Formalization of families of categorical coverages

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Abstract
Categorical coverages are popular for the presentation of thematic data in cartographic form and have gained widespread use. Routines for their preparation are included in most GIS software. Categorical coverages are simple to implement and easy to understand. The popularity of the categorical coverage is closely related to the intuitive rules that determine its behaviour. This paper presents a formalization of these rules based on the concepts of a partition of space and the refinement of such partitions of space. A family of categorical coverages is defined as all categorical coverages that can be produced from a data set through aggregations of categories. It is shown that categorical coverages are partially ordered by refinement of partitions. This ordering is preserved in the mapping from the ordering of the partition of the attribute domain to the partition of space. This formalization is not only useful for implementors of GIS, but the list of possible operations and their properties is the base for the design of a user interface to produce and manipulate categorical coverages. The thematic definition of categorical coverages is directly related to the cartographic rendering. Changes in the thematic definition, i.e., changes of the map legend, can be automatically propagated to the rendering. Limitation of cartographic rendering, e.g., resolution of the screen, limits the presentation of the result. These influences from the rendering cannot automatically be translated into changes in the subdivisions of the categories. This may explain why good graphical presentation is still an art and the production of thematic maps is not fully automated.

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