New form of geodetic coordinate system taking two length quantity as coordinate parameters

Abstract

A new form of geodetic coordinate system based on geodesic coordinates instead of geodetic longitude and latitude was proposed. The vertical and horizontal geodesic coordinates measured with length were defined as coordinate parameters, but the two families of coordinate curves were still meridians and parallel circles. The first fundamental form on the ellipsoidal surface and its three coefficients were deduced by the geodesic coordinate. The formula for the latitudinal scale factor of length for geodetic parallel lines was derived, by which the obtained result conformed to that standard value calculated from geodetic latitude, and it is applicable in the range of 400 km from north to south. Therefore, it lays the foundation for establishing the differential equation and differential relationship based on this type of coordinate parameters; and consequently, it is convenient and accurate enough to operate on the ellipsoidal surface in this new form of geodetic coordinate system.

Keywords

geodetic coordinate system / meridian and parallel circle / coordinate parameter / the first fundamental form on the ellipsoidal surface / latitudinal scale factor of length

Introduction

It is widely known that geodetic coordinate system is a curvilinear coordinate system on an ellipsoid with two families of orthogonal coordinate curves and two coordinate parameters, i.e. geodetic longitude and geodetic latitude. It is rather complicated and tedious to carry out computations on an ellipsoidal surface. So the ellipsoidal surface is always mapped onto a plane based on mathematics principles and then a kind of rectangular coordinate is applied. However, it is always accompanied with projection distortion.

On ellipsoid or on sphere, there has been a rectangular coordinate system [1] or geodetic parallel coordinate system [2] yet. Among authors who first emphasized the interest of expressing the positions by rectangular coordinate on sphere, the pioneering work of soldner is of note [1,3]. These rectangular coordinates either on ellipsoid or on sphere are expressed by length but not by angles. The transformation between the rectangular coordinate system and geodetic system can be carried out. The algorithms and formulas for direct and inverse solution of geodesic problem in the rectangular coordinate system were investigated by Grossmann [2].

However, there are still some disadvantages in the systems. First, the two families of coordinate curves are not the geodetic meridians and parallel circles which are frequently used; second, the geodesic coordinate system is only suited for limited local regions, and the transformation between the two geodesic coordinate systems in adjacent regions are performed with difficulty. Therefore, the application of the coordinate system is limited.

Recently, the foundational rectangular coordinate system on ellipsoid was discussed and called as another form of geodesic coordinate system [4,5]. With different methods, the similar and convenient formulas were obtained. There is no projection distortion in the coordinate system because the projection from ellipsoidal surface to plane is avoided. However, there are also some disadvantages in it. First, its two coordinate curves are not the geodetic meridians and parallel circles which are commonly used; second, the geodesic coordinate systems in two adjacent regions are difficult to convert, and therefore its application is limited.

New form of geodetic coordinate system

To define a new form of geodetic coordinate system in a relatively large region (the span is 400 km from south to north and without limit from west to east), P₀ is located more or less in the middle of the area that can be chosen as the origin, with geodetic latitude B₀ and geodetic longitude L₀. The geodetic meridian passing through P₀ are defined as the initial meridian, while the parallel circle that is orthogonal with the initial meridian and passing through P₀ is taken as the initial geodetic parallel circle (see Fig.1).

The two mutually orthogonal families of meridians and parallel circles constitute a grid on the ellipsoidal surface. For the point P(B, L) on the ellipsoid, s_L is the distance from the intersected point P_{B_{\mathrm{0}}} on the initial parallel circle to the point P along the meridian, i.e., P{P}_{B_{\mathrm{0}}}. s_B is the distance from the origin P₀ to the point P_{B_{\mathrm{0}}} along the initial parallel circle, i.e., P_{\mathrm{0}}{P}_{B_{\mathrm{0}}}. Therefore, the geodesic coordinates of P on the ellipsoidal surface could be denoted as (s_B, s_L). Obviously, the coordinates of the origin P₀ are (0, 0).

The parallel circle passing through P meets the initial meridian at P_{L_{\mathrm{0}}}. The length along the parallel circle from P_{L_{\mathrm{0}}} to P is denoted as s_B^{\prime }. Obviously, s_B and s_B^{\prime } are not equal in length and their difference becomes greater as the latitude difference between the two parallel circles increases. On the contrary, it is known that the distances along any meridians between two parallel circles are bound to be equal in length. Therefore, the geodetic parallel circles are geodesic parallel lines as they are defined in Differential Geometry.

