Maps communicate information about the world by using symbols to represent specific ideas or concepts. The relationship between a map symbol and the information that symbol represents must be clear and easily interpreted. The symbol design process requires first an understanding of the underlying nature of the data to be mapped (e.g., its spatial dimensions and level of measurement), then the selection of symbols that suggest those data attributes. Cartographers developed the visual variable system, a graphic vocabulary, to express these relationships on maps. Map readers respond to the visual variable system in predictable ways, enabling mapmakers to design map symbols for most types of information with a high degree of reliability.

Terrain representation is the manner by which elevation data are visualized. Data are typically stored as 2.5D grid representations, including digital elevation models (DEMs) in raster format and triangulated irregular networks (TINs). These models facilitate terrain representations such as contours, shaded relief, spot heights, and hypsometric tints, as well as automate calculations of surface derivatives such as slope, aspect, and curvature. 3D effects have viewing directions perpendicular (plan), parallel (profile), or panoramic (oblique view) to the elevation’s vertical datum plane. Recent research has focused on automating, stylizing, and enhancing terrain representations. From the user’s perspective, representations of elevation are measurable or provide a 3D visual effect, with much overlap between the two. The ones a user can measure or derive include contours, hypsometric tinting, slope, aspect, and curvature. Other representations focus on 3D effect and may include aesthetic considerations, such as hachures, relief shading, physiographic maps, block diagrams, rock drawings, and scree patterns. Relief shading creates the 3D effect using the surface normal and illumination vectors with the Lambertian assumption. Non-plan profile or panoramic views are often enhanced by vertical exaggeration. Cartographers combine techniques to mimic or create mapping styles, such as the Swiss-style.

The classic transportation problem concerns minimizing the cost of transporting a product from sources/supplies to destinations/demands. It is a network-flow problem that arises in industrial logistics and is often solved by linear programming (LP). The three inputs of the model are total units produced at each source, total units needed at each destination, and the cost to transport one unit from each source to each destination. And the objective is to minimize the total cost of transporting all units produced at sources to meet the demands at destinations. The problem solution includes three basic steps: 1) finding an initial basic feasible solution, 2) checking if the current solution is optimal (with the lowest costs), and improving the current solution through iteration. Solving such a problem relies strongly on the network data models, least-cost path algorithms, other functionalities in GIS. And an integrated framework is often adopted to utilize both GIS and non-GIS linear programming solvers to search for the optimal solution.

Local multivariate statistical models are increasingly encountered in geographical research to estimate spatially varying relationships between a response variable and its associated predictor variables. In geography and many other disciplines, such models have been largely embedded within the framework of regression and can reveal significantly more information about the determinants of observed spatial distribution of the dependent variable than their global regression model counterparts. This section introduces one type of local statistical modeling framework: Geographically Weighted Regression (GWR). Models within this framework estimate location-specific parameter estimates for each covariate, local diagnostic statistics, and bandwidth parameters as indicators of the spatial scales at which the modeled processes operate. These models provide an effective means to estimate how the same factors may evoke different responses across locations and by so doing, bring to the fore the role of geographical context on human preferences and behavior.