by some supposedly well-known characterisation rather than their physical properties. These loci may themselves be mapped to one of the lower-level models described before, i.e. either a hierarchical graph model, a topological model or a metric model. These characterisations may be compounded with other non-univocal high-level properties associated with a particular locus. These may correspond to a typing or profiling of a particular locus (e.g. authorisation, security constraints, electromagnetic compatibility).

Though these models are purely abstract, they will be used in association with a particular location-sensing or ranging technology [8], from which they retain only relevant characteristics that are mapped to their model of space. They may be used to characterise both loci and locants, as detailed in the following table.

Table 7.1: Location models
Model	Information provided	Locus concept	Locant concept
Set theory	Presence/ absence of a locant in a locus	A subset of space	Usually abstracted to a point
Fuzzy set theory	A [0 “1]degree of presence of a locant in a locus	A fuzzy subset of space	Usually abstracted to a point
Topological space ^[2]	Presence/ absence in a neighbourhood	An open set/ neighbourhood	A point, or more generally a closed set
Metric space	Relative distance to locant	An open ball	A point or closed ball
Affine	Position of locant w.r.t. absolute coordinate reference system	Point or region of space, defined from absolute coordinates	Point or region of space, defined from relative coordinates
Affine-euclidean	Position+orientation “ >mapping (translation+rotation) from absolute to relative coordinates	Point or region of space, defined from absolute coordinates	Point or region of space, defined from relative coordinates
Locus graph	Path to locus	Node of the graph & locus previously defined	A locant for the underlying model
Locant graph	Path to locant	A sub-graph (e.g. a path, a tree, a cycle))	A node (vertex) of the graph
Semantic/symbolic models	Symbolic/semantic mapping to underlying models	Semantics of a locus defined in the previous models	Semantics of a locant defined in the previous models
^[2] The spatial continuity property inherent in the topological notion of neighbourhood may itself be useful in some cases, yet places strong constraints on the underlying physical location technology. Other purely topological notions such as simple connectedness (absence of "holes" in an open set) may also be relevant.

These models are to be used in combination, each providing a different kind of information based on a different conceptual view of location.