Chapter 6 Lines and Networks

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Chapter 6 Lines and Networks

• from Geographic Information Analysis
• by OSullivan and Unwin

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Outline

• What is a line object
• Line objects and networks coded in GIS
• Length, direction and connection
• Fractal natures of some line objects
• Statistical approaches to line data

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Line Object

• The second major type of spatial entity spatial
  objects that possess just a single length
  dimension.
• Examples in geography
• Drainage network
• Transportation network road, rail, air
• Utility network electric, gas, water, sewer
• Telecom network telephone, cable, fiber optics
• Three new spatial concepts to describe lines
  Distance, Direction, and Connection

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First Issue Representing and Storing

• How to represent line entities in a digital
  database?
• Simple convention sequences of points connected
  by straight-line segments polyline, arc,
  segment, edge, or chain
• Discretization breaking a real geographical line
  entity into short straight-line segments for a
  polyline representation
• Visually obvious turning points along a line are
  encoded as points
• An effectively random selection of points along
  straighter sections
• Relying on human operators to choose
  turning points can lead to considerable
  inconsistency

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Polyline Discretization
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Better Approaches for Curving Lines

• Arcs of circles with a center and radius for
  each section
• Splines mathematical functions used to describe
  smoothly curving features common in CAD,
  available in some GIS
• Examples of spline curves

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Encoding Schemes for Line Data

• Simple list of the (x, y) coordinate pairs
• Distance-Direction Coding (x, y) of the origin
  distance and direction pair for each segment
• 2 is a type of Delta Coding only offset between
  consecutive points is stored, like
• (23.341, 45.670) (00, 13) (04, -05) (12,
  13)
• Advantage reduce data redundancy, i.e., save
  memory/drive space
• Disadvantage
• loss of generality data format problem
• need calculation to obtain the coordinates of any
  point other than the origin

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Freeman Chain Coding

• In the previous encoding schemes, any segment
  length or offset is allowed.
• Sometimes it is better to use a fixed segment
  length, known as unit length coding
• Freeman Chain Coding based on unit length
  coding, plus a fixed number of directions is
  established N, S, E, W, NE, NW,SE, SW,
  represented by a single digit between 0 and 7
  (000 111)

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Examples of Freeman Chain Coding

• Left Square Quantization Freeman Code 22020200
• Middle Circular Quantization Freeman Code 221010
• Right Grid-Intersect Quantization Freeman Code
  221100

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Line Length

• Pythagorass theorem
• Length between point S1 (x1, y1) and point S2
  (x2, y2)

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First Problem getting longer the closer we look
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How to calculate Dimensionality
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Fractal Dimension

• Fractal fraction dimensional

• From the Richardson Plot in figure 6.5, the
  fractal dimension of New Zealand coastline is
  1.4372

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2-D Lines

• Hilberts Curve a space-filling curve with a
  fractal dimension of 2

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Line Direction The Periodic Data Problem

• Direction is not a ratio quantity
• The difference between 1º and 359º is not 358º
  but 2º
• Want average direction of a set of vectors?
  Simply adding up and dividing may give bad
  results
• Solution resolve vector measurements into
  components in two perpendicular directions to get
  the mean direction.

Preferred Orientation
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Connection in Line Data

• Tree an important type of pattern, where no
  closed loops are present
• Example 1 river networks
• Example 2 transport networks around a central
  place
• Stream Ordering a classic analysis to tree
  networks

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Stream Ordering I

• Each leaf of the tree is given an order of 1
• At any junction where two leaves meet, the branch
  is assigned order 2
• The order of a branch is increased where two
  equal-order branches meet
• The network is reclassified working upstream to
  identify the main branch

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Stream Ordering II

• Allow comparisons between stream networks
• Horton found that the number of streams of
  different orders from the highest downward
  closely approximates a geometric series, like 1,
  3, 9, 27, 81
• The bifurcation ratio can be determined and used
  as a direct comparison between cases. Natural
  drainage networks tend to vary from 3 to 5.
• Ignore both length and direction, only concerned
  with the topological property of connection

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Graphs/Networks

• Tree is a special example of network
• A network or graph is a general structure without
  restriction barring closed loops
• Graph theory the mathematical theory of networks
• Like stream ordering, graphs are topological,
  only interested in connectivity and adjacencies

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Graph Theory Basics

• vertex node
• edge link
• connectivity or adjacency matrix
• two alternative representations
• of the same graph

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Adjacency Matrix

• Represent a set of objects that define a
  relational structure
• Can be undirected and directed (one way traffic,
  drainage, and etc.)
• With math manipulation (multiplication, powering,
  and etc.), it can give information more than
  immediate adjacencies, such as number of
  neighbors, topological shortest paths, and vertex
  centrality
• Often useful in GIS work

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Statistical Analysis of Line Data

• So far only limited success
• For lines difficult to devise a meaningful null
  hypothesis equivalent to CSR for point patterns
• For graphs vast number of possible graphs of
  even small numbers of vertices (20 vs. 1039)
• Some useful results for line directions
• Potential for random graphs and small-world
  networks

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Conclusion

• Line objects are quite complex
• Numerous practical applications about their
  geometry (length, direction, connection) and
  properties of flows (gas, traffic, people, water)
  along them
• Easy to declare spatial concepts, difficult to
  analyze
• GIS software packages are slowly catching up