**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

© 2012 Bian and Li, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Bian and Li, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

results indicate that the derivation efficiency can be significantly improved and formulas impossible to be obtained by hand can be easily derived with the help of Mathematica, which renovates the traditional analysis methods and enriches the mathematical theory basis of cartography to a certain extent.

Mathematical Analysis in Cartography by Means of Computer Algebra System 3

(3)

where *X* is the meridian arc; *B* is the geodetic latitude; *a* is the semi-major axis of the

(1) is an elliptic integral of the second kind and there is no analytical solution. Expanding

02 4 6 8 10 *X a e KB K B K B K B K B K B* (1 )( sin 2 sin 4 sin6 sin8 sin10 ) (2)

24 6 8 10

2 4 6 8 10

4 6 8 10

3 45 175 11025 43659 <sup>1</sup> 4 64 256 16384 65536 3 15 525 2205 72765 8 32 1024 4096 131072 15 105 2205 10395 256 1024 16384 65536 35 105 10395 3072 4096 262144

*K ee e e e*

*K ee e e e*

*Ke e e e*

3465 524288

6 8 10

*e*

2

246810

simplicity and computing efficiency, it is better to expand (6) into a power series of the eccentricity. This process is easily done by means of Mathematica. As a result, (6) becomes:

/ / / / /

*K K K K K K K K K K*

 

 

*X*

 

*BBBBB B* sin 2 sin 4 sin6 sin8 sin10 (5)

(4)

 

(6)

up to sin8*B* . For

*X*

the integrand by binomial theorem and itntegrating it item by item yield:

8

693 1310720

> 

Yang (1989, 2000) gave an expansion similar to (5) but expanded

<sup>10</sup>

*K ee e*

10

315 131072

*K e*

is defined as

*K e*

reference ellipsoid;

where

where

2

0

 

2

4

6

8

 

The rectifying latitude

Inserting (2) into (4) yields

10

2 ( )

The main contents and research results presented in this chapter are organized as follows. In Section II, we discuss the direct transformations from geodetic latitude to three kinds of auxiliary latitudes often used in cartography, and the direct transformations from these auxiliary latitudes to geodetic latitude are studied in Section III. In Section IV, the direct expansions of transformations between meridian arc, isometric latitude, and authalic functions are derived. In Section V, we discuss the non-iterative expressions of the forward and inverse Gauss projections by complex numbers. Finally in Section VI, we make a brief summary. It is assumed that the readers are somewhat conversant with Mathematica and its syntax.