Latitudinal-scale factor of length for geodesic parallel lines

First fundamental form

For the purpose of expressing the first fundamental form with the two coordinate parameters u,v(u=s_l, v=s_L), it is necessary to analyze the element dS on the ellipsoidal surface first. When P on the ellipsoidal surface moves to P′ differentially, the differential length of the geodesic PP′=dS could be factored into two differential arcs: one that passes P along the meridian PP_L=ds_L and another is on the parallel circle P{P}_B^{\prime }=d{s}_B^{\prime }. Therefore, based on the properties of geodesic parallel lines, du=P{P}_L=P_B^{\prime }{P}^{\prime }=d{s}_L can be obtained and four internal angles of the curved quadrangle P{P}_B^{\prime }, P{P}_L^{\prime } are all 90° (see Fig. 2).

Because the two families of coordinate curves are mutually orthogonal, the second coefficient F of the first fundamental form on ellipsoidal surface must be zero and the first coefficient E=1. Hence, the first fundamental form could be written as

(1)d{S}^{\mathrm{2}}=d{s}_L^{\mathrm{2}}+d{s}_B^{\prime \mathrm{2}}=Ed{u}^{\mathrm{2}}+\mathrm{2}Fdudv+Gd{v}^{\mathrm{2}}=d{s}_L^{\mathrm{2}}+Gd{s}_B^{\mathrm{2}}=d{s}_L^{\mathrm{2}}+n^{\mathrm{2}}d{s}_B^{\mathrm{2}}.

In Eq. (1), the ratio of differential arc d{s}_B^{\prime } to ds_B is denoted as n. ds_B is P_{B_{\mathrm{0}}}{P}_{B_{\mathrm{0}}}^{\prime } and two meridians passing through P and P′ are perpendicular to the initial geodetic parallel circle respectively at P_{B_{\mathrm{0}}} and P_{B_{\mathrm{0}}}^{\prime }. The symbol n here is called the latitudinal scale factor of length:

(2)n=\sqrt{G}={\displaystyle \frac{d{s}_B^{\prime }}{d{s}_B}}.

In the geodetic coordinate system which takes longitude and latitude as its coordinate parameters, the corresponding longitude differences dl on the two parallel circles between two meridians are equal in angle, therefore,

(3)n={\displaystyle \frac{d{s}_B^{\prime }}{d{s}_B}}={\displaystyle \frac{N_B\cos Bdl}{N_{\mathrm{0}}\cos B_{\mathrm{0}}dl}}={\displaystyle \frac{N_B\cos B}{N_{\mathrm{0}}\cos B_{\mathrm{0}}}},

where N₀ and N_B are radii of curvature in prime vertical section and evaluated for their corresponding latitude B₀ and B respectively. But in the new form of geodetic coordinate system, the expression of n constituted by its new coordinate parameters is need.

Expression of latitudinal scale factor of length with geodesic coordinates

It is easy to know that the new form of geodetic coordinate system is very similar to that suggested by Shi et al. [4]. If the initial geodetic parallel of the new geodetic coordinate system is the equator (B₀=0°), n could be written as

(4)n=\mathrm{1}-{\displaystyle \frac{s_L^{\mathrm{2}}}{\mathrm{2}{R}_{\mathrm{0}}^{\mathrm{2}}}}.

However, much of the region of China is not located near the equator and s_L is usually more than several thousands of kilometers. Thereupon, the higher-degree terms in the expression of n could not be neglected. Therefore, the initial geodetic parallel circle should be located close to the center of a given region to shorten s_L and limit the absolute value of s_L to less than 200 km. The third coefficient G of the first fundamental form is defined as

(5)n=\sqrt{G}=\sqrt{{\left({\displaystyle \frac{\partial x}{\partial s_B}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\partial y}{\partial s_B}}\right)}^{\mathrm{2}}+{\left({\displaystyle \frac{\partial z}{\partial s_B}}\right)}^{\mathrm{2}}}.

