4.1. The series expansions of the direct transformation between eccentric and mean anomalies

Let the mean anomaly be M. Mcan be expressed by E as follows:

$$M = E - e \sin E \tag{53}$$

Differentiating the both sides of Eq. (53) yields

Figure 2. Keplerian orbit.

ð47Þ

ð48Þ

ð49Þ

ð50Þ

ð51Þ

In case of a square grid with a unit length, Eqs. (45)–(48) can be simplified in Mathematica as.

78 Trends in Geomatics - An Earth Science Perspective

$$\frac{dE}{dM} = \frac{1}{1 - e \cos E} \tag{54}$$

where

where

b<sup>1</sup> ¼ e þ

8

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

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

Integrating at the both sides of Eq. (62) gives the series expansion

8

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

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

<sup>b</sup><sup>2</sup> <sup>¼</sup> <sup>4</sup>e<sup>2</sup> <sup>þ</sup>

<sup>b</sup><sup>3</sup> <sup>¼</sup> <sup>27</sup>e<sup>3</sup> <sup>þ</sup>

<sup>b</sup><sup>4</sup> <sup>¼</sup> <sup>256</sup>e<sup>4</sup> <sup>þ</sup>

<sup>b</sup><sup>5</sup> <sup>¼</sup> <sup>3125</sup>e<sup>5</sup> <sup>þ</sup>

<sup>b</sup><sup>6</sup> <sup>¼</sup> <sup>46656</sup>e<sup>6</sup> <sup>þ</sup>

<sup>b</sup><sup>7</sup> <sup>¼</sup> <sup>823543</sup>e<sup>7</sup> <sup>b</sup><sup>8</sup> <sup>¼</sup> <sup>16777216</sup>e<sup>8</sup>

þ α<sup>6</sup> sin 6M þ α<sup>7</sup> sin 7M þ α<sup>8</sup> sin 8M

<sup>α</sup><sup>1</sup> <sup>¼</sup> <sup>e</sup> � <sup>1</sup>

<sup>α</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> 2 e <sup>2</sup> � <sup>1</sup> 6 e 4 þ 1 48 e

<sup>α</sup><sup>3</sup> <sup>¼</sup> <sup>3</sup> 8 e <sup>3</sup> � <sup>27</sup> <sup>128</sup> <sup>e</sup> 5 þ 243 <sup>5120</sup> <sup>e</sup> 7

<sup>α</sup><sup>4</sup> <sup>¼</sup> <sup>1</sup> 3 e <sup>4</sup> � <sup>4</sup> 15 e 6 þ 4 45 e 8

<sup>α</sup><sup>5</sup> <sup>¼</sup> <sup>125</sup> <sup>384</sup> <sup>e</sup>

<sup>α</sup><sup>6</sup> <sup>¼</sup> <sup>27</sup> 80 e

<sup>α</sup><sup>7</sup> <sup>¼</sup> <sup>16807</sup> <sup>46080</sup> <sup>e</sup> 7

<sup>α</sup><sup>8</sup> <sup>¼</sup> <sup>128</sup> <sup>315</sup> <sup>e</sup> 8

8 e 3 þ 1 <sup>192</sup> <sup>e</sup>

1 2 e 3 þ 1 <sup>24</sup> <sup>e</sup> <sup>þ</sup> <sup>5</sup> <sup>61</sup>

> 13 3 e 4 þ 47 15 e 6 þ 121 <sup>63</sup> <sup>e</sup> 8

91 2 e 5 þ 1127 <sup>24</sup> <sup>e</sup> 7

2937 <sup>5</sup> <sup>e</sup> 6 þ

E ¼ M þ α<sup>1</sup> sin M þ α<sup>2</sup> sin 2M þ α<sup>3</sup> sin 3M þ α<sup>4</sup> sin 4M þ α<sup>5</sup> sin 5M

<sup>5</sup> � <sup>3125</sup> <sup>9216</sup> <sup>e</sup> 7

<sup>6</sup> � <sup>243</sup> <sup>560</sup> <sup>e</sup> 8

18173 <sup>2</sup> <sup>e</sup> 7

> 1150593 <sup>7</sup> <sup>e</sup> 8

> > <sup>5</sup> � <sup>1</sup> <sup>9216</sup> <sup>e</sup> 7

> > > <sup>6</sup> � <sup>1</sup> <sup>720</sup> <sup>e</sup> 8

<sup>720</sup> <sup>e</sup> 7

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

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

82771 <sup>105</sup> <sup>e</sup> 8

(62)

81

(63)

(64)

