Spatial Mathematics: Theory and Practice through Mapping

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ISBN 9781466505322
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  • Details the conceptual foundations that enable readers to understand projections, representing data, and much more
  • Covers spatial thinking in a theoretical and applied way
  • Supplies robust material and exercises tested in workshop situations
  • Includes software to which exercises are geared
  • Explains and includes QR (quick response) codes that link text to downloads
  • Focuses on detailed illustrations of mathematical backgrounds in GIS with an interdisciplinary interaction
  • Contains slides illustrating the concepts and exercises that educators can use in their own courses
  • Provides practical hands-on examples that the reader can immediately use to reinforce the concepts learned



In terms of statistics, GIS offers many connections. With GIS, data are gathered, displayed, summarized, examined, and interpreted to discover patterns. Spatial Mathematics: Theory and Practice through Mapping uses GIS as a platform to teach mathematical concepts and skills through visualization of numbers. It examines theory and practice from disparate academic disciplines such as geography, mathematics, physics, and general social science. This approach allows students to grapple with biodiversity, crime, natural hazards, climate, energy, water, and other relevant real-world issues of the twenty-first century.

Includes QR Codes Linked to Animated Maps, a Mapping Activity Site, or to an Interactive Webpage, Creating an Interactive Resource That Stays Relevant

The book integrates competing philosophical views of the world: synthesis and analysis. These two approaches yield different results and employ different tools. This book considers both approaches to looking at real-world issues that have mathematics as a critical, but often unseen, component. This approach shows readers how to use mathematics to consider the broad problem at hand and to explore diverse realms in the worlds of geography and mathematics and in their interface.

A truly interdisciplinary text, the book bridges the worlds of mathematics and geography and demonstrates how they are inextricably linked. It takes advantage of the convergence in citizen science, STEM education, and mapping that help readers become critical consumers of data—understanding its content, quality, limitations, and benefits. It provides thorough grounding in the analytical, statistical, and computational skills required for working in any field that uses geospatial technologies—not just surveyors and remote sensing analysts.

Table of Contents

Geometry of the Sphere
Theory: Earth Coordinate Systems
Theory: Earth’s Seasons—A Visual Display
Theory: Precision of Latitude and Longitude Values
Other Earth Models
Practice Using Selected Concepts from This Chapter
Related Theory and Practice: Access through QR Codes

Location, Trigonometry, and Measurement of the Sphere
Introduction: Relative and Absolute Location
Location and Measurement: From Antiquity to Today
Practice: Measuring the Circumference of the Earth Using GPS
Measuring Positions on the Earth Surface, and Fractions
Other Common Coordinate Systems
Practice: Coordinates Using Different Systems
Theory: Visual Trigonometry Review
Practice: Find the Length of One Degree on the Earth-Sphere
Practice: Determine Sun Angles at Different Seasons of the Year
Practice: Work with Measurement, the Graticule, and Map Projections
Summary and Looking Ahead
Related Theory and Practice: Access through QR Codes

Transformations: Analysis and Raster/Vector Formats
Partition: Point-Line-Area Transformations
Set Theory
Raster and Vector Mapping: Know Your File Formats
Practice Using Selected Concepts from this Chapter
Related Theory and Practice: Access through QR Codes

Replication of Results: Color and Number
Color Straws and Color Voxels
Color Ramps: Alternate Metrics
Algebraic Aspects of Ratios
Pixel Algebra
Preservation of the Aspect Ratio
Image Security
Theory Finale
Practice Using Selected Concepts from This Chapter
Related Theory and Practice: Access through QR Codes

Scale Change
The Dot Density Map: Theory and Example
Practice Using Selected Concepts from This Chapter
Related Theory and Practice: Access through QR Codes

Partitioning of Data: Classification and Analysis
The Choice of Data Ranges
Normalizing Data
Inside, Outside, Wrong Side Around
Making Something From Nothing?
Practice Using Selected Concepts from This Chapter
Related Theory and Practice: Access through QR Codes

Visualizing Hierarchies
Hierarchies: Census Data
Thinking Outside the Pixel
Practice Using Selected Concepts from This Chapter
Related Theory and practice: Access through QR Codes

Distribution of Data: Selected Concepts
Ann Arbor, Michigan—Tornado Siren Infill Project
Educational and Marketing Efforts to the Public
Examining the Distribution of Tornado Data
Activity: Examining the Distribution of Tornado Data
Mean Center and Standard Deviational Ellipse
Activities Using Mean Center and Standard Deviational Ellipse
Related Theory and Practice: Access through QR Codes
Appendix of Media Commentary

