New Expansions for the Distance \(s_{1.2}\) and the Longitude Difference \(\Delta L_{1.2}\) on the Earth’s Spheroid

B. F. Krutoi

(Presented in May 1966 to the mathematical section of the scientific-technical conference of the AVTF, dedicated to the 70th anniversary of TPI)

In article [1] the following closed expressions were indicated, and in article [2] obtained, as applied to the earth’s spheroid, for the distance \(s_{1.2}\) between points 1, 2 and for the longitude difference \(\Delta L_{1.2}\) of the same points 1, 2:

\[ 1)\quad \beta_{1.2}s_{1.2} = a\psi_{1.2} \int_{\varphi_1}^{\varphi_2} \frac{d\varphi}{\left(1-\kappa_{1.2}^{2}\sin^{2}\varphi\right)^{3/2}}; \]

\[ 2)\quad \beta_{1.2}\Delta L_{1.2} = (\nu\mu)_{1.2} \int_{\varphi_1}^{\varphi_2} \frac{d\varphi} {\left(1-m_{1.2}^{2}\sin^{2}\varphi\right)\sqrt{1-\kappa_{1.2}^{2}\sin^{2}\varphi}}; \]

\[ 3)\quad s_{1.2} = a\nu_{1.2} \int_{A_{1.2}}^{A'_{2.1}} \sqrt{ \frac{\sin^{2}A-e^{2}\nu_{1.2}^{2}} {\sin^{2}A-\nu_{1.2}^{2}} } \,\operatorname{cosec}^{2}A\,dA; \]

\[ 4)\quad \Delta L'_{1.2} = \int_{A_{1.2}}^{A'_{2.1}} \sqrt{ \frac{\sin^{2}A-e^{2}\nu_{1.2}^{2}} {\sin^{2}A-\nu_{1.2}^{2}} } \,dA. \]

Here \(A_{1.2}, A_{2.1}\) are the azimuths of the rectified curve \(\Gamma_{1.2}\) at points 1, 2; \(a^{2}\) and \(e^{2}\) are the semimajor axis and the eccentricity of the earth’s spheroid; \(A'_{2.1}=A_{2.1}\pm180^\circ\). As for the variable of integration \(\varphi\), the limits of integration \(\varphi_1, \varphi_2\), and the parameters \(\nu_{1.2}, \mu_{1.2}, m_{1.2}^{2}\), and \(\kappa_{1.2}^{2}\), they are functions of the latitude \(B\), of its limits \(B_1, B_2\), and of the azimuth \(A_{1.2}\) of the rectified curve \(\Gamma_{1.2}\) at point 1 of the spheroid. Finally, \(\beta_{1.2}=\operatorname{sign}\operatorname{cosec} A_{1.2}=\operatorname{sign}(B_2-B_1)=\operatorname{sign}(\varphi_2-\varphi_1)=\pm1\).

By expanding the closed expressions (1.1)—(1.2) in a series in powers of the small parameter \(\kappa^{2}\le e^{2}\), in article [2] work-

… expressions for solving the direct and inverse problems with respect to the rectified curve \(\Gamma_{1.2}\) of any length, even if the curve \(\Gamma_{1.2}\) describes a number of turns around the terrestrial spheroid. In the same paper, by using the indicated expansions for (1.1)—(1.2) in a series in powers of \(\kappa^2\), two general methods were proposed for solving the direct spheroidal resection of a point 3 determined from the given points 1,2: 1) when only the coordinates \(B_3, L_3\) of point 3 are computed; 2) when, simultaneously with the coordinates \(B_3, L_3\), the lengths \(s_{1.3}, s_{2.3}\) of the resecting sides 1.3, 2.3 are found. The solution of all these problems is substantially facilitated if one uses the tables, compiled by A. G. Lesnyak [3], of auxiliary quantities entering into the computational expressions. However, the computational expressions themselves have such a structure that they can comparatively easily be programmed for the direct solution of the indicated problems on a digital computer without the use of any tables at all.