To expand Eq. (54) into a power series of cos M, we introduce the following new variable

$$t = \cos M\tag{55}$$

therefore

$$\frac{dM}{dt} = -\frac{1}{\sin M} \tag{56}$$

and then denote

$$f(t) = \frac{dE}{dM} = \frac{1}{1 - e \cos E} \tag{57}$$

Substituting <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>π</sup> <sup>2</sup> into Eq. (53) yields

$$M\_0 = \frac{\pi}{2} - e$$

Substituting Eq. (59) into Eq. (55), one arrives at

$$t\_0 = \sin e \tag{59}$$

Making use of the chain rule of implicit differentiation

$$\begin{aligned} f'(t) &= \frac{df}{dE} \frac{dE}{dM} \frac{dM}{dt} \\ f''(t) &= \frac{df'}{dE} \frac{dE}{dM} \frac{dM}{dt} + \frac{df'}{dM} \frac{dM}{dt} \\ \dots \end{aligned}$$

It is easy to expand Eq. (58) into a power series of t<sup>0</sup>

$$f(t) = \frac{dE}{dM} = f(t\_0) + f'(t\_0)(t - t\_0) + \frac{1}{2!}f''(t\_0)(t - t\_0)^2 + \frac{1}{3!}f''(t\_0)(t - t\_0)^3 + \dotsb \tag{60}$$

One can imagine that these procedures are too complicated to be realized by hand, but will become much easier and be significantly simplified by means of Mathematica. Omitting the detailed procedure in Mathematica, one arrives at

$$\begin{aligned} \frac{dE}{dM} &= 1 + b\_1(\cos M - \sin e) + \frac{b\_2}{2!}(\cos M - \sin e)^2 + \frac{b\_3}{3!}(\cos M - \sin e)^3 + \frac{b\_4}{4!}(\cos M - \sin e)^4 \\ &+ \frac{b\_5}{5!}(\cos M - \sin e)^5 + \frac{b\_6}{6!}(\cos M - \sin e)^6 + \frac{b\_7}{7!}(\cos M - \sin e)^7 + \frac{b\_8}{8!}(\cos M - \sin e)^8 \end{aligned} \tag{61}$$

Mathematical Analysis of Some Typical Problems in Geodesy by Means of Computer Algebra http://dx.doi.org/10.5772/intechopen.81586 81

where

dE dM <sup>¼</sup> <sup>1</sup>

dM

f tðÞ¼ dE

<sup>2</sup> into Eq. (53) yields

Substituting Eq. (59) into Eq. (55), one arrives at

Making use of the chain rule of implicit differentiation

It is easy to expand Eq. (58) into a power series of t<sup>0</sup>

dM <sup>¼</sup> f tð Þþ <sup>0</sup> <sup>f</sup>

detailed procedure in Mathematica, one arrives at

f tðÞ¼ dE

dM <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup>1<sup>ð</sup> cos <sup>M</sup> � sin <sup>e</sup>Þ þ <sup>b</sup><sup>2</sup>

ð Þ cos <sup>M</sup> � sin <sup>e</sup> <sup>5</sup> <sup>þ</sup>

dE

þ b5 5! f 0 ðÞ¼ <sup>t</sup> df dE dE dM

f 00 ðÞ¼ <sup>t</sup> df <sup>0</sup> dE dE dM

0

2!

b6 6! ð Þ t<sup>0</sup> ð Þþ t � t<sup>0</sup>

ð Þ cos M � sin e

ð Þ cos <sup>M</sup> � sin <sup>e</sup> <sup>6</sup> <sup>þ</sup>

therefore

and then denote

80 Trends in Geomatics - An Earth Science Perspective

Substituting <sup>E</sup><sup>0</sup> <sup>¼</sup> <sup>π</sup>

To expand Eq. (54) into a power series of cos M, we introduce the following new variable

dt ¼ � <sup>1</sup>

dM <sup>¼</sup> <sup>1</sup>

dM dt

> dM dt <sup>þ</sup> df <sup>0</sup> dM

dt <sup>⋯</sup>

1 2! f 00

One can imagine that these procedures are too complicated to be realized by hand, but will become much easier and be significantly simplified by means of Mathematica. Omitting the

> 2 þ b3 3!

> > b7 7!

dM

2 þ 1 3! f

ð Þ cos <sup>M</sup> � sin <sup>e</sup> <sup>3</sup> <sup>þ</sup>

ð Þ cos M � sin e

ð Þ t<sup>0</sup> ð Þ t � t<sup>0</sup>

<sup>M</sup><sup>0</sup> <sup>¼</sup> <sup>π</sup>

<sup>1</sup> � <sup>e</sup> cos <sup>E</sup> (54)

t ¼ cos M (55)

sin <sup>M</sup> (56)

<sup>1</sup> � <sup>e</sup> cos <sup>E</sup> (57)

<sup>2</sup> � <sup>e</sup> (58)

‴ð Þ t<sup>0</sup> ð Þ t � t<sup>0</sup>

b4 4!

