How much math does a GIS Analyst need to know?(转)

• Trigonometry and spherical trigonometry. Let me surprise you: this stuff is overused. In many cases trig can be avoided altogether by using simpler, but slightly more advanced, techniques, especially basic vector arithmetic.

• Elementary differential geometry. This is the investigation of smooth curves and surfaces. It was invented by C. F. Gauss in the early 1800's specifically to support wide-area land surveys, so its applicability to GIS is obvious. Studying the basics of this field prepares the mind well to understand geodesy, curvature, topographic shapes, and so on.

• Topology. No, this does not mean what you think it means: the word is consistently abused in GIS. This field emerged in the early 1900's as a way to unify otherwise difficult concepts with which people had been grappling for centuries. These include concepts of infinity, of space, of nearness, of connectedness. Among the accomplishments of 20th century topology was the ability to describe spaces and calculate with them. These techniques have trickled down into GIS in the form of vector representations of lines, curves, and polygons, but that merely scratches the surface of what can be done and of the beautiful ideas lurking there. (For an accessible account of part of this history, read Imre Lakatos' Proofs and Refutations. This book is a series of dialogs within a hypothetical classroom that is pondering questions that we would recognize as characterizing the elements of a 3D GIS. It requires no math beyond grade school but eventually introduces the reader to homology theory.)

  Differential geometry and topology also deal with "fields" of geometric objects, including the vector and tensor fields Waldo Tobler has been talking about for the latter part of his career. These describe extensive phenomena within space, such as temperatures, winds, and crustal movements.

• Calculus. Many people in GIS are asked to optimize something: find the best route, find the best corridor, the best view, the best configuration of service areas, etc. Calculus underlies all thinking about optimizing functions that depend smoothly on their parameters. It also offers ways to think about and calculate lengths, areas, and volumes. You don't need to know much Calculus, but a little will go a long way.

• Numerical analysis. We often have difficulties solving problems with the computer because we run into limits of precision and accuracy. This can cause our procedures to take a long time to execute (or be impossible to run) and can result in wrong answers. It helps to know the basic principles of this field so that you can understand where the pitfalls are and work around them.

• Computer science. Specifically, some discrete mathematics and methods of optimization contained therein. This includes some basic graph theory, design of data structures, algorithms, and recursion, as well as a study of complexity theory.

• Geometry. Of course. But not Euclidean geometry: a tiny bit of spherical geometry, naturally; but more important is the modern view (dating to Felix Klein in the late 1800's) of geometry as the study of groups of transformations of objects. This is the unifying concept to moving objects around on the earth or on the map, to congruence, to similarity.

• Statistics. Not all GIS professionals need to know statistics, but it is becoming clear that a basic statistical way of thinking is essential. All our data are ultimately derived from measurements and heavily processed afterwards. The measurements and the processing introduce errors that can only be treated as random. We need to understand randomness, how to model it, how to control it when possible, and how to measure it and respond to it in any case. That does not mean studying t-tests, F-tests, etc; it means studying the foundations of statistics so that we can become effective problem solvers and decision makers in the face of chance. It also means learning some modern ideas of statistics, including exploratory data analysis and robust estimation as well as principles of constructing statistical models.

Please note that I am not advocating that all GIS practitioners need to learn all this stuff! Also, I am not suggesting that the different topics should be learned in isolation by taking separate courses. This is merely an (incomplete) compendium of some of the most powerful and beautiful ideas that many GIS people would deeply appreciate (and be able to apply) were they to know them. What I suspect we need is to learn enough about these subjects to know when they might be applicable, to know where to go for help, and to know how to learn more if it should be needed for a project or a job. From that perspective, taking a lot of courses would be overkill and would likely tax the patience of the most dedicated student. But for anyone who has an opportunity to learn some mathematics and has a choice of what to learn and how to learn it, this list might provide some guidance.