Solving the Direct Ray Intersection on the Terrestrial Spheroid for Target Distances Not Exceeding 1000–1500 km

B. F. Krutoi

(Presented by the scientific seminar of the Departments of Geodesy and Mine Surveying in January 1960.)

In my article [1], two methods were given for solving the direct ray intersection on the terrestrial spheroid at arbitrary distances to the target. In the first of these methods, only the geodetic coordinates of the target are found. In the second method, however, the coordinates of the target are determined together with the distances to it from the initial points.

In the present article a somewhat different, simpler method is set forth for solving the direct ray intersection on the terrestrial spheroid; it was found by me earlier than the general methods [1] and is suitable when the distances to the target do not exceed 1000–1500 km. The following problem is posed.

The geodetic coordinates \(B_1, L_1\) and \(B_2, L_2\) of the initial points 1, 2 of the spheroid are known, and the geodetic azimuths \(A_{1.3}, A_{2.3}\) at points 1, 2 are also known for the straightened rays \(\overset{\uparrow}{\odot}L_{1.3}\), \(\overset{\uparrow}{\odot}L_{2.3}\), intersecting the point 3 of the spheroid to be determined. Find the geodetic coordinates \(B_3, L_3\) of point 3.

The proposed particular method for solving the direct spheroidal intersection is based on the same considerations as the first general method [1]. We shall present these considerations.

Suppose that by some method, for example from a map, we have determined the approximate position \(B_3^{(0)}\) of the latitude of the point 3 to be determined. Then, along each intersecting ray \(\overset{\uparrow}{\odot}L_{1.3}\), \(\overset{\uparrow}{\odot}L_{2.3} \in \overset{\uparrow}{\odot}L_{i3}\), we can compute, quite precisely for the corresponding latitude \(B_3^{(0)}\), the approximate longitude differences \(\Delta L_{i3}^{(0)}\). As a result we obtain two approximate positions \(3_1^{(0)}, 3_2^{(0)} \in 3_i^{(0)}\) of the point 3 to be determined, which lie on the corresponding rays \(\overset{\uparrow}{\odot}L_{1.3}\), \(\overset{\uparrow}{\odot}L_{2.3}\), have one and the same latitude \(B_3^{(0)}\), and longitudes \(L_{3}^{i(0)}\) differing somewhat from one another, where

\[ L_{3}^{i(0)} = L_i + \Delta L_{i3}^{(0)} . \tag{1} \]

For the exact longitude differences \(\Delta L_{i3} = L_3 - L_i\) between the control points \(i = 1, 2\) and the point 3 to be determined, the obvious identity will hold:

\[ \Delta L_{1.3} - \Delta L_{2.3} = \Delta L_{1.2}. \tag{2} \]

If, however, in this identity, instead of the exact values \(\Delta L_{i3}\), we substitute the approximate values \(\Delta L_{i3}^{(0)}\) found above, then we obtain a certain residual \(w_L\):

\[ w_L=\left(\Delta L_{1.3}^{(0)}-\Delta L_{2.3}^{(0)}\right)-\Delta L_{1.2}, \tag{3} \]

which is caused, evidently, by the errors \(\delta(\Delta L_{i3}^{(0)})\) of the indicated approximate values \(\Delta L_{i3}^{(0)}\). Assuming, as usual,

\[ \Delta L_{i3}^{(0)}=\Delta L_{i3}+\delta(\Delta L_{i3}^{(0)}), \tag{4} \]

we obtain, taking into account identity (1), that

\[ w_L=\delta(\Delta L_{1.3}^{(0)})-\delta(\Delta L_{2.3}^{(0)}). \tag{5} \]

The errors \(\delta(\Delta L_{i3}^{(0)})\) of the approximate longitude differences \(\Delta L_{i3}^{(0)}\) are, of course, caused by the error \(\delta B_3^{(0)}\) in the adopted value \(B_3^{(0)}\) of the latitude of point 3, where by definition

\[ \delta B_3^{(0)}=B_3^{(0)}-B_3. \tag{6} \]

