Spatial Relationship: Distance

4. Spatial Relationship: Distance#

Note: see frechet and hausdorff distance function in shapely.

4.1. Topics#

  • Co-locational relationship

    • Qualitative (topological) and quantitative relationships

  • Interactional relationship

    • Connection between two locations

    • Exchange of matter, energy and information links locations

  • Distance relationship

    • As a measure/proxy of interactional relationship

    • Quantify the disjoint relationship (beyond co-location)

4.2. Spatial Interactional Relationships#

  • Interactional relationships

    • Exchange of matter, energy and information between locations

    • Beyond distance

    • Domain dependent

    • Viewshed, watershed, hydro-connectivity, …

  • Some may be described by physical processes

  • Some are still mysterious/unknown

    • Teleconnection, gravity, …

  • Connections or links between locations

4.3. Why Care About Spatial Relationships#

  • Yoneda Lemma

    • If you want to understand a mathematical object, all of the information about that object is contained in the totality of relationships that object has with all other objects in its environment

  • We can learn a lot about a location by seeing how the location interacts with other locations

    • May vary from location to location

  • Meta-map (a map of maps)

    • Location -> {(location, {attributes})}

    • Map of spatial relationships

4.4. Distance as the Proxy of Interactional Relationship#

  • “Everything is related to everything else, but near things are more related than distant things.” (Waldo Tobler, 1970)

  • Distance is used to represent the movement of matter, energy and information when the underlying physical processes are not well understood

  • Distance is the least movement cost between two locations

4.5. Types of Distance Relationships#

  • Distance to specific location(s)

  • Buffers

  • Proximity regions (Thiessen Polygons/Voronoi Diagram)

4.6. Distance to Specific Location(s)#

  • Every location in a study area has a distance to a specific location

  • Examples:

    • Distance to the coastline

    • Distance to the nearest hospital

    • Distance to the nearest highway

4.7. Buffers#

  • A buffer is a region within a specified distance of a geographic feature

  • Buffer types:

    • Points, lines, or polygons

    • Fixed distance vs. variable distance

    • Single vs. multiple rings

4.8. Proximity Regions (Voronoi Diagram)#

  • Every location within a proximity region is closer to the associated generator point than to any other generator point

  • Boundaries are the perpendicular bisectors of the lines connecting neighboring points

4.9. Calculating Distance#

  • Movement cost in different space/media

  • Euclidean space (vacuum/abstract space)

    • Euclidean distance (straight line)

  • Non-Euclidean space

    • On a spheroid (Great circle distance)

    • In a network (Road/river networks)

    • On a friction surface (Terrain/Least cost path)

4.10. Euclidean Distance#

  • \(d_{ij} = \sqrt{(x_i - x_j)^2 + (y_i - y_j)^2}\)

  • Straight line distance in a flat plane (PCS)

  • Manhattan distance (\(L_1\) norm): \(d_{ij} = |x_i - x_j| + |y_i - y_j|\)

4.11. Distance on a Spheroid#

  • Great circle distance

  • The shortest distance between two points on the surface of a sphere

  • Calculated using latitude and longitude (GCS)

4.12. Network Distance#

  • Distance is constrained by a network of interconnected lines (links) and points (nodes)

  • Shortest path algorithm (e.g., Dijkstra’s algorithm)

  • Examples: travel time, fuel consumption

4.13. Distance on a Friction Surface#

  • Movement cost varies across space due to different “friction” or “impedance”

  • Cost Surface/Friction Surface:

    • A raster map where each cell represents the cost of moving across it

  • Least Cost Path:

    • The path between two locations that minimizes the cumulative cost

  • Accumulated Cost Surface:

    • A map showing the minimum cost to reach a source from any location

4.14. Isotropic vs. Anisotropic Friction#

  • Isotropic: Cost is the same regardless of the direction of movement (e.g., land cover)

  • Anisotropic: Cost depends on the direction of movement (e.g., wind, slope/terrain)

4.15. Summary#

  • Distance is the least movement cost between two locations

    • Affects the exchange of matter, energy and information

    • Proxy for interactional relationship

  • Types of distance for a location/MU

    • Distance to specific locations/MUs

    • Buffers

    • Proximity regions

  • Distance in different space/media

    • Euclidean space

    • Non-Euclidean space (Spheroid, Network, Friction Surface)

4.16. Map of Distance Relationship (meta-map)#

  • A map where each MU has a “distance” map

  • A “distance” map stores locations and distance (and other attributes)

    • To pre-identified locations, buffers, proximity regions

    • Euclidean and non-Euclidean

    • Vector and raster MUs

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