Spatial Mathematics: Theory and Practice through Mapping, 1st Edition (Hardback) book cover

Spatial Mathematics

Theory and Practice through Mapping, 1st Edition

By Sandra Lach Arlinghaus, Joseph J. Kerski

CRC Press

300 pages | 208 B/W Illus.

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Hardback: 9781466505322
pub: 2013-06-26
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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.


"…this book does not contain any equations or math, as one would expect from the title… As such, this book will be of use to those who want to start out with an effort to learn GIS, without having to struggle through the actual mathematics behind it all."

—Harold Schuch, GeoCounsel, Inc., Littleton, Colorado, USA

"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

"The form of the book (theoretical analysis, examples, and exercises) and the extent of its contents provide a useful reference for scientists, engineers, and the general public regarding entry level GIS concepts. A big congratulation to the authors."

Photogrammetric Engineering & Remote Sensing, October 2016

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

About the Originator

Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Earth Sciences / General
TECHNOLOGY & ENGINEERING / Remote Sensing & Geographic Information Systems