**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.

**Figure 1.** Gauss Projection, where *x* and *y* are the vertical and horizontal axes after the projection respectively, *O* is the origin of the projection coordinates.

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

① conformal mapping;

18 Cartography – A Tool for Spatial Analysis

*q*

 

latitude

where

 

2

 

*F R*

arcsin( )

arctanh(sin )

2

*F R*

arcsin( )

1

 

 

**4.4 Accuracies of the direct expansions** 

3

5

7

9

246810 246810

 

Expanding *q* as a power series of the eccentricity at *e* 0 by means of Mathematica yields the direct expansion of the transformation form authalic latitude function to isometric

arctanh(sin ) sin sin 3 sin 5 sin7 sin9

1 1 11 107 1513 3 30 1890 32400 1663200 1 61 2321 1021 90 11340 907200 831600

*lee e e e*

*q ll l l l*

6 8 10

*le e e e*

1 5 151 756 4032 166320 71 41 302400 123200

*l ee e*

8 10

The accuracies of the indirect and direct expansions given by Yang(1989, 2000) derived by

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 <sup>2</sup> km , while the accuracy of the direct expansion (59) is higher than 5 2 10 km . The accuracy of the indirect expansion of the transformation from authalic latitude function to meridian arc is higher than 10-2 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 <sup>2</sup> km , while the accuracy of the direct expansion (65) is

10

the author has been examined choosing the CGCS2000 reference ellipsoid.

61 1197504

*l e*

*l ee*

*Bc c c c c*

 

sin 2 sin 4 sin6 sin8 sin10 sin 2 sin 4 sin6 sin8 sin10

 

13 5 7 9

2 4 6 8 10

4 6 8 10

  (67)

(68)

(69)

 

*BBBBB B*

② the central meridian mapped as a straight line (usually chosen as a vertical axis of *x* ) after projection;

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**

Let *w* be complex numbers consisting of isometric latitude *q* and longitude difference *l* before projection, *z* be complex numbers consisting of corresponding coordinates *x* , *y* after projection.

$$\begin{cases} \mathbf{z}v = q + il \\ \mathbf{z} = \mathbf{x} + i\mathbf{y} \end{cases} \tag{70}$$

Mathematical Analysis in Cartography by Means of Computer Algebra System 21

1 1 *w q il f z f x iy* () ( ) (75)

  (74)

0 2 <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup>

*z x iy a j j j j j j*

(74) is the solution of the forward Gauss projection by complex numbers. Its correctness can

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 *w q il* to *z x iy* meets the conformal mapping constraint. When *l* 0 , the imaginary part disappears and (74) restores to (73). Therefore, (74) meets the second and third constraints of Gauss projection when *l* 0 . Hence, it is clear

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

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

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

where <sup>1</sup> *f* is the inverse function of *f* . According to the second constraint, when *l* 0 ,

<sup>1</sup> *qf x*( ) (76)

Finally, from the third constraint, one knows that *x* in (76) should be the meridian arc *X* , and (76) is essentially consist with the direct expansion of the transformation from meridian arc to isometric latitude as (53) shows. Substituting *X* in (53) with *x* gives the explicit form of (76)

arctanh(sin ) sin sin 3 sin 5 sin7 sin9

or substitutes *x* with *z x iy* , the original real number rectifying latitude

 

If one extends the definition of *x* in a real number variable to a complex numbers variable,

automatically extended as a complex numbers variable. We denote the corresponding complex number latitude as , and insert it into (77). Rewriting a real variable *q* at the lefthand of the second equation in (77) as a complex numbers variable *w q il* , one arrives at

13 5 7 9

 

 

 

(77)

will be

sin 2 sin 4 sin6 sin8 sin10

arcsin(tanh )

be explained as follows:

**numbers** 

isometric latitude (53).

in the following form

*w*

that (74) is the solution of the forward Gauss projection indeed.

imaginary part disappears and only real part exists, (75) becomes

2 0

> 

 

(1 )

*q*

 

*x a eK* where *i* 1 .