7 þ b8 8! <sup>3</sup> <sup>þ</sup> <sup>⋯</sup> (60)

ð Þ cos M � sin e

ð Þ cos <sup>M</sup> � sin <sup>e</sup> <sup>8</sup>

4

(61)

t<sup>0</sup> ¼ sin e (59)

$$\begin{cases} b\_1 = e + \frac{1}{2}e^3 + \frac{1}{24}e + 5\frac{61}{720}e^7 \\\\ b\_2 = 4e^2 + \frac{13}{3}e^4 + \frac{47}{15}e^6 + \frac{121}{63}e^8 \\\\ b\_3 = 27e^3 + \frac{91}{2}e^5 + \frac{1127}{24}e^7 \\\\ b\_4 = 256e^4 + \frac{2937}{5}e^6 + \frac{82771}{105}e^8 \\\\ b\_5 = 3125e^5 + \frac{18173}{2}e^7 \\\\ b\_6 = 46656e^6 + \frac{1150593}{7}e^8 \\\\ b\_7 = 823543e^7 \\\\ b\_8 = 16777216e^8 \end{cases} \tag{62}$$

Integrating at the both sides of Eq. (62) gives the series expansion

$$\begin{aligned} E &= M + a\_1 \sin M + a\_2 \sin 2M + a\_3 \sin 3M + a\_4 \sin 4M + a\_5 \sin 5M \\ &+ a\_6 \sin 6M + a\_7 \sin 7M + a\_8 \sin 8M \end{aligned} \tag{63}$$

where

$$\begin{cases} \alpha\_1 = e - \frac{1}{8}e^3 + \frac{1}{192}e^5 - \frac{1}{9216}e^7 \\\\ \alpha\_2 = \frac{1}{2}e^2 - \frac{1}{6}e^4 + \frac{1}{48}e^6 - \frac{1}{720}e^8 \\\\ \alpha\_3 = \frac{3}{8}e^3 - \frac{27}{128}e^5 + \frac{243}{5120}e^7 \\\\ \alpha\_4 = \frac{1}{3}e^4 - \frac{4}{15}e^6 + \frac{4}{45}e^8 \\\\ \alpha\_5 = \frac{125}{384}e^5 - \frac{3125}{9216}e^7 \\\\ \alpha\_6 = \frac{27}{80}e^6 - \frac{243}{560}e^8 \\\\ \alpha\_7 = \frac{16807}{46080}e^7 \\\\ \alpha\_8 = \frac{128}{315}e^8 \end{cases} (64)$$

#### 4.2. The series expansions of the direct transformation between eccentric and true anomalies

The true anomaly υ can be expressed by E as follows:

$$
\tan\frac{\upsilon}{2} = \sqrt{\frac{1+e}{1-e}} \tan\frac{E}{2} \tag{65}
$$

E ¼ υ þ γ<sup>1</sup> sin υ þ γ<sup>2</sup> sin 2υ þ γ<sup>3</sup> sin 3υ þ γ<sup>4</sup> sin 4υ þ γ<sup>5</sup> sin 5υ

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

(70)

83

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

(71)

(72)

(73)

þ γ<sup>6</sup> sin 6υ þ γ<sup>7</sup> sin 7υ þ γ<sup>8</sup> sin 8υ

<sup>γ</sup><sup>1</sup> ¼ �<sup>e</sup> � <sup>1</sup>

<sup>γ</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> 4 e 2 þ 1 8 e 4 þ 5 64 e 6 þ 7 <sup>128</sup> <sup>e</sup> 8

8

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

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

<sup>γ</sup><sup>3</sup> ¼ � <sup>1</sup> 12 e <sup>3</sup> � <sup>1</sup> 16 e <sup>5</sup> � <sup>3</sup> 64 e 7

<sup>γ</sup><sup>4</sup> <sup>¼</sup> <sup>1</sup> 32 e 4 þ 1 32 e 6 þ 7 <sup>256</sup> <sup>e</sup> 8

<sup>γ</sup><sup>5</sup> ¼ � <sup>1</sup> 80 e <sup>5</sup> � <sup>1</sup> 64 e 7

<sup>γ</sup><sup>6</sup> <sup>¼</sup> <sup>1</sup> <sup>192</sup> <sup>e</sup> 6 þ 1 <sup>128</sup> <sup>e</sup> 8

<sup>γ</sup><sup>7</sup> ¼ � <sup>1</sup>

<sup>γ</sup><sup>8</sup> <sup>¼</sup> <sup>1</sup> <sup>1024</sup> <sup>e</sup> 8

