geometric measures

FC-17 - Proximity and Distance Decay

Distance decay is an essential concept in geography. At its core, distance decay describes how the relationship between two entities generally gets weaker as the separation between them increases. Inspired by long-standing ideas in physics, the concept of distance decay is used by geographers to analyze two kinds of relationships. First, the term expresses how measured interactions (such as trade volume or migration flow) generally decrease as the separation between entities increases, as is analyzed by spatial interaction models. Second, the term is used to describe how the implicit similarity between observations changes with separation, as measured by variograms. For either type of relationship, we discuss how "separation" must be clearly articulated according to the mechanism of the relationship under study. In doing this, we suggest that separation need not refer to positions in space or time, but can involve social or behavioral perceptions of separation, too. To close, we present how the "death of distance" is transforming distance decay in uneven ways.

FC-16 - Area and Region
  • List reasons why the area of a polygon calculated in a GIS might not be the same as the real world object it describes
  • Demonstrate how the area of a region calculated from a raster data set will vary by resolution and orientation
  • Outline an algorithm to find the area of a polygon using the coordinates of its vertices
  • Explain how variations in the calculation of area may have real world implications, such as calculating density
  • Delineate regions using properties, spatial relationships, and geospatial technologies
  • Exemplify regions found at different scales
  • Explain the relationship between regions and categories
  • Identify the kinds of phenomena commonly found at the boundaries of regions
  • Explain why general-purpose regions rarely exist
  • Differentiate among different types of regions, including functional, cultural, physical, administrative, and others
  • Compare and contrast the opportunities and pitfalls of using regions to aggregate geographic information (e.g., census data)
  • Use established analysis methods that are based on the concept of region (e.g., landscape ecology)
  • Explain the nature of the Modifiable Areal Unit Problem (MAUP)
FC-18 - Adjacency and connectivity
  • List different ways connectivity can be determined in a raster and in a polygon dataset
  • Explain the nine-intersection model for spatial relationships
  • Demonstrate how adjacency and connectivity can be recorded in matrices
  • Calculate various measures of adjacency in a polygon dataset
  • Create a matrix describing the pattern of adjacency in a set of planar enforced polygons
  • Describe real world applications where adjacency and connectivity are a critical component of analysis
FC-17 - Proximity and distance decay
  • Describe real world applications where distance decay is an appropriate representation of the strength of spatial relationships (e.g., shopping behavior, property values)
  • Explain the rationale for using different forms of distance decay functions
  • Explain how a semi-variogram describes the distance decay in dependence between data values
  • Outline the geometry implicit in classical “gravity” models of distance decay
  • Plot typical forms for distance decay functions
  • Write typical forms for distance decay functions
  • Write a program to create a matrix of pair-wise distances among a set of points
  • Describe real world applications where distance decay would not be an appropriate representation of the strength of spatial relationships (e.g., distance education, commuting, telecommunications)
FC-14 - Directional Operations

In the same manner as distance, direction plays an equally important role in GIS. This article first summarizes different ways of measuring direction, either quantitatively or qualitatively. Formulas and examples are provided. In the following discussion, fundamental differences between distance and direction in describing spatial relations is examined; properties of angles are emphasized in the context of GIS; and the classification of both cardinal and projective direction is illustrated. With a focus on quantitative operations, various directional operations are categorized and elaborated based on factors such as the underlying data model (vector or raster) and whether direction effect is explicitly or implicitly embedded in the data.

FC-18 - Adjacency and connectivity
  • List different ways connectivity can be determined in a raster and in a polygon dataset
  • Explain the nine-intersection model for spatial relationships
  • Demonstrate how adjacency and connectivity can be recorded in matrices
  • Calculate various measures of adjacency in a polygon dataset
  • Create a matrix describing the pattern of adjacency in a set of planar enforced polygons
  • Describe real world applications where adjacency and connectivity are a critical component of analysis
FC-17 - Proximity and distance decay
  • Describe real world applications where distance decay is an appropriate representation of the strength of spatial relationships (e.g., shopping behavior, property values)
  • Explain the rationale for using different forms of distance decay functions
  • Explain how a semi-variogram describes the distance decay in dependence between data values
  • Outline the geometry implicit in classical “gravity” models of distance decay
  • Plot typical forms for distance decay functions
  • Write typical forms for distance decay functions
  • Write a program to create a matrix of pair-wise distances among a set of points
  • Describe real world applications where distance decay would not be an appropriate representation of the strength of spatial relationships (e.g., distance education, commuting, telecommunications)
FC-14 - Directional Operations

In the same manner as distance, direction plays an equally important role in GIS. This article first summarizes different ways of measuring direction, either quantitatively or qualitatively. Formulas and examples are provided. In the following discussion, fundamental differences between distance and direction in describing spatial relations is examined; properties of angles are emphasized in the context of GIS; and the classification of both cardinal and projective direction is illustrated. With a focus on quantitative operations, various directional operations are categorized and elaborated based on factors such as the underlying data model (vector or raster) and whether direction effect is explicitly or implicitly embedded in the data.

FC-16 - Area and Region
  • List reasons why the area of a polygon calculated in a GIS might not be the same as the real world object it describes
  • Demonstrate how the area of a region calculated from a raster data set will vary by resolution and orientation
  • Outline an algorithm to find the area of a polygon using the coordinates of its vertices
  • Explain how variations in the calculation of area may have real world implications, such as calculating density
  • Delineate regions using properties, spatial relationships, and geospatial technologies
  • Exemplify regions found at different scales
  • Explain the relationship between regions and categories
  • Identify the kinds of phenomena commonly found at the boundaries of regions
  • Explain why general-purpose regions rarely exist
  • Differentiate among different types of regions, including functional, cultural, physical, administrative, and others
  • Compare and contrast the opportunities and pitfalls of using regions to aggregate geographic information (e.g., census data)
  • Use established analysis methods that are based on the concept of region (e.g., landscape ecology)
  • Explain the nature of the Modifiable Areal Unit Problem (MAUP)
FC-14 - Directional Operations

In the same manner as distance, direction plays an equally important role in GIS. This article first summarizes different ways of measuring direction, either quantitatively or qualitatively. Formulas and examples are provided. In the following discussion, fundamental differences between distance and direction in describing spatial relations is examined; properties of angles are emphasized in the context of GIS; and the classification of both cardinal and projective direction is illustrated. With a focus on quantitative operations, various directional operations are categorized and elaborated based on factors such as the underlying data model (vector or raster) and whether direction effect is explicitly or implicitly embedded in the data.

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