As we translate real world phenomena into data structures that we can store in a computer, we must determine the most appropriate spatial representation and how it relates to the characteristics of such a phenomenon. All spatial representations are derivatives of graph theory and should therefore be described in such terms. This then helps to understand the principles of low-level GIS operations. A constraint-driven approach allows the reader to evaluate implementations of the geo-relational principle in terms of the hierarchical level of mathematical space adopted.
Representations of space and time are central to GIScience. Field-based representations conceptualize space and time as continuous surfaces where each location is associated with measurable attribute values. Traditional models, such as raster grids and Triangulated Irregular Networks (TINs), discretize continuous fields for computational efficiency. However, these models often rely on rigid pixel assumptions and linear interpolations that fail to capture subtle curvatures and variations in real-world phenomena. To address these shortcomings, surface adjustment methods refine spatial measurements by constructing local terrain models that better represent spatial variations. Beyond static spatial fields, higher-dimensional models integrate space, time, and scale into a unified framework. Time-geography introduces the space-time cube, where space and time are integrated into a 3D field. Additionally, the Triangle Model (TM) and Pyramid Model (PM) incorporate scale into temporal and spatial analysis, respectively. These models allow for more nuanced analysis, such as tracking objects, assessing interaction probabilities, and exploring cross-scale relationships in space and time. Taken together, these models form a multi-scale spatio-temporal framework with four key dimensions: spatial location (s), spatial scale (s′), temporal location (t) and temporal scale (t′), providing a systematic approach to analyze dynamic geographic phenomena across multiple dimensions.