In contrast to the closed expressions (1.1)—(1.2), considered above, for \(s_{1.2}\) and \(\Delta L_{1.2}\), the expressions (1.3) and (1.4) for these quantities can be used in solving the direct and inverse problems only as verification formulas, since their direct application for the indicated purpose proves inconvenient. The expressions (1.3)—(1.4) are more suitable for solving the direct spheroidal resection, since these expressions contain only one easily computed parameter \(v_{1.2}\), while the upper limit \(A'_{2.1}\) can be found exactly and simply from the approximate latitude \(B_3^{(0)}\) of the point 3 being determined. The actual solution of the direct spheroidal resection is carried out, as usual, by successive approximations. In all these cases it is necessary to develop effective methods for computing \(s_{1.2}\) and \(\Delta L_{1.2}\) according to (1.3)—(1.4), which is the aim of the present paper.

The closed expressions (1.3)—(1.4) under consideration for the distance \(s_{1.2}\) and the longitude difference \(\Delta L_{1.2}\) are elliptic integrals of the general form. The calculation of these integrals can be performed in several ways and, first of all, by representing them through Legendre’s reduced elliptic integrals \(F, E\), and \(\Pi\), for which exact and sufficiently detailed tables are available. However, the corresponding expressions of the general integrals (1.3)—(1.4) through the indicated reduced integrals \(F, E\), and \(\Pi\) prove to be rather complicated. Moreover, computing the reduced integrals themselves from tables is far from simple, especially for the integrals \(\Pi\), since it requires interpolation with differences of the 4th–6th order if the values of the integrals \(F, E\), and \(\Pi\) are to be obtained to 8 decimal places.

A more acceptable procedure is the calculation of elliptic integrals of the general form (1.3)—(1.4) by means of one of the existing methods of numerical integration, for example by Gauss’s method. However, here too the computations turn out to be rather complicated if they are to be carried out with an accuracy of up to 8 decimal places.

Therefore, if it is taken into account that \(e^2 = 0.0066934216\) for Krasovsky’s spheroid and that, according to [1],

\[ v_{1.2} = \frac{r_1}{a}\sin A_{1.2} = \frac{\sqrt{1+e'^2}\cos B_1\sin A_{1.2}}{V_1} = \]

\[ = \frac{\sqrt{1+e'^2}}{\sqrt{1+e'^2\cos^2 B_1}}\cos B_1\sin A_{1.2} \leq 1, \tag{2} \]

then it will be more advantageous to compute the integrals (1.3)—(1.4) by expanding their integrands in a series in powers of \(e^2 \vartheta_{1.2}^2\). Carrying out such an expansion, we obtain:

\[ \begin{aligned} 1)\quad s_{1.2} &= a\psi_{1.2}\sum_{x=0}^{n}\binom{1/2}{x} \left(-e^2\vartheta_{1.2}^2\right)^x \int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{2x+1} A}{\sqrt{\sin^2 A-\vartheta_{1.2}^2}}\,dA,\\[6pt] 2)\quad \beta_{1.2}\Delta L_{1.2} &= \sum_{x=0}^{n}\binom{1/2}{x} \left(-e^2\vartheta_{1.2}^2\right)^x \int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{2x-1} A}{\sqrt{\sin^2 A-\vartheta_{1.2}^2}}\,dA, \end{aligned} \tag{3} \]

where, according to [1],

\[ 1)\quad \binom{m}{x} = \frac{m(m-1)\ldots [m-(x-1)]}{1\cdot 2\ldots x}, \qquad \binom{m}{0}=\binom{m}{m}=1; \]

\[ 2)\quad \psi_{1.2}=\frac{1-e^2}{\sqrt{1-e^2\vartheta_{1.2}^2}}. \tag{4} \]

Let us compute the integrals entering into the expansion (3).

We shall first calculate the first term of the expansion (3.2), which is obtained for \(x=0\). Introducing the substitution \(x=\cos A\), we find:

\[ \Delta\lambda = \int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{-1}A\,dA}{\sqrt{\sin^2 A-\vartheta_{1.2}^2}} = \arc\sin\frac{\cos A_{1.2}}{\sqrt{1-\vartheta_{1.2}^2}} - \arc\sin\frac{\cos A'_{2.1}}{\sqrt{1-\vartheta_{1.2}^2}} = \]

\[ = \arc\sin\frac{\cos A_{1.2}}{\cos A^{э}_{1.2}} - \arc\sin\frac{\cos A'_{2.1}}{\cos A^{э}_{1.2}}, \tag{5} \]

where \(A^{э}_{1.2}\) is the azimuth of the adjusted curve \(\Gamma_{1.2}\) at the equator, since, according to (2), the parameter \(\vartheta_{1.2}=\sin A^{э}_{1.2}\) for \(B_1=0\).