The whole formulae for the transformation from M to υ are as follows

8 >>>>>><

>>>>>>:

where

þ α<sup>6</sup> sin 6M þ α<sup>7</sup> sin 7M þ α<sup>8</sup> sin 8M

þ β<sup>6</sup> sin 6E þ β<sup>7</sup> sin 7E þ β<sup>8</sup> sin 8E

<sup>448</sup> <sup>e</sup> 7

4.3. The series expansions of the direct transformation between mean and true anomalies

E ¼ M þ α<sup>1</sup> sin M þ α<sup>2</sup> sin 2M þ α<sup>3</sup> sin 3M þ α<sup>4</sup> sin 4M þ α<sup>5</sup> sin 5M

υ ¼ E þ β<sup>1</sup> sin E þ β<sup>2</sup> sin 2E þ β<sup>3</sup> sin 3E þ β<sup>4</sup> sin 4E þ β<sup>5</sup> sin 5E

Since the coefficients αi,β<sup>i</sup> (i ¼ 1, 2, ⋯8) are expressed in a power series of the eccentricity, one could expand υ as a power series of the eccentricity at e ¼ 0 in order to obtain the direct expansion of the transformation from M to υ. Omitting the main operations by means of

υ ¼ M þ δ<sup>1</sup> sin M þ δ<sup>2</sup> sin 2M þ δ<sup>3</sup> sin 3M þ δ<sup>4</sup> sin 4M þ δ<sup>5</sup> sin 5M

Mathematica, one arrives at the direct expansion of the transformation from M to υ

þδ<sup>6</sup> sin 6M þ δ<sup>7</sup> sin 7M þ δ<sup>8</sup> sin 8M

4 e <sup>3</sup> � <sup>1</sup> 8 e <sup>5</sup> � <sup>5</sup> 64 e 7

where

Therefore, it holds

$$\upsilon = 2 \arctan\left(\sqrt{\frac{1+e}{1-e}} \tan\frac{E}{2}\right) \tag{66}$$

One could expand υ as a power series of the eccentricity at e ¼ 0 in order to obtain the direct series expansion of the transformation from E to υ. Omitting the detailed procedure in Mathematica, one arrives at

$$\begin{aligned} \upsilon &= E + \beta\_1 \sin E + \beta\_2 \sin 2E + \beta\_3 \sin 3E + \beta\_4 \sin 4E + \beta\_5 \sin 5E \\ &+ \beta\_6 \sin 6E + \beta\_7 \sin 7E + \beta\_8 \sin 8E \end{aligned} \tag{67}$$

where

$$\begin{cases} \beta\_1 = e + \frac{1}{4}e^3 + \frac{1}{8}e^5 + \frac{5}{64}e^7 \\ \beta\_2 = \frac{1}{4}e^2 + \frac{1}{8}e^4 + \frac{5}{64}e^6 + \frac{7}{128}e^8 \\ \beta\_3 = \frac{1}{12}e^3 + \frac{1}{16}e^5 + \frac{3}{64}e^7 \\ \beta\_4 = \frac{1}{32}e^4 + \frac{1}{32}e^6 + \frac{7}{256}e^8 \\ \beta\_5 = \frac{1}{80}e^5 + \frac{1}{64}e^7 \\ \beta\_6 = \frac{1}{192}e^6 + \frac{1}{128}e^8 \\ \beta\_7 = \frac{1}{448}e^7 \\ \beta\_8 = \frac{1}{1024}e^8 \end{cases} \tag{68}$$

From Eq. (67), one knows

$$E = 2 \arctan\left(\sqrt{\frac{1-e}{1+e}} \tan\frac{\upsilon}{2}\right) \tag{69}$$

Expanding E as a power series of the eccentricity at e ¼ 0 by means of Mathematica yields the direct series expansion of the transformation from υ to E

Mathematical Analysis of Some Typical Problems in Geodesy by Means of Computer Algebra http://dx.doi.org/10.5772/intechopen.81586 83

$$\begin{aligned} E &= \nu + \gamma\_1 \sin \nu + \gamma\_2 \sin 2\nu + \gamma\_3 \sin 3\nu + \gamma\_4 \sin 4\nu + \gamma\_5 \sin 5\nu \\ &+ \gamma\_6 \sin 6\nu + \gamma\_7 \sin 7\nu + \gamma\_8 \sin 8\nu \end{aligned} \tag{70}$$

where

4.2. The series expansions of the direct transformation between eccentric and true

ffiffiffiffiffiffiffiffiffiffi 1 þ e 1 � e

> ffiffiffiffiffiffiffiffiffiffi 1 þ e 1 � e

!

r

One could expand υ as a power series of the eccentricity at e ¼ 0 in order to obtain the direct series expansion of the transformation from E to υ. Omitting the detailed procedure in Mathematica,

