## 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 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.

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.

## 2. The forward expansions of the rectifying, conformal and authalic latitudes

Cartographers prefer to adopt sphere as a basis of the map projection for convenience since calculation on the ellipsoid are significantly more complex than on the sphere. Formulas for the spherical form of a given map projection may be adapted for use with the ellipsoid by substitution of one of various “auxiliary latitudes” in place of the geodetic latitude. In using them, the ellipsoidal earth is, in effect, transformed to a sphere under certain restraints such as conformality or equal area, and the sphere is then projected onto a plane (Snyder, 1987). If the proper auxiliary latitudes are chosen, the sphere may have either true areas, true distances in certain directions, or conformality, relative to the ellipsoid. Spherical map projection formulas may then be used for the ellipsoid solely with the substitution of the appropriate auxiliary latitudes.

The rectifying, conformal and authalic latitudes are often used as auxiliary ones in cartography. The direct transformations form geodetic latitude to these auxiliary ones are expressed as transcendental functions or non-integrable ones. Adams (1921), Yang (1989, 2000) had derived forward expansions of these auxiliary latitudes form geodetic one through complicated formulation. Due to historical condition limitation, the derivation processes were done by hand and orders of these expansions could not be very high. Due to these reasons, the forward expansions for these auxiliary latitudes are reformulated by means of Mathematica. Readers will see that new expansions are expressed in a power series of the eccentricity of the reference ellipsoid

### 2.1. The forward expansion of the rectifying latitude

The meridian arc from the equator

where

(1) is an elliptic integral of the second kind and there is no analytical solution. Expanding the integrand by binomial theorem and itntegrating it item by item yield:

where

(3) |

The rectifying latitude

where

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

### 2.2. The forward expansion of the conformal latitude

Omitting the derivation process, the explicit expression for the isometric latitude

For the sphere, putting

Comparing (9) with (8) leads to

Therefore, it holds

Since the eccentricity is small, the conformal latitude is close to the geodetic one. Though (11) is an analytical solution of

as the conventional usage in mathematical cartography.

Through the tedious expansion process, Yang (1989, 2000) gave a power series of the eccentricity

Due to that (11) is a very complicated transcendental function, the coefficients

In fact, Mathematica works perfectly in solving derivatives of any complicated functions. By means of Mathematica, the new derived forward expansion expanded to tenth-order terms of

The derived coefficients in (13) and (14) are listed in Table 1 for comparison.

Table 1 shows that the eighth order terms of

### 2.3. The forward expansion of the authalic latitude

From the knowledge of mapping projection theory, the area of a section of a lune with a width of a unit interval of longitude

where

Denote

Suppose that there is an imaginary sphere with a radius

and whose area from the spherical equator

Therefore, it yields

where

(19) is a complicated transcendental function. It is almost impossible to derive its eighth-order derivate by hand. There are some mistakes in the high order terms of coefficients

The derived coefficients in (20) and (21) are listed in Table 2 for comparison.

Table 2 shows that the eighth-orders terms of

### 2.4. Accuracies of the forward expansions

In order to validate the exactness and reliability of the forward expansions of rectifying, conformal and authalic latitudes derived by the author, one has examined their accuracies choosing the CGCS2000 (China Geodetic Coordinate System 2000) reference ellipsoid with parameters^{-5}″, while the accuracy of the forward expansion (5) derived by the author is higher than 10^{-7}″. The accuracies of the forward expansion of conformal and authalic latitudes derived by Yang (1989, 2000) are higher than 10^{-4}″, while the accuracies of the forward expansions derived by the author are higher than 10^{-8}″. The accuracies of forward expansions derived by the author are improved by 2~4 orders of magnitude compared to forward expansions derived by Yang (1989, 2000).