Map Projections
In the news
Looking at Maps and Their Underlying Projections
Sampling Projection Distortion
Some Projection Characteristics
Pseudo or Miscellaneous Projections
Contemporary Approach: Web Mercator Auxiliary Sphere Projection
Sampling the Environment: The Degree Confluence Project
Practice Using Selected Concepts from This Chapter
Around the Theoretical Corner?
Related Theory and Practice: Access through QR Codes

Integrating Past, Present, and Future Approaches
From the Classics to the Modern: Past and Present
A Non-Euclidean Future?
Practice Using Selected Concepts from the Chapter
Graph Theory and Topology: Discrete and Continuous Spatial Mathematics
Putting It All Together: Theory
Related Theory and Practice: Access through QR Codes

References, Further Reading, and Related Materials

Author Bio(s)

Editorial Reviews

"this colourful book should be useful for complementing technical geography lectures using Google Earth and Esri’s software explorations in view of attracting high-school and general science students to spatial geography and computer mapping."
—Rod Blais, GEOMATICA, Vol. 67, No. 4, 2013

"In this important work, Drs. Arlinghaus and Kerski fill an important need in the geospatial literature with their accessible introduction to spatial mathematics. … this book frames concepts in a way that is accessible to new learners who may not have an advanced math background. This accessibility should not be confused with simplicity, however. In successive chapters, the authors build an intellectually challenging description of the math underlying geospatial analysis. Throughout, easy-to-follow activities help make the concepts clear and relevant. If we are to build a generation of spatial thinkers (and not just people trained in GIS and GPS), we need foundational understanding like this book supports. Highly recommended!"
—Bob Coulter, Missouri Botanical Garden

"Two ancient texts had a profound and lasting impact on the literate world. … Now, in this book, additional insight for the mathematical solution of geographical tasks is provided. The pedagogical orientation is especially worthy of comment."
—Waldo Tobler, Professor Emeritus of Geography, University of California at Santa Barbara

"If a picture is worth a thousand words, then a map is worth exponentially more. Underneath the colors of a modern electronic map is an enormous amount of math and science, from how to represent a three dimensional globe on a two dimensional surface to how to produce and represent color itself in a way to communicate meaning and make sense of complex data. This book is both all about the map and all about the math behind the map, using what has become ubiquitous on our smart phones and in our vehicles as a vehicle itself to teach complex math concepts in accessible ways. The book’s use of modern, socially pressing issues as the basis of understanding spatial math further augments its goal to make complex concepts accessible, meaningful, and useful for students."
—Marc Schlossberg, Professor of Planning, Public Policy & Management, University of Oregon

"Teaching mathematics can be tough but here is a book that is a gentle introduction to the mathematics of the spatial world through the medium of mapping. The use of QR codes to access additional map-based material is clever and innovative, and provides a nice link to the very technologies that this mathematics supports."
—Michael Batty, University College London

"An excellent textbook mainstreaming geoinformation perspectives through mathematical education, Spatial Mathematics has a clearly developed theoretical approach with great exercises and practical hands-on activities."
—Karl Donert, President, European Association of Geographers (EUROGEO)

"… timely and welcome. The wealth of practical examples and the enthusiasm of its authors will fill an important niche in a mapping literature that often underplays the importance and relevance of mathematics. … particularly refreshing to see the book begin with the mathematics of the Earth’s curved surface, and deal only later with the complexities of flattening the Earth through the use of map projections. … Google Earth did much to restructure our view of the Earth when it was released in 2005; this book will without doubt make another step in that direction."
—Michael F. Goodchild, University of California, Santa Barbara

"… sets the standard for explaining the relationship between mathematics, geography, and GIS technologies. … A benchmark book for interdisciplinary teaching and learning."
—Mark C. Hogrebe, Washington University in St. Louis

"… this book is worth reading by anyone interested in overlaps of geography and mathematics. It contributes to the progress of quantitative geography and GIScience by melding together appropriate topics from geography and mathematics that focus on cartographic themes. … It succeeds in using the visualization capabilities of a GIS to motivate readers to familiarize themselves with and explore formal mathematical subject matter that interfaces with geography."
—The AAG Review OF BOOKS