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

$$z = x + iy = f(w) = f(q + il) \tag{71}$$

where *f* is an arbitrary analytical function in the complex numbers domain. According to the second constraint, when *l* 0 , imaginary part disappears and only real part exists, (71) becomes

$$\mathbf{x} = f(\mathbf{q}) \tag{72}$$

(72) shows that the central meridian is a straight line after the projection when *l* 0 .

Finally, from the third constraint, "scale is true along the central meridian", one knows that *x* in (72) should be nothing else but the meridian arc *X* , and (72) is essentially consist with the direct expansion of the transformation from isometric latitude to meridian arc (56). Substituting *X* in (56) with *x* gives the explicit form of (72)

$$\begin{cases} \varphi = \arcsin(\tanh q) \\ \mathbf{x} = a \left( j\_0 \rho + j\_2 \sin 2\rho + j\_4 \sin 4\rho + j\_6 \sin 6\rho + j\_8 \sin 8\rho + j\_{10} \sin 10\rho \right) \end{cases} \tag{73}$$

(73) defines the functional relationship between meridian arc and isometric latitude. If one extends the definition of *q* in a real number variable to a complex numbers variable, or substitutes *q* with *w q il* , the original real number conformal latitude will be automatically extended as a complex numbers variable. We denote the corresponding complex numbers latitude as , and insert it into (73). Rewriting a real variable *x* at the left-hand of the second equation in (73) as a complex numbers variable *z x iy* , one arrives at

$$\begin{cases} \Phi = \arcsin(\tanh w) \\ \mathbf{z} = \mathbf{x} + \dot{\mathbf{z}}\mathbf{y} = a \begin{pmatrix} j\_0 \rho + j\_2 \sin 2\rho + j\_4 \sin 4\rho + j\_6 \sin 6\rho + j\_8 \sin 8\rho + j\_{10} \sin 10\rho \end{pmatrix} \end{cases} \tag{74}$$

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

20 Cartography – A Tool for Spatial Analysis

*w q il*

projection should be in the following form

numbers are derived.

**numbers** 

after projection.

where *i* 1 .

becomes

arrives at

Ⅳ, the non-iterative expressions of the forward and inverse Gauss projections by complex

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

Let *w* be complex numbers consisting of isometric latitude *q* and longitude difference *l* before projection, *z* be complex numbers consisting of corresponding coordinates *x* , *y*

> *z x iy*

In terms of complex functions theory, analytical functions meet conformal mapping naturally. Therefore, to meet the conformal mapping constraint, the forward Gauss

*z x iy f w f q il* () ( ) (71)

where *f* is an arbitrary analytical function in the complex numbers domain. According to the second constraint, when *l* 0 , imaginary part disappears and only real part exists, (71)

*x fq* ( ) (72)

Finally, from the third constraint, "scale is true along the central meridian", one knows that *x* in (72) should be nothing else but the meridian arc *X* , and (72) is essentially consist with the direct expansion of the transformation from isometric latitude to meridian arc (56).

0 2 <sup>4</sup> <sup>6</sup> <sup>8</sup> <sup>10</sup>

(73) defines the functional relationship between meridian arc and isometric latitude. If one extends the definition of *q* in a real number variable to a complex numbers variable, or

automatically extended as a complex numbers variable. We denote the corresponding complex numbers latitude as , and insert it into (73). Rewriting a real variable *x* at the left-hand of the second equation in (73) as a complex numbers variable *z x iy* , one

sin 2 sin 4 sin6 sin8 sin10

(72) shows that the central meridian is a straight line after the projection when *l* 0 .

*x aj j j j j j*

substitutes *q* with *w q il* , the original real number conformal latitude

Substituting *X* in (56) with *x* gives the explicit form of (72)

*q*

arcsin(tanh )

(70)

  (73)

will be

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 *w q il* to *z x iy* meets the conformal mapping constraint. When *l* 0 , the imaginary part disappears and (74) restores to (73). Therefore, (74) meets the second and third constraints of Gauss projection when *l* 0 . Hence, it is clear that (74) is the solution of the forward Gauss projection indeed.