## 3. The inverse expansions of rectifying, conformal and authalic latitudes

The inverse expansions of these auxiliary latitudes are much more difficult to derive than their forward ones. In this case, the differential equations are usually expressed as implicit functions of the geodetic latitude. There are neither any analytical solutions nor obvious expansions. For the inverse cases, to find geodetic latitude from auxiliary ones, one usually adopts iterative methods based on the forward expansions or an approximate series form. Yang (1989, 2000) had given the direct expansions of the inverse transformation by means of Lagrange series method, but their coefficients are expressed as polynomials of coefficients of the forward expansions, which are not convenient for practical use. Adams (1921) expressed the coefficients of inverse expansions as a power series of the eccentricity

### 3.1. The inverse expansions using the power series method

The processes to derive the inverse expansions using the power series method are as follows:

To obtain their various order derivatives in terms of the chain rule of implicit differentation;

To compute the coefficients of their power series expansions;

To integrate these series item by item and yield the final inverse expansions.

One can image that these procedures are quite complicated. Mathematica output shows that the expression of the sixth order derivative is up to 40 pages long! Therefore, it is unimaginable to derive the so long expression by hand. These procedures, however, will become much easier and be significantly simplified by means of Mathematica. As a result, the more simple and accurate expansions yield.

#### 3.1.1. The inverse expansion of the rectifying Latitude

Differentiation to the both sides of (1) yields

Inserting (23) into (22) yields

To expand (24) into a power series of

therefore

and then denote

Making use of the chain rule of implicit differentiation

It is easy to expand (27) into a power series of

Omitting the detailed procedure, one arrives at

where

(31) |

Multiplying

where

#### 3.1.2. The inverse expansion of the conformal latitude

Differentiating the both sides of (10) yields

Therefore, it holds

For the same reason, we introduce the following new variable

and then denote

Using the same procedure as described in the former section, one arrives at

where

(39) |

Integrating the both sides of (38) gives the inverse expansion of conformal latitude as

where

#### 3.1.3. The inverse expansion of the authalic latitude

Inserting (18) into (15) yields

Differentiating the both sides of (42) yields

For the same reason, we introduce the folllowing new variable

and then denote

(45) can be expanded into a power series of

where

(47) |

To get the inverse expansion of the authalic latitude, one integrates (46) and arrives at

where

### 3.2. The inverse expansions using the Hermite interpolation method

In mathematical analysis, interpolation with functional values and their derivative values is called Hermite interpolation. The processes to derive the inverse expansions using this method are as follows:

To suppose the inverse expansions are expressed in a series of the sines of the multiple arcs with coefficients to be determined;

To compute the functional values and their derivative values at specific points;

To solve linear equations according to interpolation constraints and obtain the coefficients.

The detailed derivation processes are given by Li (2008, 2010). Confined to the length of the chapter, they are omitted. Comparing the results derived by this method with those in 3.1, one will find that they are consistent with each other even though they are formulated in different ways. This fact substantiates the correctness of the derived formulas.

### 3.3. The inverse expansions using the Lagrange’s theorem method

We wish to investigate the inversion of an equation such as

with

The proof of this theorem is given by Whittaker (1902) and Peter (2008).

The processes to derive the inverse expansions using the Lagrange series method are as follows:

To apply the Lagrange theorem to a trigonometric series;

To write the inverse expansions of the rectifying, conformal and authalic latitude;

To express the coefficients of the inverse expansions as a power series of the eccentricity.

The detailed derivation processes are given by Li (2010). Confined to the length of the chapter, they are also omitted. Comparing the results derived by this method with those in 3.1 and 3.2, one will find that they are all consistent with each other even though they are also formulated in different ways. This fact substantiates the correctness of the derived formulas, too.

### 3.4. Accuracies of the inverse expansions

The accuracies of the inverse expansions derived by Yang (1989, 2000) and the author has been examined choosing the CGCS2000 reference ellipsoid.

The results show that the accuracy of the inverse expansion of rectifying latitude is higher than 10^{-5}″, while the accuracy of the inverse expansion (32) derived by the author is higher than 10^{-7}″. The accuracies of the inverse expansion of conformal and authalic latitudes derived by Yang (1989, 2000) are higher than 10^{-4}″, while the accuracies of the inverse expansions derived by the author are higher than 10^{-8}″. The accuracies of inverse expansions derived by the author are improved by 2~4 orders of magnitude compared to those derived by Yang (1989, 2000).

