This paper provides a new representation for fields (continuous surfaces) in Geographical Information Systems (GIS), based on the notion of spatial functions and their combinations. Following Tomlin’s (1990) Map Algebra, the term “Map Calculus” is used for this new representation. In Map Calculus, GIS layers are stored as functions, and new layers can be created by combinations of other functions. This paper explains the principles of Map Calculus and demonstrates the creation of function-based layers and their supporting management mechanism. The proposal is based on Church’s (1941) Lambda Calculus and elements of functional computer languages (such as Lisp or Scheme).

about geographic space and geographic phenomena. Virtually all aspects of a GIS have inherent spatial, graphical, and visual characteristics. While the database and analytical aspects of GIS have enjoyed considerable advancement in recent areas, a user's access to and interaction with spatial information has not. For such a highly visual system, GIS is often characterized by its distinctly non-visual user interfaces, where command-line and window-icon-menu-pointer (WIMP) user interfaces are most common, whereas visual, direct-manipulation user interfaces are rare. Direct-manipulation user interfaces based on metaphor offer increased usability for GIS. This paper extends the Geographer's Desktop, an innovative direct-manipulation environment for viewing data in a GIS, by integrating a new method for GIS Map Algebra operations. Used by planners, geographers, and other spatial scientists, Map Algebra facilitates the analysis of geographic phenomena. Historically, Map Algebra was performed manually by overlaying thematic map layers, a process that offers a rich source domain for user interface metaphors. Visual Map Algebra is a direct-manipulation query language that allows users to construct arbitrarily complex combinations of map layers by stacking iconic representations of thematic map layers onto an interface object called the computational platform. Users visualize such calculated map layers by moving them onto an interface object called the viewing platform that manages cartographic display parameters and is associated with a viewing window. Visual Map Algebra enables exploratory analysis by changing parameters of the overlay and immediately observing the outcome, and adding or removing map layers on the fly.