υ ¼ E þ β<sup>1</sup> sin E þ β<sup>2</sup> sin 2E þ β<sup>3</sup> sin 3E þ β<sup>4</sup> sin 4E þ β<sup>5</sup> sin 5E

tan E

> tan E 2

<sup>þ</sup> <sup>β</sup><sup>6</sup> sin 6<sup>E</sup> <sup>þ</sup> <sup>β</sup><sup>7</sup> sin 7<sup>E</sup> <sup>þ</sup> <sup>β</sup><sup>8</sup> sin 8<sup>E</sup> (67)

<sup>2</sup> (65)

(66)

(68)

(69)

r

tan υ 2 ¼

υ ¼ 2arctan

β<sup>1</sup> ¼ e þ

<sup>β</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> 4 e 2 þ 1 8 e 4 þ 5 64 e 6 þ 7 <sup>128</sup> <sup>e</sup> 8

8

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

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

direct series expansion of the transformation from υ to E

<sup>β</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup> 12 e 3 þ 1 16 e 5 þ 3 64 e 7

<sup>β</sup><sup>4</sup> <sup>¼</sup> <sup>1</sup> 32 e 4 þ 1 32 e 6 þ 7 <sup>256</sup> <sup>e</sup> 8

<sup>β</sup><sup>5</sup> <sup>¼</sup> <sup>1</sup> 80 e 5 þ 1 64 e 7

<sup>β</sup><sup>6</sup> <sup>¼</sup> <sup>1</sup> <sup>192</sup> <sup>e</sup> 6 þ 1 <sup>128</sup> <sup>e</sup> 8

<sup>β</sup><sup>7</sup> <sup>¼</sup> <sup>1</sup> <sup>448</sup> <sup>e</sup> 7

<sup>β</sup><sup>8</sup> <sup>¼</sup> <sup>1</sup> <sup>1024</sup> <sup>e</sup> 8

E ¼ 2arctan

ffiffiffiffiffiffiffiffiffiffi 1 � e 1 þ e

!

tan υ 2

r

Expanding E as a power series of the eccentricity at e ¼ 0 by means of Mathematica yields the

1 4 e 3 þ 1 8 e 5 þ 5 64 e 7

The true anomaly υ can be expressed by E as follows:

82 Trends in Geomatics - An Earth Science Perspective

anomalies

Therefore, it holds

one arrives at

From Eq. (67), one knows

where

$$\begin{cases} \mathcal{V}\_{1} = -e - \frac{1}{4}e^{3} - \frac{1}{8}e^{5} - \frac{5}{64}e^{7} \\\\ \mathcal{V}\_{2} = \frac{1}{4}e^{2} + \frac{1}{8}e^{4} + \frac{5}{64}e^{6} + \frac{7}{128}e^{8} \\\\ \mathcal{V}\_{3} = -\frac{1}{12}e^{3} - \frac{1}{16}e^{5} - \frac{3}{64}e^{7} \\\\ \mathcal{V}\_{4} = \frac{1}{32}e^{4} + \frac{1}{32}e^{6} + \frac{7}{256}e^{8} \\\\ \mathcal{V}\_{5} = -\frac{1}{80}e^{5} - \frac{1}{64}e^{7} \\\\ \mathcal{V}\_{6} = \frac{1}{192}e^{6} + \frac{1}{128}e^{8} \\\\ \mathcal{V}\_{7} = -\frac{1}{448}e^{7} \\\\ \mathcal{V}\_{8} = \frac{1}{1024}e^{8} \end{cases} \tag{71}$$

#### 4.3. The series expansions of the direct transformation between mean and true anomalies

The whole formulae for the transformation from M to υ are as follows

$$\begin{cases} E = M + \alpha\_1 \sin M + \alpha\_2 \sin 2M + \alpha\_3 \sin 3M + \alpha\_4 \sin 4M + \alpha\_5 \sin 5M \\\\ \quad + \alpha\_6 \sin 6M + \alpha\_7 \sin 7M + \alpha\_8 \sin 8M \\\\ \nu = E + \beta\_1 \sin E + \beta\_2 \sin 2E + \beta\_3 \sin 3E + \beta\_4 \sin 4E + \beta\_5 \sin 5E \\\\ \quad + \beta\_6 \sin 6E + \beta\_7 \sin 7E + \beta\_8 \sin 8E \end{cases} \tag{72}$$