In Eq. (5), x, y, z are 3-dimension coordinates of P in a local geodetic coordinate system, which takes P_{B_{\mathrm{0}}} as its origin. The location (x, y, z) can be expressed by the parameter equations with two parameters i.e., geodesic length S and geodetic azimuth A, both of which took P_{B_{\mathrm{0}}} as their start point. Three partial derivatives with respect to s_B in Eq. (5) are considered. Obviously, their changes are irrelevant to that of the coordinate parameters s_L. Therefore,

(6){\left({\displaystyle \frac{\partial x}{\partial s_B}}{\displaystyle \frac{\partial y}{\partial s_B}}{\displaystyle \frac{\partial z}{\partial s_B}}\right)}^T={\left({\displaystyle \frac{\partial x}{\partial A}}{\displaystyle \frac{\partial A}{\partial s_B}}{\displaystyle \frac{\partial y}{\partial A}}{\displaystyle \frac{\partial A}{\partial s_B}}{\displaystyle \frac{\partial z}{\partial A}}{\displaystyle \frac{\partial A}{\partial s_B}}\right)}^T

is obtained.

The equations of geodesic line in the local geodetic coordinate system can be found readily in some literatures. From the well-known Weingarten series expansion, we have

(7)$$\begin{gathered} x=S\cos A-{\displaystyle \frac{S^{\mathrm{3}}\cos A(\mathrm{1}+{\eta }^{\mathrm{2}}+{\eta }^{\mathrm{2}}{\cos }^{\mathrm{2}}A)}{\mathrm{6}{N}^{\mathrm{2}}}}+{\displaystyle \frac{S^{\mathrm{4}}(\mathrm{1}+\mathrm{8}{\cos }^{\mathrm{2}}A){\eta }^{\mathrm{2}}t}{\mathrm{24}{N}^{\mathrm{3}}}}+{\displaystyle \frac{S^{\mathrm{5}}\cos A}{\mathrm{120}{N}^{\mathrm{4}}}}+\cdots , \\ y=S\sin A-{\displaystyle \frac{S^{\mathrm{3}}\sin A(\mathrm{1}+{\eta }^{\mathrm{2}}{\cos }^{\mathrm{2}}A)}{\mathrm{6}{N}^{\mathrm{2}}}}+{\displaystyle \frac{S^{\mathrm{4}}\sin A\cos A{\eta }^{\mathrm{2}}t}{\mathrm{3}{N}^{\mathrm{3}}}}+{\displaystyle \frac{S^{\mathrm{5}}\sin A}{\mathrm{120}{N}^{\mathrm{4}}}}+\cdots , \\ z={\displaystyle \frac{S^{\mathrm{2}}(\mathrm{1}+{\eta }^{\mathrm{2}}{\cos }^{\mathrm{2}}A)}{\mathrm{2}N}}-{\displaystyle \frac{S^{\mathrm{3}}\cos A{\eta }^{\mathrm{2}}t}{\mathrm{2}{N}^{\mathrm{2}}}}-{\displaystyle \frac{S^{\mathrm{4}}}{\mathrm{24}{N}^{\mathrm{3}}}}+\cdots . \end{gathered}$$

The geodetic azimuth A of the meridian, which takes P_{B_{\mathrm{0}}} as its origin, is 0° or 180° and S here is S_L. This approach yields the following form:

(8){\left({\displaystyle \frac{\partial x}{\partial A}}{\displaystyle \frac{\partial y}{\partial A}}{\displaystyle \frac{\partial z}{\partial A}}\right)}^T={\left(\mathrm{0}{s}_L\left[\mathrm{1}-{\displaystyle \frac{s_L^{\mathrm{2}}}{\mathrm{6}{N}_{\mathrm{0}}^{\mathrm{2}}}}(\mathrm{1}+{\eta }_{\mathrm{0}}^{\mathrm{2}})+{\displaystyle \frac{s_L^{\mathrm{3}}}{\mathrm{3}{N}_{\mathrm{0}}^{\mathrm{3}}}}{\eta }_{\mathrm{0}}^{\mathrm{2}}{t}_{\mathrm{0}}+{\displaystyle \frac{s_L^{\mathrm{4}}}{\mathrm{120}{N}_{\mathrm{0}}^{\mathrm{4}}}}\right]\mathrm{0}\right)}^T.

where the auxiliary quantities are t_{\mathrm{0}}=\tan B_{\mathrm{0}}, {\eta }_{\mathrm{0}}^{\mathrm{2}}=e^{\prime \mathrm{2}}{\cos }^{\mathrm{2}}{B}_{\mathrm{0}}. The expression for the partial derivative {\displaystyle \frac{\partial A}{\partial s_B}} is deduced in the following. Due to the properties of the geodesic parallels, the lengths of the meridians between the initial parallel circle P_{B_{\mathrm{0}}}{P}_{B_{\mathrm{0}}}^{\prime } and the parallel circle P{P}_B^{\prime } are equal, which can be written as P{P}_{B_{\mathrm{0}}}=P_{B_{\mathrm{0}}}^{\prime }{P}_B^{\prime }=s_L.