## 4. The direct expansions of transformations between meridian arc, isometric latitude and authalic latitude function

The meridian arc, isometric latitude and authalic latitude function are functions of rectifying, conformal and authalic latitudes correspondingly. The transformations between the three variables are indirectly realized by computing the geodetic latitude in the past literatures such as Yang (1989, 2000), Snyder (1987). The computation processes are tedious and time-consuming. In order to simplify the computation processes and improve the computation efficiency, the direct expansions of transformations between meridian arc, isometric latitude and authalic latitude function are comprehensively derived by means of Mathematica.

### 4.1. The direct expansions of transformations between meridian arc and isometric latitude

#### 4.1.1. The direct expansion of the transformation from meridian arc to isometric latitude

Inserting the known meridian arc

(52) |

Since the coefficients

where

#### 4.1.2. The direct expansion of the transformation from isometric latitude to meridian arc

The whole formulas for the transformation from isometric latitude to meridian arc are as follows:

(55) |

Expanding

where

### 4.2. The direct expansions of transformations between meridian arc and authalic latitude function

#### 4.2.1. The direct expansion of the transformation from meridian arc to authalic latitude function

The whole formulas for the transformation from meridian arc to authalic latitude function are as follows:

(58) |

Expanding

where

#### 4.2.2. The direct expansion of the transformation from authalic latitude function to meridian arc

The whole formulas for the transformation from authalic latitude function to meridian arc are as follows:

(61) |

Expanding

where

(63) |

### 4.3. The direct expansions of transformations between isometric latitude and authalic latitude function

#### 4.3.1. The direct expansion of the transformation from isometric latitude to authalic latitude function

The whole formulas for the transformation from isometric latitude to authalic latitude function are as follows:

(64) |

Expanding

where

#### 4.3.2. The direct expansion of the transformation from authalic latitude function to isometric latitude

The whole formulas for the transformation from authalic latitude function to isometric latitude are as follows:

(67) |

Expanding

where

### 4.4. Accuracies of the direct expansions

The accuracies of the indirect and direct expansions given by Yang(1989, 2000) derived by the author has been examined choosing the CGCS2000 reference ellipsoid.

The results show that the accuracy of the indirect expansion of the transformation from meridian arc to isometric latitude is higher than 10^{-3}″, while the accuracy of the direct expansion (53) is higher than 10^{-7}″. The accuracy of the indirect expansion of the transformation from isometric latitude to meridian arc is higher than 10^{-2} m, while the accuracy of the direct expansion (56) is higher than 10^{-7} m. The accuracy of the indirect expansion of the transformation from meridian arc to authalic latitude function is higher than 0.1^{-2} m, while the direct expansion (62) is higher than 10^{-4} m. The accuracy of the indirect expansion of the transformation from isometric latitude to authalic latitude function is higher than 0.1^{-2}″, while the accuracy of the direct expansion (67) is higher than 10^{-6}″. The accuracies of the direct expansions derived by the author are improved by 2~6 orders of magnitude compared to the indirect ones derived by Yang (1989, 2000).

## 5. The non-iterative expressions of the forward and inverse Gauss projections by complex numbers

Gauss projection plays a fundamental role in ellipsoidal geodesy, surveying, map projection and geographical information system (GIS). It has found its wide application in those areas.

As shown in Figure 1, Gauss projection has to meet the following three constraints:

① conformal mapping;

② the central meridian mapped as a straight line (usually chosen as a vertical axis of

③ scale being true along the central meridian.