Since the coefficients αi,β<sup>i</sup> (i ¼ 1, 2, ⋯8) are expressed in a power series of the eccentricity, one could expand υ as a power series of the eccentricity at e ¼ 0 in order to obtain the direct expansion of the transformation from M to υ. Omitting the main operations by means of Mathematica, one arrives at the direct expansion of the transformation from M to υ

$$\begin{aligned} \upsilon &= M + \delta\_1 \sin M + \delta\_2 \sin 2M + \delta\_3 \sin 3M + \delta\_4 \sin 4M + \delta\_5 \sin 5M \\ &+ \delta\_6 \sin 6M + \delta\_7 \sin 7M + \delta\_8 \sin 8M \end{aligned} \tag{73}$$

where

$$\begin{cases} \delta\_1 = 2\varepsilon - \frac{1}{4}\varepsilon^3 + \frac{5}{96}\varepsilon^5 + \frac{107}{4608}\varepsilon^7\\ \delta\_2 = \frac{5}{4}\varepsilon^2 - \frac{11}{24}\varepsilon^4 + \frac{17}{192}\varepsilon^6 + \frac{43}{5760}\varepsilon^8\\ \delta\_3 = \frac{13}{12}\varepsilon^3 - \frac{43}{64}\varepsilon^5 + \frac{95}{512}\varepsilon^7\\ \delta\_4 = \frac{103}{96}\varepsilon^4 - \frac{451}{480}\varepsilon^6 + \frac{4123}{11520}\varepsilon^8\\ \delta\_5 = \frac{1097}{960}\varepsilon^5 - \frac{5957}{4608}\varepsilon^7\\ \delta\_6 = \frac{1223}{960}\varepsilon^6 - \frac{7913}{4480}\varepsilon^8\\ \delta\_7 = \frac{47273}{32256}\varepsilon^7\\ \delta\_8 = \frac{556403}{322560}\varepsilon^8 \end{cases} \tag{74}$$

4.4. The accuracy of the derived series expansions

Table 2. Errors of the derived series expansions.

series expansions, which are denoted as ΔE1, Δυ1= <sup>00</sup> ð Þ, ΔM1= <sup>00</sup> ð Þ,

expansions is correspondingly higher than 10�10<sup>00</sup> , 10�3<sup>00</sup> and 0:1

reference ellipsoid and extended up to its tenth-order terms.

tions, and the nonsingular expressions are systematically derived.

results presented in this chapter are as follows:

5. Conclusions

space, these errors when e is equal to 0.05 are only listed in Table 2.

In order to validate the exactness of the derived series expansions, one has examined their

One makes use of Eq. (53) and Eq. (67) to obtain the theoretical value M<sup>0</sup> and υ<sup>0</sup> at given geodetic latitude E0. Substituting M<sup>0</sup> into Eq. (64) and Eq. (74), one arrives at the computation value E<sup>1</sup> and υ1. Substituting υ<sup>0</sup> into Eq. (71) and Eq. (77), one arrives at the computation value E<sup>2</sup> and M1. Substituting E<sup>0</sup> into Eq. (68), one arrives at the computation value υ2. The differences between the computation and theoretical values indicate the accuracies of the derived

From Table 2, one could find that the accuracy of derived series expansions is higher than 10�5<sup>00</sup> , which could satisfy practical application. Other numerical examples indicate that when the orbital eccentricity e is respectively equal to 0.01, 0.01 and 0.2, the accuracy of derived series

Some typical mathematical problems in geodesy are solved by means of computer algebra analysis method and computer algebra system Mathematica. The main contents and research

1. The forward and inverse expansions of the meridian arc often used in geometric geodesy are derived. Their coefficients are expressed in a power series of the first eccentricity of the

2. The singularity existing in the integral expressions of height anomaly, deflections of the vertical and gravity gradient is eliminated using the nonsingular integration transforma-

ΔE2= <sup>00</sup> ð Þ,

00 . Δυ2= <sup>00</sup> ð Þ. Due to limited

accuracies when the orbital eccentricity e is respectively equal to 0.01, 0.05, 0.1 and 0.2.

<sup>E</sup>0<sup>=</sup> <sup>∘</sup> ð Þ <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup>