Let us now calculate the other integrals entering into (3). Introducing into these integrals the substitution \(\sin^2 A=x\) and denoting \(2x-1,\; 2x+1\) by \(2s+1\), where \(s=0,1,2,\ldots\), we obtain first of all:

\[ I_s = \int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{2s+1}A\,dA}{\sqrt{\sin^2 A-\vartheta_{1.2}^2}} = \frac{1}{2}\int_{x_1}^{x_2} \frac{dx}{x^{s+1}\sqrt{(1-x)(x-\vartheta_{1.2}^2)}}, \tag{6} \]

where \(x_1=\sin^2 A_{1.2}\), \(x_2=\sin^2 A'_{2.1}\). Let us now perform the second substitution \(x=t^{-1}\); then

\[ I_s = -\frac{1}{2}\int_{t_1}^{t_2} \frac{t^s\,dt}{\sqrt{-\vartheta_{1.2}^2 t^2+(1+\vartheta_{1.2}^2)t-1}} = \]

\[ = -\frac{1}{2}\int_{t_1}^{t_2} \frac{t^s\,dt}{\sqrt{a t^2+bt+c}} = -\frac{1}{2}\int_{t_1}^{t_2} \frac{t^s\,dt}{\sqrt{R(t)}}, \tag{7} \]

where

\[ \begin{gathered} 1)\ t=x^{-1}=\csc^{2} A,\quad 2)\ t_{1}=\csc^{2} A_{1.2},\quad 3)\ t_{2}=\csc^{2} A'_{2.1},\\ 4)\ a=-\gamma_{1.2}^{2},\quad 5)\ b=1+\gamma_{1.2}^{2},\quad 6)\ c=-1,\quad 7)\ R(t)=at^{2}+bt+c. \end{gathered} \tag{8} \]

According to [4], for the integrals \(I_s\) we shall have the following expressions:

\[ \begin{gathered} 1)\ I_{0}=\int_{A_{1.2}}^{A'_{2.1}} \frac{\csc A\,dA}{\sqrt{\sin^{2} A-\gamma_{1.2}^{2}}} = S(t)\bigg|_{t_{1}}^{t_{2}}, \qquad S(t)=\frac{1}{2\gamma_{1.2}^{2}} \arcsin\frac{(1+\gamma_{1.2}^{2})-2\gamma_{1.2}^{2}t}{1-\gamma_{1.2}^{2}}; \\[1.2em] 2)\ I_{1}=\int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{3} A\,dA}{\sqrt{\sin^{2} A-\gamma_{1.2}^{2}}} = -\frac{1}{2\gamma_{1.2}^{2}} \left[\sqrt{R(t)}+(1+\gamma_{1.2}^{2})S(t)\right]\bigg|_{t_{1}}^{t_{2}}. \\[1.2em] 3)\ I_{2}=\int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{5} A\,dA}{\sqrt{\sin^{2} A-\gamma_{1.2}^{2}}} = \frac{1}{4\gamma_{1.2}^{2}} \left\{ \left[t+\frac{3(1+\gamma_{1.2}^{2})}{2\gamma_{1.2}^{2}}\right]\sqrt{R(t)} +\left[\frac{3(1+\gamma_{1.2}^{2})^{2}}{2\gamma_{1.2}^{2}}-2\right]S(t) \right\}\bigg|_{t_{1}}^{t_{2}}. \end{gathered} \tag{9} \]

\[ \begin{gathered} 4)\ I_{3}=\int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{7} A\,dA}{\sqrt{\sin^{2} A-\gamma_{2.1}^{2}}} = \frac{1}{2\gamma_{1.2}^{2}} \left\{ \left[ \frac{t^{2}}{3} +\frac{5(1+\gamma_{1.2}^{2})t}{12\gamma_{1.2}^{2}} +\frac{5(1+\gamma_{1.2}^{2})^{2}}{8\gamma_{1.2}^{4}} -\frac{2}{3\gamma_{1.2}^{2}} \right]\sqrt{R(t)} \right.\\ \left. -\frac{(1+\gamma_{1.2}^{2})}{2\gamma_{1.2}^{2}} \left[ 3-\frac{5(1+\gamma_{1.2}^{2})^{2}}{4\gamma_{1.2}^{2}} \right]S(t) \right\}\bigg|_{t_{1}}^{t_{2}}. \end{gathered} \]

