attribute transformation begins only here. Various operations, logical and mathematical, are used to transform attributes and relate them—for exam- ple, evaluating soil type and soil moisture to determine crop suitability. Ras- ter GI performs these attribute transformation as the overlay transforma- tion, assuming both raster data sets use the same raster size and origin point (otherwise some complex geometric transformations must first take place).
Chapter 14 covers these issues and the overlay operation in more detail.
Summary
This chapter examined GI representation types and transformations. GI rep- resentation types are the formats available for GI: positions, networks, and fields. Positions and networks rely on vector data formats; fields rely on ras- ter data formats. Positional GI is stored in a GIS as points, lines, or areas (also known as polygons), most often following the georelational model that uses topology. Networks also use these data formats, but areas are of very limited use in a network. Points, called nodes in networks, are much more important.
Transformations are operations on GI representation types that change the information content. A buffer transforms a point through a distance measure into an impacted area.
residents of Königsberg (now Kaliningrad). The puzzle sought a solution about how to cross seven bridges that connected two islands in the middle of the city without crossing any bridge twice.
Euler’s solution was to abstract the problem into a set of relationships between vertices (also called nodes), edges, and faces. This is called a graph.
Euler established that a graph has a path traversing each edge exactly once if exactly two vertices link an odd number of edges. Since this isn’t the case in Königsberg there isn’t a route that crosses each bridge once and only once.
The mathematics of this relationship are simple. To determine if there is single relationship, count the number of vertices connecting three edges. If the number of vertices is two, then there is a single way around. Otherwise, at least one vertice must be crossed twice.
Euler contributed an immense body of work, over 775 papers, half of which were written after he went blind at the age of 59. The Königsberg prob- lem is related to Euler’s polyhedral formula, which is the basis for determining topology in a GIS:
v – f + e = 2
v stands for the number of vertices, f for the number of faces, and e for the number of edges. Regardless of the type of polygon, this number will always be two.
Topology was extended by numerous mathematicians in the late 19th cen- tury, and although most people learn little about it, it has been immensely sig- nificant for many technological developments.
Topology focuses on connectivity. In regards to GIS, topology is impor- tant for three reasons. First, it can be calculated to determine if all polygons are closed, lines connected by nodes, and nodes connected to lines. This allows for the determination of errors in digitized or scanned vector data. Second, it can be used in network GI to determine network routing. Finally, because it allows that the same line (edge) is used for neighboring polygons, the number of lines stored in a GIS can be greatly reduced.
Review Questions
1. What sets GI apart from maps in terms of discrete and nondiscrete information?
2. Why are multiple types of data structures needed?
3. What is Tobler’s transformational concept?
Figure showing Euler’s seven bridges of Königsberg problem.
4. What is the main difference between discrete and nondiscrete GI?
5. What is the main difference between topological and nontopological vector data?
6. What is a quad-tree?
7. What is a triangular irregular network (TIN)?
8. How can the GI storage format impact GI representation?
9. How does a buffer operation transform a geographic representation?
10. Why can’t maps be transformed?
Answers
1. What sets GI apart from maps in terms of discrete and nondiscrete information?
GI offers multiple ways to store and transform data that can be used to make meaningful representations of things and events as GI. Maps can show both discrete and nondiscrete information, but the information can- not be transformed.
2. Why are multiple types of data structures needed?
Different types of data structures make it possible to adequately geographi- cally and cartographically represent observations of things and events.
3. What is Tobler’s transformational concept?
Tobler’s transformational concept is the development and application of the mathematical transformation concept to cartography. With this concept comes an understanding of GI as sets of associations with particular repre- sentations that can be converted to create other sets of associations.
4. What is the main difference between discrete and nondiscrete GI?
Discrete GI shows things with fixed boundaries; nondiscrete GI shows pro- cesses or states of processes.
5. What is the main difference between topological and nontopological vector data?
Topological vector data has a set of relationships between nodes and links;
nontopological vector data maintains only start, possibly intermediate, and end points.
6. What is a quad-tree?
A quad-tree is a data structure for the efficient storing of raster data follow- ing a hierarchy based on areas of contiguous attribute values.
