Transcript Document

Chapter 6
Lines and Networks
from Geographic Information Analysis
by O’Sullivan and Unwin
4/12/2005
1
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
4/12/2005
2
Line Object
•
•
•
The second major type of spatial entity – spatial objects
that possess just a single length dimension.
Examples in geography:
1. Drainage network
2. Transportation network: road, rail, air
3. Utility network: electric, gas, water, sewer
4. Telecom network: telephone, cable, fiber optics
Three new spatial concepts to describe lines: Distance,
Direction, and Connection
4/12/2005
3
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
4/12/2005
4
Polyline Discretization
4/12/2005
5
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:
4/12/2005
6
Encoding Schemes for Line Data
1.
2.
3.
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:
1.
2.
4/12/2005
loss of generality – data format problem
need calculation to obtain the coordinates of any point other than the
origin
7
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)
4/12/2005
8
Examples of Freeman Chain Coding
• Left: Square Quantization Freeman Code 22020200
• Middle: Circular Quantization Freeman Code 221010
• Right: Grid-Intersect Quantization Freeman Code 221100
4/12/2005
9
Line Length
• Pythagoras’s theorem
Length between point S1 (x1, y1) and point S2 (x2, y2)
d 12 || s2  s1 || ( x2  x1 ) 2  ( y 2  y1 ) 2
4/12/2005
10
First Problem: getting longer the closer we look
4/12/2005
Resolution
# of segs
L (km)
N
Total
Length
(km)
2.5
132
330
5.0
52
260
10.0
8
180
11
How to calculate Dimensionality
L1 D
N 2  N1 ( )
L2
L1  2, L2  1,
N1  1, N 2  8
2 D
8  1  ( )  2D
1
D  3
4/12/2005
12
Fractal Dimension
• Fractal = fraction + dimensional
D
log(N1
log(
L1
N2
L2
)
)
• From the Richardson Plot in figure 6.5, the
fractal dimension of New Zealand coastline
is 1.4372
4/12/2005
13
2-D Lines
Hilbert’s Curve – a space-filling curve with a fractal
dimension of 2
4/12/2005
14
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.
VN   cos i
VE   sin  i
Preferred Orientation
4/12/2005
VE
t an R 
VN
15
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
4/12/2005
16
Stream Ordering I
1.
2.
3.
4.
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
4/12/2005
17
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
4/12/2005
18
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
4/12/2005
19
Graph Theory Basics
vertex = node
edge = link
connectivity or adjacency
matrix
two alternative representations
of the same graph
4/12/2005
20
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
4/12/2005
21
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
4/12/2005
22
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
4/12/2005
23