Geography 625 Intermediate Geographic Information Science Week 9: Lines and Networks Instructor: Changshan Wu Department of Geography The University of Wisconsin-Milwaukee Fall 2006
Outline • Introduction • Representing and storing linear entities • Line length and direction • Connecting in line data: trees and graphs • Statistical analysis of geographic line data • Applications in transportation studies
1. Introduction • Line objects • Drainage networks: rivers • Transportation networks: road, rail, air, utility • Communication networks: Internet Concepts • Length or distance • Direction • Connection
2. Representing and storing linear entities How to represent lines in a digital database? • Most GISs store lines as sequences of points connected by straight-line segments. • Polyline makes the distinction between individual segments and the combined entity. • Breaking a real geographical line entity into short straight-line segments for a polyline representation is called discretization and usually performed semiautomatically by human operators working with digitizing equipment.
2. Representing and storing linear entities • Obvious and simple • More points in curved portions and less points in smoothed portions
2. Representing and storing linear entities Smoothly curving lines can be more efficiently stored and manipulated as curves. 1) Curve sections are represented as arcs of circles (center and radius of curvature for each section are stored) 2) Curve sections are represented as splines. Splines are mathematical functions that can be used to describe smoothly curving features Circle arcs and splines are common in computer-aided design (CAD) and vector graphics drawing package, but not popular in GIS.
2. Representing and storing linear entities - Encoding schemes of Line Data Simple coding: a simple list of the (x,y) coordinate pairs of the digitized points in a polyline (x1,y1) (x2, y2) (x3, y3) (x4, y4)…… Distance direction coding: an origin point is specified, followed by a series of distance – direction pairs that indicate how far the polyline extends on a particular bearing to the next stored point. (x1,y1) (dist2, direction2) (dist3, direction3)……
2. Representing and storing linear entities - Encoding schemes of Line Data Deta coding: an origin point is specified, followed by a series of offsets in x and y direction. (x1,y1) (Δx2, Δy2) (Δx3, Δy3)…… Unit length (Freeman chain) coding: an origin point, followed by a step length, followed by a series of coded directions (usually eight directions)
3. Line length and direction - Straight Lines
3. Line length and direction - Lines in real world The coastline is getting longer the closer we look What is the correct length?
3. Line length and direction - Fractal Dimension Mathematic idea for dealing with this difficulty: fractal dimension A line may be somewhere between one- and two dimensional, with fractal dimension (e.g. 1.2) The dimensionality describes how its total size (length, area, or volume) increases as its linear size, measured in one dimension, increases.
3. Line length and direction - Fractal Dimension Resolution 1, 1/2, 1/3 Segments: 1, 2, 3 Resolution 1, 1/2, 1/3 Segments: 1, 4, 9 Resolution 1, 1/2, 1/3 Segments: 1, 8, 27
3. Line length and direction - Fractal Dimension L: the linear size or scale of our yardstick element N: the number of elements L= 1, 1/2, 1/3 N= 1, 8, 27
3. Line length and direction - Fractal Dimension An example: Koch's snowflake What is the fractal dimension?
3. Line length and direction - Fractal Dimension Hilbert Curve What is the fractal dimension?
3. Line length and direction - Fractal Dimension Richardson plot: estimate the fractional dimension by plotting the relationship between Log N and Log L
3. Line length and direction - Fractal Dimension The dimensionality of a geometric entity expresses the extent to which it fills the next whole dimension up from its own topological dimension. A line with fractal dimension of 1.0 is an ideal Euclidean geometric line with no area, so take up no space in the plane. However, a line with fractal dimension 1.1 or 1.15 begins to fill up the two topological dimensions of the plane in which it is drawn.
