## geometric measures

##### 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-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 - Distance, Length, and Direction • Describe several different measures of distance between two points (e.g., Euclidean, Manhattan, network distance, spherical)
• Explain how different measures of distance can be used to calculate the spatial weights matrix
• Explain why estimating the fractal dimension of a sinuous line has important implications for the measurement of its length
• Explain how fractal dimension can be used in practical applications of GIS
• Explain the differences in the calculated distance between the same two places when data used are in different projections
• Outline the implications of differences in distance calculations on real world applications of GIS, such as routing and determining boundary lengths and service areas
• Estimate the fractal dimension of a sinuous line
• Describe operations that can be performed on qualitative representations of direction
• Explain any differences in the measured direction between two places when the data are presented in a GIS in different projections
• Compute the mean of directional data
• Compare and contrast how direction is determined and stated in raster and vector data
• Define “direction” and its measurement in different angular measures
##### 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-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 - Distance, Length, and Direction • Describe several different measures of distance between two points (e.g., Euclidean, Manhattan, network distance, spherical)
• Explain how different measures of distance can be used to calculate the spatial weights matrix
• Explain why estimating the fractal dimension of a sinuous line has important implications for the measurement of its length
• Explain how fractal dimension can be used in practical applications of GIS
• Explain the differences in the calculated distance between the same two places when data used are in different projections
• Outline the implications of differences in distance calculations on real world applications of GIS, such as routing and determining boundary lengths and service areas
• Estimate the fractal dimension of a sinuous line
• Describe operations that can be performed on qualitative representations of direction
• Explain any differences in the measured direction between two places when the data are presented in a GIS in different projections
• Compute the mean of directional data
• Compare and contrast how direction is determined and stated in raster and vector data
• Define “direction” and its measurement in different angular measures
##### 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-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