So there is an ellipsoidal triangle, which is formed by the geodesic passing through P_{B_0}, P_{B_0}^{\prime } and the meridian passing through P_{B_{\mathrm{0}}}^{\prime }, P_B^{\prime } . Because the latitude difference between P_{B_{\mathrm{0}}} and P_{B_{\mathrm{0}}}^{\prime } is zero, according to the so-called inverse problem of the ellipsoidal solutions, where the Gauss mid-latitude solution is adopted, the geodesic length dq passing through P_{B_{\mathrm{0}}}, P_{B_0}^{\prime } is obtained:

(9)dq\cos A_m=\mathrm{0},

(10)dq\sin A_m=N_{\mathrm{0}}\cos B_{\mathrm{0}}dl-{\displaystyle \frac{\mathrm{1}}{\mathrm{24}}}{N}_{\mathrm{0}}\cos B_{\mathrm{0}}{\sin }^{\mathrm{2}}{B}_{\mathrm{0}}d{l}^{\mathrm{3}}.

From Eqs.(9),(10), the following formulas can be derived:

(11)A_m=\mathrm{90}°or270°,

(12)dq=d{s}_B-{\displaystyle \frac{\mathrm{1}}{\mathrm{24}}}{N}_{\mathrm{0}}\cos B_{\mathrm{0}}{\sin }^{\mathrm{2}}{B}_{\mathrm{0}}d{l}^{\mathrm{3}}=d{s}_B-{\displaystyle \frac{{\tan }^{\mathrm{2}}{B}_{\mathrm{0}}}{\mathrm{24}{N}_{\mathrm{0}}^{\mathrm{2}}}}d{s}_B^{\mathrm{3}}.

So there is no need to distinguish the difference between the length of dq and dB.

Different with parallel circles P_{B_{\mathrm{0}}}, P_B^{\prime } , the geodesic P_{B_{\mathrm{0}}}, P_B^{\prime } is not orthogonal with the meridian in P_{B_{\mathrm{0}}} or P_B^{\prime }. Derived from the direct solution of geodetic problem, the difference between forward and back azimuths on points P_{B_{\mathrm{0}}} and P_B^{\prime } respectively can be written as

(13)a=A_{\mathrm{21}}-A_{\mathrm{12}}\pm \mathrm{180}°={\displaystyle \frac{\tan B_m}{N_{\mathrm{0}}}}dq\left[\mathrm{1}+{\displaystyle \frac{d{q}^{\mathrm{2}}(\mathrm{1}+{\cos }^{\mathrm{2}}{B}_m)}{\mathrm{24}{N}_{\mathrm{0}}^{\mathrm{2}}{\cos }^{\mathrm{2}}{B}_m}}\right]={\displaystyle \frac{\tan B_m}{N_{\mathrm{0}}}}dq,

in which the third term is negligible. Equation (14) can be obtained:

(14)A_{\mathrm{12}}=A_m-{\displaystyle \frac{a}{\mathrm{2}}}=\mathrm{90}°-{\displaystyle \frac{\tan B_m}{\mathrm{2}{N}_{\mathrm{0}}}}dq,

(15)A_{\mathrm{21}}=A_m+{\displaystyle \frac{a}{\mathrm{2}}}+\mathrm{180}°=\mathrm{270}°+{\displaystyle \frac{\tan B_m}{\mathrm{2}{N}_{\mathrm{0}}}}dq.

It is known that the ellipsoidal surface within local area can be replaced by Gaussian sphere, of which radius R is the mean radius of curvature:

(16)R=\sqrt{MN}.

Here M and N are respectively the radius of curvature of the ellipsoid in the meridian and in prime vertical section, which are evaluated for the geodetic latitude (B+2B₀)/3. So the ellipsoidal triangle P_{B_0}{P}_{B_0}^{\prime }{P}_B^{\prime } becomes a spherical triangle. Therefore, the spherical excess of the triangle P_{B_{\mathrm{0}}}^{\prime }{P}_B^{\prime }{P}_{B_{\mathrm{0}}} can be expressed by

(17)ϵ={\displaystyle \frac{dq{s}_L\sin \left[\mathrm{90}°-{\displaystyle \frac{\tan B_m}{\mathrm{2}{N}_{\mathrm{0}}}}dq\right]}{\mathrm{2}{R}^{\mathrm{2}}}}.

