Customer Reviews

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

5 star:		(0)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

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.

Comment Comment | Permalink