Topological Data Structures for Surfaces: An Introduction to Geographical Information Science

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In Geography and GIS, surfaces can be analysed and visualised through various data structures, and topological data structures describe surfaces in the form of a relationship between certain surface-specific features. Drawn from many disciplines with a strong applied aspect, this is a research-led, interdisciplinary approach to the creation, analysis and visualisation of surfaces, focussing on topological data structures.

Topological Data Structures for Surfaces: an introduction for Geographical Information Science describes the concepts and applications of these data structures. The book focuses on how these data structures can be used to analyse and visualise surface datasets from a range of disciplines such as human geography, computer graphics, metrology, and physical geography. Divided into two Parts, Part I defines the topological surface data structures and explains the various automated methods used for their generation. Part II demonstrates a number of applications of surface networks in diverse fields, ranging from sub-atomic particle collision visualisation to the study of population density patterns. To ensure that the material is accessible, each Part is prefaced by an overview of the techniques and application.

• Provides GI scientists and geographers with an accessible overview of current surface topology research.
• Algorithms are presented and explained with practical examples of their usage.
• Features an accompanying website developed by the Editor - http://geog.le.ac.uk/sanjayrana/surface-networks/

This book is invaluable for researchers and postgraduate students working in departments of GI Science, Geography and Computer Science. It also constitutes key reference material for Masters students working on surface analysis projects as part of a GI Science or Computer Science programme.

Topological Data Structures for Surfaces: An Introduction to Geographical Information Science Review

Geography used to be a rather sleepy discipline. But the advent of ever cheaper computing power has given rise to Geographic Information Science. Within this, a basic issue has arisen about how best to represent geographic surfaces in a computer memory. What Rana shows in the amassed chapters (contributed by different authors) is how to possibly use various topological descriptions.

Many of the chapters deal with a surface network. Here the original geographic surface is assumed to be twice differentiable and that resultant function is assumed to be continuous; ie. C^2. Theorems from real analysis are used to define critical points (where the first derivative is 0), and where the Hessian (second derivative) is assumed to be non-singular at those points. (Cf. Marsden's Elementary Classical Analysis for a more detailed exposition on this point).

From this arises the mountaineer's equation, or the Euler-Poincare equation, #max + #min - #saddle = 2. Very pretty.

Those readers already versed in graph theory will see a familiar relationship. The book in several places then takes the logical next step by forming a topological graph, where a maximum, minimum or saddle point in the original graph becomes a node. While a ridge between a maximum and a saddle, or a channel between a saddle and a minimum defines a edge between two nodes.

Another key idea in the book is the Reeb graph. Both this and the surface network are elegant formulations that help capture a number of the essential features of the original metric map.

The book takes you beyond Marsden's general approach to scalar functions of several variables, by showing applications to real world (literally!) data.

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