Regardless of the higher-order terms, Eq.(18) can be rewritten as

(18)ϵ={\displaystyle \frac{dq{s}_L}{2R^2}}.

From the law of sine from spherical trigonometry, we have

{\displaystyle \frac{s_L}{\sin \left(\mathrm{90}°-dA-{\displaystyle \frac{\tan B_m}{\mathrm{2}{N}_{\mathrm{0}}}}dq-{\displaystyle \frac{dq{s}_L}{\mathrm{6}{R}^{\mathrm{2}}}}\right)}}={\displaystyle \frac{dq}{sin\left({\displaystyle \frac{dq{s}_L}{\mathrm{2}{R}^{\mathrm{2}}}}+dA+{\displaystyle \frac{\tan B_m}{\mathrm{2}{N}_{\mathrm{0}}}}dq+{\displaystyle \frac{\tan B_m}{\mathrm{2}{N}_{\mathrm{0}}}}dq-{\displaystyle \frac{dq{s}_L}{{6R}^2}}\right)}}.

The tiny difference between dq and dB can be neglected considering Eq. (12) and we finally obtain

(19){\displaystyle \frac{dA}{d{s}_B}}={\displaystyle \frac{dA}{dq}}={\displaystyle \frac{\mathrm{1}}{s_L}}\left(\mathrm{1}-{\displaystyle \frac{\tan B_{\mathrm{0}}{s}_L}{N^{\mathrm{0}}}}-{\displaystyle \frac{s_L^{\mathrm{2}}}{\mathrm{3}{R}^{\mathrm{2}}}}\right).

Combined with Eqs. (6), (8) and (19), we arrive at

$$\begin{gathered} n=\left({\displaystyle \frac{\partial y}{\partial A}}\right)\left({\displaystyle \frac{\partial A}{\partial s_B}}\right)=\left[\mathrm{1}-{\displaystyle \frac{(\mathrm{1}+{\eta }_{\mathrm{0}}^{\mathrm{2}})s_L^{\mathrm{2}}}{\mathrm{6}{N}_{\mathrm{0}}^{\mathrm{2}}}}+{\displaystyle \frac{{\eta }_{\mathrm{0}}^{\mathrm{2}}\tan B_{\mathrm{0}}{s}_L^{\mathrm{3}}}{\mathrm{3}{N}_{\mathrm{0}}^{\mathrm{3}}}}+{\displaystyle \frac{s_L^{\mathrm{4}}}{\mathrm{120}{N}_{\mathrm{0}}^{\mathrm{4}}}}\right]\left(\mathrm{1}-{\displaystyle \frac{\tan B_{\mathrm{0}}{s}_L}{N_{\mathrm{0}}}}-{\displaystyle \frac{s_L^{\mathrm{2}}}{\mathrm{3}{R}^{\mathrm{2}}}}\right) \\ =\mathrm{1}-{\displaystyle \frac{\tan B_{\mathrm{0}}{s}_L}{N_{\mathrm{0}}}}-{\displaystyle \frac{s_L^{\mathrm{2}}}{\mathrm{2}{R}^{\mathrm{2}}}}-{\displaystyle \frac{{\eta }_{\mathrm{0}}^{\mathrm{2}}{s}_L^{\mathrm{2}}}{\mathrm{6}{N}_{\mathrm{0}}^{\mathrm{2}}}}+{\displaystyle \frac{\tan B_{\mathrm{0}}(\mathrm{1}+\mathrm{3}{\eta }_{\mathrm{0}}^{\mathrm{2}})s_L^{\mathrm{3}}}{\mathrm{6}{N}_{\mathrm{0}}^{\mathrm{3}}}}+{\displaystyle \frac{(\mathrm{23}-\mathrm{120}{\eta }_{\mathrm{0}}^{\mathrm{2}}{\tan }^{\mathrm{2}}{B}_{\mathrm{0}}+\mathrm{20}{\eta }_{\mathrm{0}}^{\mathrm{2}})s_L^{\mathrm{4}}}{\mathrm{360}{N}_{\mathrm{0}}^{\mathrm{4}}}}. \end{gathered}$$

Up to the three-order term and quadratic term respectively, the formulas of n can be given as