Let us also note that the integral \(\Delta\lambda\), determined according to (5), can likewise be represented in the form (8), if in \(\Delta\lambda\) one introduces the substitution \(\sin^{2}A=x\) and then in (6) sets \(s=-1\). Then we obtain:

\[ \Delta\lambda = \int_{A_{1.2}}^{A'_{2.1}} \frac{\csc^{-1}A\,dA}{\sqrt{\sin^{2} A-\gamma_{1.2}^{2}}} = \frac{1}{2}\int_{x_{1}}^{x_{2}} \frac{dx}{\sqrt{(1-x)(x-\gamma_{1.2}^{2})}} = \frac{S(x)}{2}\bigg|_{x_{1}}^{x_{2}} = I_{-1}, \tag{10} \]

where

\[ 1)\ x_{1}=\sin^{2} A_{1.2},\qquad 2)\ x_{2}=\sin^{2} A'_{2.1}. \tag{11} \]

Relying on the working expressions (5), (9) obtained above, we can now present in the following final form the expansions (3) for the distance \(s_{1.2}\) and the longitude difference \(\Delta L_{1.2}\) of points 1, 2 of the spheroid:

\[ 1)\ s_{1.2} = a\gamma_{1.2}\sum_{x=0}^{n} \binom{1/2}{x}(-e^{2}\gamma_{1.2}^{2})^{x}I_{x} \doteq a\gamma_{1.2} \left[ I_{0} -\frac{e^{2}\gamma_{1.2}^{2}}{2}I_{1} -\frac{e^{4}\gamma_{1.2}^{4}}{8}I_{2} -\frac{e^{6}\gamma_{1.2}^{6}}{16}I_{3} -\ldots \right]; \tag{12} \]

\[ \begin{aligned} 2)\quad \beta_{1.2}\Delta L_{1.2} &= \Delta\lambda + \sum_{x=0}^{n} \binom{1/2}{x+1} \left(-e^2\nu_{1.2}^2\right)^{x+1} I_x \\ &= \Delta\lambda - \frac{e^2\nu_{1.2}^2}{2} I_0 - \frac{e^4\nu_{1.2}^4}{8} I_1 - \frac{e^6\nu_{1.2}^6}{16} I_2 - \cdots \, . \end{aligned} \tag{13} \]

The expansions (12) are new, and they can be used successfully to check the values of \(s_{1.2}\) and \(\Delta L_{1.2}\) found according to (1.1)—(1.2). It should be noted, however, that expression (12) is unsuitable for computing \(s_{1.2}\) along the meridian, i.e., when \(A_{1.2}=A_{2.1}=0\), while expressions (12) and (13) are unsuitable for computing \(s_{1.2}\) and \(\Delta L_{1.2}\) along the equator, i.e., when \(A_{1.2}=A'_{2.1}=90^\circ\). Naturally, also for short arcs \(\Delta T_{1.2}\) close in direction to the meridian or the equator, the expansions (12) likewise give insufficient accuracy.

As for the application of expansions (12) to the solution of the direct spheroidal intersection problem, it differs in no essential respect from the use, for this purpose, of expansions obtained on the basis of the closed expressions (1.1)—(1.2). One may only note that the computation, along the intersecting sides \(i_3\), of the values \(s_{i3}\) and \(\Delta L_{i3}\) according to (12) requires less work than when using the expansions obtained from (1.1)—(1.2).

REFERENCES

1. B. F. Krutoi. General methods for solving the principal computational problems on the surface of the terrestrial spheroid. Tomsk, Izvestiya TPI, vol. 118, 1963.

2. B. F. Krutoi. General methods for solving the principal computational problems on the terrestrial spheroid. Tomsk, Izvestiya TPI, vol. 131, 1965.

3. A. G. Lesnyak. Auxiliary tables for solving the principal geodetic problems on the terrestrial spheroid by the method of B. F. Krutoi. Tomsk, Izvestiya TPI, vol. 154, 1966.

4. I. S. Gradshteyn and I. M. Ryzhik. Tables of integrals, sums, series, and products. Moscow, Fizmatgiz, 1963.