<sup>Δ</sup>E1<sup>=</sup> <sup>00</sup> ð Þ <sup>8</sup>:<sup>2</sup> � <sup>10</sup>�<sup>8</sup> �1:<sup>7</sup> � <sup>10</sup>�<sup>7</sup> 1.6 � <sup>10</sup>�<sup>7</sup> <sup>2</sup>:<sup>2</sup> � <sup>10</sup>�<sup>8</sup> <sup>Δ</sup>υ1<sup>=</sup> <sup>00</sup> ð Þ <sup>3</sup>:<sup>5</sup> � <sup>10</sup>�<sup>7</sup> �7:<sup>4</sup> � <sup>10</sup>�<sup>7</sup> <sup>6</sup>:<sup>8</sup> � <sup>10</sup>�<sup>7</sup> <sup>9</sup>:<sup>9</sup> � <sup>10</sup>�<sup>8</sup> <sup>Δ</sup>M1<sup>=</sup> <sup>00</sup> ð Þ 3.5 � <sup>10</sup>�<sup>8</sup> �5:<sup>2</sup> � <sup>10</sup>�<sup>9</sup> �7:<sup>1</sup> � <sup>10</sup>�<sup>9</sup> �3:<sup>7</sup> � <sup>10</sup>�<sup>9</sup> <sup>Δ</sup>E2<sup>=</sup> <sup>00</sup> ð Þ <sup>2</sup>:<sup>8</sup> � <sup>10</sup>�<sup>8</sup> <sup>2</sup>:<sup>1</sup> � <sup>10</sup>�<sup>8</sup> <sup>1</sup>:<sup>4</sup> � <sup>10</sup>�<sup>8</sup> <sup>1</sup>:<sup>2</sup> � <sup>10</sup>�<sup>8</sup> <sup>Δ</sup>υ2<sup>=</sup> <sup>00</sup> ð Þ �2:<sup>9</sup> � <sup>10</sup>�<sup>8</sup> �2:<sup>4</sup> � <sup>10</sup>�<sup>8</sup> �1:<sup>5</sup> � <sup>10</sup>�<sup>8</sup> �1:<sup>3</sup> � <sup>10</sup>�<sup>8</sup>

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

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

85

The whole formulae for the transformation from υ to Mare as follows

$$\begin{cases} E = \nu + \gamma\_1 \sin \nu + \gamma\_2 \sin 2\nu + \gamma\_3 \sin 3\nu + \gamma\_4 \sin 4\nu + \gamma\_5 \sin 5\nu \\ \quad + \gamma\_6 \sin 6\nu + \gamma\_7 \sin 7\nu + \gamma\_8 \sin 8\nu \\ M = E - e \sin E \end{cases} \tag{75}$$

Expanding Mas a power series of the eccentricity at e ¼ 0 by means of Mathematica yields the direct series expansion of the transformation from υ to M

$$\begin{aligned} M &= \upsilon + \varepsilon\_1 \sin \upsilon + \varepsilon\_2 \sin 2\upsilon + \varepsilon\_3 \sin 3\upsilon + \varepsilon\_4 \sin 4\upsilon \\ &+ \varepsilon\_5 \sin 5\upsilon + \varepsilon\_6 \sin 6\upsilon + \varepsilon\_7 \sin 7\upsilon + \varepsilon\_8 \sin 8\upsilon \end{aligned} \tag{76}$$

where

$$\begin{cases} \varepsilon\_1 = -2\varepsilon \\ \varepsilon\_2 = \frac{3}{4}\varepsilon^2 + \frac{1}{8}\varepsilon^4 + \frac{3}{64}\varepsilon^6 + \frac{3}{128}\varepsilon^8 \\ \varepsilon\_3 = -\frac{1}{3}\varepsilon^3 - \frac{1}{8}\varepsilon^5 - \frac{7}{16}\varepsilon^7 \\ \varepsilon\_4 = \frac{5}{32}\varepsilon^4 + \frac{3}{32}\varepsilon^6 + \frac{15}{256}\varepsilon^6 \\ \varepsilon\_5 = -\frac{3}{40}\varepsilon^5 - \frac{1}{16}\varepsilon^7 \\ \varepsilon\_6 = \frac{7}{192}\varepsilon^6 + \frac{5}{128}\varepsilon^6 \\ \varepsilon\_7 = -\frac{1}{56}\varepsilon^7 \\ \varepsilon\_8 = \frac{9}{1024}\varepsilon^8 \end{cases} \tag{77}$$

Mathematical Analysis of Some Typical Problems in Geodesy by Means of Computer Algebra http://dx.doi.org/10.5772/intechopen.81586 85


Table 2. Errors of the derived series expansions.

<sup>δ</sup><sup>1</sup> <sup>¼</sup> <sup>2</sup><sup>e</sup> � <sup>1</sup>

<sup>δ</sup><sup>2</sup> <sup>¼</sup> <sup>5</sup> 4 e <sup>2</sup> � <sup>11</sup> 24 e 4 þ 17 <sup>192</sup> <sup>e</sup> 6 þ 43 <sup>5760</sup> <sup>e</sup> 8

8

84 Trends in Geomatics - An Earth Science Perspective

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

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

M ¼ E � e sin E

direct series expansion of the transformation from υ to M

8 ><

>:

where

<sup>δ</sup><sup>3</sup> <sup>¼</sup> <sup>13</sup> 12 e

<sup>δ</sup><sup>4</sup> <sup>¼</sup> <sup>103</sup> <sup>96</sup> <sup>e</sup>

<sup>δ</sup><sup>5</sup> <sup>¼</sup> <sup>1097</sup> <sup>960</sup> <sup>e</sup>

<sup>δ</sup><sup>6</sup> <sup>¼</sup> <sup>1223</sup> <sup>960</sup> <sup>e</sup>