3. Line length and direction - Fractal Dimension Many natural phenomena, from clouds and trees, to coastlines, topography, and drainage basins exhibit the unsettling property of changing their size depending on how close you look. The fractal dimension of surface topography can be related to other geological or socio-economic characteristics. Fractal dimension is indicative of how measures of an object will change with scale and generalization, and in general, scale-varying properties of phenomena can be related to their fractal dimension. Related to “self-similarity”. A geographic entity is strongly self-similar (coastlines: big bays have little bays within them, and so on).
3. Line length and direction - Other length measures Mean length Standard deviation Path density
3. Line length and direction - Line direction North V VN θi Average direction VE East
4. Trees and graphs Trees: no closed loops Graphs (networks): closed loops Transportation network River
4. Trees and graphs Graph theory: mathematical theory of networks (a branch of topology) Vertices: junctions or nodes Links: the lines joining the vertices Connectivity or adjacency • Directed graph • Undirected graph
4. Trees and graphs Minimally connected network: each vertex is connected to the network, and none is isolated from it The number of edges emin = v - 1
4. Trees and graphs Maximally connected network: every vertex is directly linked to each of the other vertex. The number of edges emax = v(v – 1)/2 = 6*5/2 = 15
4. Trees and graphs Nonplanar graph: There are no intersections among edges (e.g. airplane network) Planar graph: all linkages are in the same plane and all intersections not corresponding to the original vertices would have to be eliminated. Planar Nonplanar
4. Trees and graphs Planar graph The maximum number of edges emax = 3(V-2)
4. Trees and graphs - Connectivity Diameter: the number of linkages or steps needed to connect the two most remote nodes in the network. The better connected the network, the lower the diameter. What is the diameter?
4. Trees and graphs - Connectivity The gamma index (γ): the ratio between the number of edges (linkages) actually in a given network and the maximum number possible in that network. For planar network Spinal: 1/3 <= γ <= 1/2 where V > 4 Grid: 1/2 <= γ <= 2/3 where V > 4 Delta: 2/3 <= γ <= 1.0 where V > 3 What is the gamma index for the above planar network?
4. Trees and graphs - Connectivity Matrix A represents one-step connectivity
4. Trees and graphs - Connectivity Matrix A2 represents two-step connectivity Diameter = 2
4. Trees and graphs - Connectivity Total connectivity (T) n is the diameter of the graph 4 7 8 7 A B C D T= A1 + A2 = = +
4. Trees and graphs Shortest Path Algorithm • Dijkstra’s Shortest Path Algorithm (http://carbon.cudenver.edu/~hgreenbe/sessions/dijkstra/DijkstraApplet.html) C 2 B 3 1 F 2 2 A 1 E 2 D
4. Trees and graphs Shortest Path Algorithm C C 2 2 B B 3 3 1 F 1 F 2 2 2 2 A A 1 1 E E 2 2 D D Step 1: start from a vertex (A), scan the connected vertices, activate the one with the nearest distance, put the distance for the active vertex (B with a distance of 1 in this example).
4. Trees and graphs Shortest Path Algorithm C 2 B C B (1) 2 3 1 F 2 3 1 F 2 2 2 A 2 2 A(0) 1 E 2 1 E 2 D D (2) Step 2: from active vertices (A and B in the example), calculate the distance, put the vertex with the smallest distance into the active list (D in this example). Step 3: check whether an active vertex is connected to potential vertex, if not, close it. Step 4: repeat step 2 and 3
4. Trees and graphs Shortest Path Algorithm C C B (1) 2 B (1) 2 3 3 1 F 2 1 F 2 2 2 2 2 A(0) A(0) 1 1 E 2 2 E (2) D (2) D (2) C (3) B (1) 2 B (1) 2 C (3) 3 1 2 3 F(4) 1 F 2 2 2 2 2 A(0) 1 1 2 E (2) 2 E (2) D (2) D (2)
5. Statistical analysis of geographic line data Not widely applied. Probability of different line lengths across a rectangle area
6. Applications in transportation studies • Traffic demand modeling • Best route selection • Location analysis (retail stores) • Scheduling • Etc…