Traditional expressions of the forward and inverse Gauss projections are real functions in a power series of longitude difference. Though real functions are easy to understand for most people, they make Gauss projection expressions very tedious. Mathematically speaking, there is natural relationship between the conformal mapping and analytical complex functions which automatically meet the differential equation of the conformal mapping, or the “Cauchy-Riemann Equations”. Complex functions, a powerful mathematical method, play a very special and key role in the conformal mapping. Bowring (1990) and Klotz (1993) have discussed Gauss projection by complex numbers. But the expressions they derived require iterations, which makes the computation process very fussy. In terms of the direct expansions of transformations between meridian arc and isometric latitude given in section Ⅳ, the non-iterative expressions of the forward and inverse Gauss projections by complex numbers are derived.

### 5.1. The non-iterative expressions of the forward Gauss projection by complex numbers

Letwhere

.In terms of complex functions theory, analytical functions meet conformal mapping naturally. Therefore, to meet the conformal mapping constraint, the forward Gauss projection should be in the following form

where

(72) shows that the central meridian is a straight line after the projection when

Finally, from the third constraint, “scale is true along the central meridian”, one knows that

(73) defines the functional relationship between meridian arc and isometric latitude. If one extends the definition of

(74) is the solution of the forward Gauss projection by complex numbers. Its correctness can be explained as follows:

The two equations in (74) are all elementary complex functions. Because elementary functions in their basic interval are all analytical ones in the complex numbers domain, the mapping defined by (74) form

### 5.2. The non-iterative expressions of the inverse Gauss projection by complex numbers

In principle, the inverse Gauss projection can be iteratively solved in terms of the forward Gauss projection (74). In order to eliminate the iteration, one more practical approach is proposed based on the direct expansion of the transformation from meridian arc to isometric latitude (53).

In order to meet the conformal mapping constraint, the inverse Gauss projection should be in the following form

where

Finally, from the third constraint, one knows that

If one extends the definition of

Therefore, the isometric latitude

Then one can compute the geodetic latitude through the inverse expansion of the conformal latitude (40).

(77) is the solution of the inverse Gauss projection by complex numbers. Its correctness can be explained as follows:

The two equations in (78) are all elementary complex functions, so the mapping defined by (78) form

## 6. Conclusions

Some typical mathematical problems in map projections are solved by means of computer algebra system which has powerful function of symbolical operation. The main contents and research results presented in this chapter are as follows:

Forward expansions of rectifying, conformal and authalic latitudes are derived, and some mistakes once made in the high orders of traditional forward formulas are pointed out and corrected. Inverse expansions of rectifying, conformal and authalic latitudes are derived using power series expansion, Hermite interpolation and Language’s theorem methods respectively. These expansions are expressed in a series of the sines of the multiple arcs. Their coefficients are expressed in a power series of the first eccentricity of the reference ellipsoid and extended up to its tenth-order terms. The accuracies of these expansions are analyzed through numerical examples. The results show that the accuracies of these expansions derived by means of computer algebra system are improved by 2~4 orders of magnitude compared to the formulas derived by hand.

Direct expansions of transformations between meridian arc, isometric latitude and authalic latitude function are derived. Their coefficients are expressed in a power series of the first eccentricity of the reference ellipsoid, and extended up to its tenth-order terms. Numerical examples show that the accuracies of these direct expansions are improved by 2~6 orders of magnitude compared to the traditional indirect formulas.

Gauss projection is discussed in terms of complex numbers theory. The non-iterative expressions of the forward and inverse Gauss projections by complex numbers are derived based on the direct expansions of transformations between meridian arc and isometric latitude, which enriches the theory of conformal projection. In USA, Universal Transverse Mercator Projection (or UTM) is usually implemented. Mathematically speaking, there is no essential difference between UTM and Gauss projections. The only difference is that the scale factor of UTM is 0.9996 rather than 1. With a slight modification, the non-iterative expressions of the forward and inverse Gauss projections can be extended to UTM projection accordingly.

### Acknowledgement

This work was financially supported by 973 Program (2012CB719902), National Natural Science Foundation of China (No. 41071295 and 40904018), and Key Laboratory of Surveying and Mapping Technology on Island and Reef, State Bureau of Surveying, Mapping and Geoinformation, China (No.2010B04).