<sup>δ</sup><sup>7</sup> <sup>¼</sup> <sup>47273</sup> <sup>32256</sup> <sup>e</sup> 7

<sup>δ</sup><sup>8</sup> <sup>¼</sup> <sup>556403</sup> <sup>322560</sup> <sup>e</sup> 8

þ γ<sup>6</sup> sin 6υ þ γ<sup>7</sup> sin 7υ þ γ<sup>8</sup> sin 8υ

ε<sup>1</sup> ¼ �2e <sup>ε</sup><sup>2</sup> <sup>¼</sup> <sup>3</sup> 4 e 2 þ 1 8 e 4 þ 3 64 e 6 þ 3 <sup>128</sup> <sup>e</sup> 8

8

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

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

<sup>ε</sup><sup>3</sup> ¼ � <sup>1</sup> 3 e <sup>3</sup> � <sup>1</sup> 8 e <sup>5</sup> � <sup>7</sup> 16 e 7

<sup>ε</sup><sup>4</sup> <sup>¼</sup> <sup>5</sup> 32 e 4 þ 3 32 e 6 þ 15 <sup>256</sup> <sup>e</sup> 6

<sup>ε</sup><sup>5</sup> ¼ � <sup>3</sup> 40 e <sup>5</sup> � <sup>1</sup> 16 e 7

<sup>ε</sup><sup>6</sup> <sup>¼</sup> <sup>7</sup> <sup>192</sup> <sup>e</sup> 6 þ 5 <sup>128</sup> <sup>e</sup> 6

<sup>ε</sup><sup>7</sup> ¼ � <sup>1</sup> 56 e 7

<sup>ε</sup><sup>8</sup> <sup>¼</sup> <sup>9</sup> <sup>1024</sup> <sup>e</sup> 8

The whole formulae for the transformation from υ to Mare as follows

4 e 3 þ 5 96 e 5 þ 107 <sup>4608</sup> <sup>e</sup> 7

<sup>3</sup> � <sup>43</sup> 64 e 5 þ 95 <sup>512</sup> <sup>e</sup> 7

<sup>4</sup> � <sup>451</sup> <sup>480</sup> <sup>e</sup> 6 þ

<sup>5</sup> � <sup>5957</sup> <sup>4608</sup> <sup>e</sup> 7

<sup>6</sup> � <sup>7913</sup> <sup>4480</sup> <sup>e</sup> 8

E ¼ υ þ γ<sup>1</sup> sin υ þ γ<sup>2</sup> sin 2υ þ γ<sup>3</sup> sin 3υ þ γ<sup>4</sup> sin 4υ þ γ<sup>5</sup> sin 5υ

Expanding Mas a power series of the eccentricity at e ¼ 0 by means of Mathematica yields the

M ¼ υ þ ε<sup>1</sup> sin υ þ ε<sup>2</sup> sin 2υ þ ε<sup>3</sup> sin 3υ þ ε<sup>4</sup> sin 4υ þε<sup>5</sup> sin 5υþ ε<sup>6</sup> sin 6υ þ ε<sup>7</sup> sin 7υ þ ε<sup>8</sup> sin 8υ

4123 <sup>11520</sup> <sup>e</sup> 8

(74)

(75)

(76)

(77)

### 4.4. The accuracy of the derived series expansions

In order to validate the exactness of the derived series expansions, one has examined their accuracies when the orbital eccentricity e is respectively equal to 0.01, 0.05, 0.1 and 0.2.

One makes use of Eq. (53) and Eq. (67) to obtain the theoretical value M<sup>0</sup> and υ<sup>0</sup> at given geodetic latitude E0. Substituting M<sup>0</sup> into Eq. (64) and Eq. (74), one arrives at the computation value E<sup>1</sup> and υ1. Substituting υ<sup>0</sup> into Eq. (71) and Eq. (77), one arrives at the computation value E<sup>2</sup> and M1. Substituting E<sup>0</sup> into Eq. (68), one arrives at the computation value υ2. The differences between the computation and theoretical values indicate the accuracies of the derived series expansions, which are denoted as ΔE1, Δυ1= <sup>00</sup> ð Þ, ΔM1= <sup>00</sup> ð Þ, ΔE2= <sup>00</sup> ð Þ, Δυ2= <sup>00</sup> ð Þ. Due to limited space, these errors when e is equal to 0.05 are only listed in Table 2.

From Table 2, one could find that the accuracy of derived series expansions is higher than 10�5<sup>00</sup> , which could satisfy practical application. Other numerical examples indicate that when the orbital eccentricity e is respectively equal to 0.01, 0.01 and 0.2, the accuracy of derived series expansions is correspondingly higher than 10�10<sup>00</sup> , 10�3<sup>00</sup> and 0:1 00 .