#### **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

$$\text{i } w = q + \text{i}\mathbf{i} = f^{-1}(\mathbf{z}) = f^{-1}(\mathbf{x} + \mathbf{i}\mathbf{y}) \tag{75}$$

where <sup>1</sup> *f* is the inverse function of *f* . According to the second constraint, when *l* 0 , imaginary part disappears and only real part exists, (75) becomes

$$q = f^{-1}(\mathbf{x})\tag{76}$$

Finally, from the third constraint, one knows that *x* in (76) should be the meridian arc *X* , and (76) is essentially consist with the direct expansion of the transformation from meridian arc to isometric latitude as (53) shows. Substituting *X* in (53) with *x* gives the explicit form of (76)

$$\begin{cases} \mathcal{W} = \frac{\mathcal{X}}{a(1 - e^2)K\_0} \\\\ \eta = \operatorname{arctanh}(\sin \psi) + \tilde{\xi}\_1 \sin \psi + \tilde{\xi}\_3 \sin 3\psi + \tilde{\xi}\_5 \sin 5\psi + \tilde{\xi}\_7 \sin 7\psi + \tilde{\xi}\_9 \sin 9\psi \end{cases} \tag{77}$$

If one extends the definition of *x* in a real number variable to a complex numbers variable, or substitutes *x* with *z x iy* , the original real number rectifying latitude will be automatically extended as a complex numbers variable. We denote the corresponding complex number latitude as , and insert it into (77). Rewriting a real variable *q* at the lefthand of the second equation in (77) as a complex numbers variable *w q il* , one arrives at

$$\begin{cases} \Psi = \frac{\mathbf{x} + i\mathbf{y}}{a(1 - e^2)K\_0} \\\\ \varpi = q + il = \operatorname{arctanh}(\sin \Psi) + \underline{\xi}\_1 \sin \Psi + \underline{\xi}\_3 \sin 3\Psi + \underline{\xi}\_5 \sin 5\Psi + \underline{\xi}\_7 \sin 7\Psi + \underline{\xi}\_9 \sin 9\Psi \end{cases} \tag{78}$$

Therefore, the isometric latitude *q* and longitude *l* is known. Inserting *q* into (78) yields the conformal latitude

$$
\varphi = \arcsin(\tanh q) \tag{80}
$$

Mathematical Analysis in Cartography by Means of Computer Algebra System 23

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.

*Department of Navigation, Naval University of Engineering, Wuhan, China* 

*Department of Navigation, Naval University of Engineering, Wuhan, China* 

*Surveying, Mapping and Geoinformation, Qingdao, China* 

Mapping and Geoinformation, China (No.2010B04).

Pub. No.67, U. S. Coast and Geodetic Survey.

Professional Paper 1395, Washington.

Applications. Taylor and Francis, London.

Geodesy and Geoinformatics. Springer, Berlin.

*Key Laboratory of Surveying and Mapping Technology on Island and Reef, State Bureau of* 

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,

[1] Adam O S (1921). Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projection. Spec.

[2] Snyder J P (1987). Map Projections-a Working Manual. U. S. Geological Survey

[3] Yang Qihe (1989). The Theories and Methods of Map Projection. PLA Press, Beijing. (in

[4] Yang Qihe, Snyder J P, Tobler W R (2000). Map Projection Transformation: Principles and

[5] Awange J L, Grafarend E W (2005). Solving Algebraic Computational Problems in

[6] Bian S F, Chen Y B (2006). Solving an Inverse Problem of a Meridian Arc in Terms of Computer Algebra System. Journal of Surveying Engineering, 132(1): 153-155. [7] Chen Junyong (2008). Chinese Modern Geodetic Datum-Chinese Geodetic Coordinate System 2000 (CGCS2000) and its Frame. Acta Geodaetica et Cartographica Sinica, 37(3):

[8] Yang Y X (2009). Chinese Geodetic Coordinate System 2000. Chinese Science Bulletin,

**Author details** 

**Acknowledgement** 

**7. References** 

Chinese)

269-271. (in Chinese)

54(16): 2714-2721.

Shao-Feng Bian

Hou-Pu Li

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 *z x iy* to *w q il* meets the conformal mapping constraint. When *l* 0 , the imaginary part disappears and (78) restores to (77). Therefore, (78) meets the second and third constraints of Gauss projection when *l* 0 . Hence, it is clear that (78) is the solution of the inverse Gauss projection indeed.