(20)n\mathrm{2}=\mathrm{1}-{\displaystyle \frac{\tan B_{\mathrm{0}}{s}_L}{N_{\mathrm{0}}}}-{\displaystyle \frac{s_L^{\mathrm{2}}}{\mathrm{2}{R}^{\mathrm{2}}}}-{\displaystyle \frac{{\eta }_{\mathrm{0}}^{\mathrm{2}}{s}_L^{\mathrm{2}}}{\mathrm{6}{N}_{\mathrm{0}}^{\mathrm{2}}}}+{\displaystyle \frac{\tan B_{\mathrm{0}}(\mathrm{1}+\mathrm{3}{\eta }_{\mathrm{0}}^{\mathrm{2}})s_L^{\mathrm{3}}}{\mathrm{6}{R}_{\mathrm{0}}^{\mathrm{3}}}},

(21)n\mathrm{1}=\mathrm{1}-{\displaystyle \frac{\tan B_{\mathrm{0}}{s}_L}{N_{\mathrm{0}}}}-{\displaystyle \frac{s_L^{\mathrm{2}}}{\mathrm{2}{R}_{\mathrm{0}}^{\mathrm{2}}}}.

Numerical examples

The calculated latitudinal scale factor of length for geodesic parallel lines is denoted by symbols n1 and n2, which are respectively computed according to Eqs. (21) and (20). When P_i with latitude B_i is respectively in the north and the south of the initial parallel circle with latitude B₀, the calculated results are listed in Tables 1 and 2.

For comparison, the standard values n₀ calculated by Eq. (3) with latitude B₀ and B_i are also listed in the two tables.

From Tables 1 and 2, in the region, (the span is 400 km from south to north and unlimitedly from west to east), compared the standard values of latitudinal scale factor of length obtained by Eq. (3) with the calculated value of latitudinal scale factor of length calculated by Eq. (20), the maximal difference between them is less than 1.3×10^-6 . Even for that calculated from Eq. (21), the maximal difference in the region is less than 8.6×10^-6.

In the north of the equator, the lower the latitude is, the higher the calculation precision is. The calculated values and standard values in the south and the north of the initial latitude are not of absolute symmetry and calculation precision in the south of the initial latitude is slightly higher than that in the north of the initial latitude.

Intrinsic properties of new geodetic coordinate system

1) The coordinate parameters s_L, s_B measured with length are naturally different from the geodetic coordinates (B, L) measured with angles, so it is useful and convenient to express the positions determined by surveying and modeling of GIS in the new proposed geodetic coordinates.

2) The new form of geodetic coordinate system also consists of two families of coordinate curves i.e. geodetic meridians and parallel circles.

3) The latitudinal scale factor of length n=\sqrt{G} of one point mainly depends on s_L at the point itself. Its longitude can be omitted. It is interesting to note that the latitudinal scale factor of length has the theoretical value. When a point is in the north of the initial parallel circle, n at this point is less than 1, otherwise it is more than 1, compared with the case in the geodetic parallel coordinate system on ellipsoid [2], n (named as the reducing factor for abscissa) is always less than 1 or equal 1.

4) The coordinate transformation between the new geodetic coordinate system expressed by s_L, s_B and the conventional geodetic coordinate system expressed by longitude and latitude can be achieved. The algorithms and formulas for the direct and inverse solutions to geodesic problems can also be obtained. The direct transformation between the two different new geodetic coordinate systems can be carried out easily.

References

[1] Kneissl M. Mathematisch Geodäsie (Landevermessung), Die geodätischen Berechnungen auf der Kugel und auf dem Ellipsoid. Jordan-Eggert-Kneißl. Handbuch der Vermessungskunde, Band Ⅳ 10. Auf J. B. Metzlersche,Verlagsbuchhandlung, Stuttgart:1959: 704-753, 1067-1093

[2] Grossmann W. Geodaetische Rechnungen und Abbildungen in der Landesvermessung, Dritte, Verlag Konrad Wittwer in Stuttgart, 1976: 31-56, 127-136

[3] Heck H. Rechenverfahren und Auswertemodelle der Landesvermessung. 3rd ed. Herbert Wichmann, Heidelbe: Springer, 2003: 123-152, 187-204

[4] Shi Yimin, Feng Yan. Establishment and application of another form of geodesic coordinate system on the earth ellipsoid. Journal of Tongji University (Natural Science), 2001, 29(11): 1282-1285 (in Chinese)

[5] Shi Yimin, Zhu Ziyang. The differential equations and differential relationship of geodesic lines in the geodesic coordinate system. Journal of Tongji University (Natural Science), 2003, 31(1): 40-43 (in Chinese)

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