7. What is a triangular irregular network (TIN)?
A Tin is a data structure for storing GI based on distance relationships and single values; it is most widely used for storing and modeling elevation data.
8. How can the geographic data structure impact GI representation?
It allows certain attributes and relationships to be better stored than others;
transformations make it possible to convert GI to other formats that may resolve the limitations with one particular type of data structure.
9. How does a buffer operation transform a geographic representation?
Based on existing geometry (point, line, area) and attribute value(s), it cre- ates a new area that represents a new geographic representation with a new thing or event.
10. Why can’t maps be transformed?
Maps cannot be transformed because of the cartographic representation and recording in the fixed media of a map. Maps cannot be directly trans- formed into other representations. Information collected from maps through digitization can, however, be transformed.
Chapter Readings
The second edition of this text contains a wealth of new and additional information, but the first edition is still a classic. See
Burrough, P. A. (1987). Principles of Geographical Information Systems for Land Resource Assessment. Oxford, UK: Oxford University Press.
From the computer science perspective, this is a key book documenting the develop- ment of GIS:
Worboys, M. F. (1995). GIS: A Computing Perspective. London: Taylor & Francis.
This book presents the use of databases for representing GI:
Rigaux, P., M. Scholl, et al. (2002). Introduction to Spatial Databases: Applications to GIS.
San Francisco: Morgan Kaufmann.
Web Resources
For an introduction to some of the fundamental GI representation issues, see the Wikipedia entry online at http://en.wikipedia.org/wiki/Geographic_information_system For a paper that discusses some of the limitations of the widely used types of GI representation, see the website www.ucgis.org/priorities/research/research_white/1998%20Papers/
extensions.html
A basic GIS tutorial can be found online at www.gisdevelopment.net/tutorials/
Some examples of how animations help visualize the temporal aspects of geographic things and events in current GIS can be found at the website www.farmresearch.com/
gis/gallery/animations.asp
Exercises
1. Euler’s Seven Bridges Problem Description
Topology is a field of mathematics where distance is not relevant. In this exercise, you will examine some of the basic concepts of topology.
Exercise Instructions
On this rough map illustrating the seven bridges of Königsberg problem that moti- vated the mathematician Leonard Euler to develop topology, try to draw with your pencil in one continuous line a way to walk around the city crossing each bridge only once.
Questions
Why do you think this is so difficult?
Is it possible?
A second case: assume a flood washes out one of the bridges in Königsberg, leav- ing six. Draw a route around the city now using one continuous pencil line.
Questions
Does it matter which bridge you take away?
What if you add bridges?
2. EXTENDED EXERCISE: Networks, Topologies, and Route Overview
In this exercise, you will use GPS equipment to determine the locations of several key points near campus, determine the time to walk between locations, and prepare a network graph where distance corresponds to time.
Objectives
Learn how to apply topography in geography.
Exercise Steps and Questions
1. Configure the Equipment
Make sure the GPS is working properly. Please check the battery status on the main menu (you get there by pressing the main menu button twice). If the power indicator is significantly below 25%, please see the instructor to get new batteries.
2. Collect Location and Travel-Time Data
In this step, you will need to collect locational data for each point and the time it takes to walk between each location. You should plan on taking 1 hour to collect this data. On the table on the next page, first write down the names of seven places (the first place should be in front of the main entrance to the building where class normally meets) you will collect location data for. Go outside the main entrance and wait until you have excellent GPS satellite reception (your accuracy should be less than 30 ft). Write down the coordi- nates displayed on the GPS receiver, the departure time, and then start to walk at a comfortable pace to your second location. When you get there write down the arrival time and location information. When done writing this information, write down the departure time and proceed to the next point.
Please note:
• Each point should be at least 200 m from any other point—further is even better.
• Each connection between points should only be recorded once.
• If you walk in the order of your locations, your arrival and departure times are always related. However, if you change the order, you will need to make a note of that on the worksheet.
Comments/Observations:
3. Make a Map of the Locations
On a separate sheet of paper, prepare a drawing showing the geographic locations of the data you observed above and the routes you traveled to
Description Easting Northing Elevation
Arrival
Time Departure Time Difference 1
2 3 4 5 6 7