1. Introduction

Geodesy is the science of accurately measuring and understanding three fundamental properties of the Earth: its geometric shape, its orientation in space, and its gravity field, as well as the changes of these properties with time. There are many fussy symbolic problems to be dealt with in geodesy, such as the power series expansions of the ellipsoid's eccentricity, high order derivation of complex and implicit functions, expansions of special functions and integral

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is 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.

transformation. Many geodesists have made great efforts to solve these problems, see [1–8]. Due to historical condition limitation, they mainly disposed of these problems by hand, which were not perfectly solved yet. Traditional algorithms derived by hand mainly have the following problems: (1) The expressions are complex and lengthy, which makes the computation process very complicated and time-consuming. (2) Some approximate disposal is adopted, which influences the computation accuracy. (3) Some formulae are numerical and only apply to a specific reference ellipsoid, which are not convenient to be generalized.

where X is the meridian arc; Bis the geodetic latitude; a is the semi-major axis of the reference

Mathematical Analysis of Some Typical Problems in Geodesy by Means of Computer Algebra

http://dx.doi.org/10.5772/intechopen.81586

69

Expanding the integrand in Eq. (1) and integrating it item by item using Mathematica as

<sup>2</sup> � �ð Þ <sup>K</sup>0<sup>B</sup> <sup>þ</sup> <sup>K</sup><sup>2</sup> sin 2<sup>B</sup> <sup>þ</sup> <sup>K</sup><sup>4</sup> sin 4<sup>B</sup> <sup>þ</sup> <sup>K</sup><sup>6</sup> sin 6<sup>B</sup> <sup>þ</sup> <sup>K</sup><sup>8</sup> sin 8<sup>B</sup> <sup>þ</sup> <sup>K</sup><sup>10</sup> sin 10<sup>B</sup> (2)

<sup>6</sup> � <sup>2205</sup> <sup>4096</sup> <sup>e</sup>

8 þ

2205 <sup>16384</sup> <sup>e</sup>

10

<sup>8</sup> � <sup>10395</sup> <sup>262144</sup> <sup>e</sup>

11025 <sup>16384</sup> <sup>e</sup>

8 þ

10395 <sup>65536</sup> <sup>e</sup> 10

10

<sup>8</sup> � <sup>72765</sup> <sup>131072</sup> <sup>e</sup>

43659 <sup>65536</sup> <sup>e</sup> 10

10

(3)

ellipsoid; e is the first eccentricity of the reference ellipsoid.

follows:

Then one arrives at

where

X ¼ a 1 � e

K<sup>0</sup> ¼ 1 þ

8

>>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>>:

<sup>K</sup><sup>2</sup> ¼ � <sup>3</sup> 8 e <sup>2</sup> � <sup>15</sup> 32 e

<sup>K</sup><sup>4</sup> <sup>¼</sup> <sup>15</sup> <sup>256</sup> <sup>e</sup> 4 þ 105 <sup>1024</sup> <sup>e</sup> 6 þ

<sup>K</sup><sup>6</sup> ¼ � <sup>35</sup>

<sup>K</sup><sup>8</sup> <sup>¼</sup> <sup>315</sup> <sup>131072</sup> <sup>e</sup>

<sup>K</sup><sup>10</sup> ¼ � <sup>693</sup>

3 4 e 2 þ 45 64 e 4 þ 175 <sup>256</sup> <sup>e</sup> 6 þ

<sup>3072</sup> <sup>e</sup>

<sup>4</sup> � <sup>525</sup> <sup>1024</sup> <sup>e</sup>

3465 <sup>524288</sup> <sup>e</sup>

10

<sup>6</sup> � <sup>105</sup> <sup>4096</sup> <sup>e</sup>

8 þ

<sup>1310720</sup> <sup>e</sup>

In computational mathematics, computer algebra, also called symbolic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Software applications that perform symbolic calculations are called computer algebra systems, which are more popular today. Computer algebra systems, like Mathematica, Maple and Mathcad, are powerful tools to solve some mathematical derivation in geodesy, see [9–11]. By means of computer algebra analysis method and computer algebra system Mathematica, we have already solved many complicated mathematical problems in special fields of geodesy in the past few years; see [12–15].

The main contents and research results presented in this chapter are organized as follows: In Section 2, we discuss the forward and inverse expansions of the meridian arc often used in geometric geodesy. In Section 3, the nonsingular expressions of singular integration in physical geodesy are derived. In Section 4, we discuss series expansions of direct transformations between three anomalies in satellite geodesy. Finally in Section 5, we make a brief summary.
