Modelling space for cartometric analysis: a grid-based approach

The paper describes research and preliminary results relating to an auxiliary data structure for modelling space in map generalisation research. The representation is based on a regular grid that is then aligned with linear cartographic features, such that the topology of the grid and the topology of the features is shared. The author first outlines the motivation for using the structure in relation to their research. How the model is generated is then described with reference to previous work by researchers in the field of computer graphics. Next, the author’s own experiences in adapting the model to the cartographic domain are discussed. Finally, preliminary results are presented and the outlook for continuing and future research described. Introduction In map generalisation a model of the map space is necessary for a number of reasons: to identify the spatial relationships that should be preserved at different map scales, to determine where and when feature conflicts are likely to occur due to a chosen symbolisation scheme, to evaluate extrinsic relationships between generalisation states and, to make space more manageable by discretisation. Various techniques exist to meet these needs. They differ in terms how they represent space and the spatial relationships they abstract. On the one hand, space-primary representations model space and its properties explicitly, usually using a geometric tessellation. Dutton (1998) and Li and Openshaw (1993) provide examples of these in map generalisation. On the other hand, object-primary models represent space as the relationships amongst geographic features (Molenaar, 1998). Examples of these are; minimal spanning trees (Regnauld, 2001), partitions (Brazile and Edwardes, 1999), Voronoi diagrams (Hangouët 2000) and Delaunay triangulations (Ruas 1998, Ware and Jones 1998), though the latter could be used in either sense. A principle of this research has been to consider generalisation as mediating a dynamic between communicating information on a map as symbols and describing it through spatial relationships (Sloman, 1985) at different scales. To model this dynamic, space can be represented explicitly as a deformable surface (Edwardes and Burghardt, 2004). However, a consideration of objects also needs to be made to allow the effect of different symbology specifications to be considered. To meet these twin goals a data-structure has been developed that embeds map features within a regular quadrilateral tessellation. This is based on research made in the field of computer graphics to embed sharp features in a subdivision surface (Biermann et al, 2001; Boier-Martin et al 2004). In the next section the paper describes the motivations for using such data structure. It then goes on to describe the data model itself, the methods used traverse it and the ways in which implementation in the cartographic domain differed from that described previously in the literature. Finally, the paper summarises how the data structure is to be used further in research. Motivation Maps are analogue representational media (Palmer, 1978) that communicate information in two ways; symbolically and spatially (Berendt et al, 1998). Symbolic information is represented explicitly by presenting attribute information of features pictographically. Spatial information is represented implicitly by using the spatial characteristics that constrain the map medium as analogous to those constraining geographic space (Sloman, 1985). Hence spatial relationships in geographic space can be said to be represented more or less “faithfully” in the map space. These two forms of information description do not generally sit happily together. When features on a map are depicted with symbols larger than their real world footprint, as inevitably happens at increasingly smaller map scales, the ability of the map to represent 8th ICA WORKSHOP on Generalisation and Multiple Representation, A Coruña, July 7-8th, 2005 2 spatial relationships is impacted. This is because the space available to describe spatial relationships has been reduced by the increased symbol footprint. Redressing this balance involves the map’s generalisation. On one side, this is by removing less important features and representing others at higher levels of abstraction. This rectifies the relationship by reducing the pressure of symbolic communication. On the other side, the space itself can be thought of as being generalised, either by the selection and emphasis of certain spatial relationships or by abstracting spatial properties using a less constraining metric (Habel, 2003; Barkowsky and Freska, 1997). The commitment that a map, or region of map, makes to describing spatial relationships at a particular level of geometric or topological abstraction has been termed representational commitment (Habel, 2003). For example, a large scale topographic map might commit to representing all distances and angles at the highest level of accuracy, a small scale visitors’ map might commit to showing distances more roughly and only 8-neighbourhood angles, a transport map might only commit to a faithful topological representation. Essentially then, it describes how the space and spatial properties are made more general rather than the semantics or form of individual or groups of features. The dynamic between symbolisation and spatial relationships can be modeled in two different ways according to whether space is viewed as relative or absolute. Viewing space as absolute, a symbol is thought of as occupying a region of space, or consuming “white space”. Viewing space as relative, the impact of symbology is to distort space around a feature. Figure 1. illustrates these perspectives for a set of point symbols viewed in terms of absolute and relative space. The grid in the background describes the underlying space as a coordinate system in which points are located. If we consider only those points of space where the grid lines cross, we can see that for the absolute view there are regions of space covered by the symbol. For the relative view these same regions are preserved because they have been pushed away from the symbol. Figure 1. Distortions of space and spatial relations caused by symbology, in absolute (left) and relative (right)