Geography Reference
In-Depth Information
Mathematical Analysis in Cartography
by Means of Computer Algebra System
Shao-Feng Bian and Hou-Pu Li
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/50159
1. Introduction
Theory of map projections is a branch of cartography studying the ways of projecting the
curved surface of the earth and other heavenly bodies into the plane, and it is often called
mathematical cartography. There are many fussy symbolic problems to be dealt with in map
projections, such as power series expansions of elliptical functions, high order differential of
transcendental functions, elliptical integrals and the operation of complex numbers. Many
famous cartographers such as Adams (1921), Snyder (1987), Yang (1989, 2000) have made
great efforts to solve these problems. Due to historical condition limitation, there were no
advanced computer algebra systems at that time, so they had to dispose of these problems
by hand, which had often required a paper and a pen. Some derivations and
computations were however long and labor intensive such that one gave up midway.
Briefly reviewing the existing methods, one will find that these problems were not
perfectly and ideally solved yet. Formulas derived by hand often have quite complex and
prolix forms, and their orders could not be very high. The most serious problem is that
some higher terms of the formulas are erroneous because of the adopted approximate
disposal.
With the advent of computers, the paper and pen approach is slowly being replaced by
software developed to undertake symbolic derivations tasks. Specially, where symbolic
rather than numerical solutions are desired, this software normally comes in handy.
Software which performs symbolic computations is called computer algebra system.
Nowadays, computer algebra systems like Maple, Mathcad, and Mathematica are widely
used by scientists and engineers in different fields(Awang, 2005; Bian, 2006). By means of
computer algebra system Mathematica, we have already solved many complicated
mathematical problems in special fields of cartography in the past few years. Our research
 
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