Therefore, if we take into account that

\[ \Delta L_{i3}=F(B_i,A_{i3},B_3)=F(B_i,A_{i3},B_3^{(0)}-\delta B_3^{(0)})=F_i, \tag{7} \]

then, from this, when the error \(\delta B_3^{(0)}\) is small, we may write:

\[ \Delta L_{i3}=F(B_i,A_{i3},B_3^{(0)})-\left(\frac{\partial F_i}{\partial B_3}\right)_0\delta B_3^{(0)} =F_i^{(0)}-\frac{\partial F_i^{(0)}}{\partial B_3}\delta B_3^{(0)}. \]

But from (7) it follows that

\[ F(B_i,A_{i3},B_3^{(0)})=F_i^{(0)}=\Delta L_{i3}^{(0)} \]

and, consequently,

\[ \Delta L_{i3}=\Delta L_{i3}^{(0)}-\left[\frac{\partial(\Delta L_{i3})}{\partial B_3}\right]_0\delta B_3^{(0)}. \tag{8} \]

Having determined from this the two values \(\Delta L_{i3}=\Delta L_{1.3}^{(0)},\ \Delta L_{2.3}^{(0)}\) and substituting them into expression (3) for the residual \(w_L\), we obtain an equation for computing the error \(\delta B_3^{(0)}\) in the adopted value \(B_3^{(0)}\) of the latitude of point 3:

\[ \left[\frac{\partial(\Delta L_{1.3})}{\partial B_3} -\frac{\partial(\Delta L_{2.3})}{\partial B_3}\right]_0 \delta B_3^{(0)}-w_L =(a_{1.3}-a_{2.3})\delta B_3^{(0)}-w_L=0, \tag{9} \]

where the symbol \({}^{\circ}\) reminds us that the derivatives must be computed at \(B_3=B_3^{(0)}\). From comparison of (8) with (4) we then find the errors \(\delta(\Delta L_{i3}^{(0)})\) in the approximate longitude differences \(\Delta L_{i3}^{(0)}\):

\[ \delta(\Delta L_{i3}^{(0)})= -\frac{\partial(\Delta L_{i3}^{(0)})}{\partial B_3}\delta B_3^{(0)} =-a_{i3}\delta B_3^{(0)}. \tag{10} \]

In conclusion, we compute the refined coordinates \(B_3^{(1)}, L_3^{(1)}\) of the determined point 3:

\[ \begin{aligned} &1)\quad B_3^{(1)}=B_3^{(0)}-\delta B_3^{(0)}\simeq B_3;\\ &2)\quad L_3^{(1)}=L_3^{(0)}-\delta(\Delta L_{1.3}^{(0)}) =L_3^{(0)}-\delta(\Delta L_{2.3}^{(0)})\simeq L_3, \end{aligned} \tag{11} \]

which may be adopted as final \(B_3, L_3\), if the initial value \(B_3^{(0)}\) was taken sufficiently accurately. If, however, the initial value \(B_3^{(0)}\) of the latitude of point 3 was too rough, then it will be necessary

the procedure described here for finding the coordinates \(B_3, L_3\) should be repeated one or two more times, taking as the initial value \(B_3^{(1)}\), obtained from the first approximation.

In the method set forth here for solving the direct intersection, two questions have remained unresolved: 1) the exact computation of the longitude differences \(\Delta L_{i3}^{(0)}\), corresponding to the adopted value \(B_3^{(0)}\) of the latitude of point 3; 2) the computation of the coefficients \(a_{i3}\) of equation (9).

As for the coefficients \(a_{i3}\), they can be found as follows:

\[ \begin{gathered} 1)\quad A'_{3i}=A_{3i}\pm180^\circ;\qquad 2)\quad \sin A'_{3i}=\frac{V_3\cos B_i}{V_i\cos B_3}\sin A_{i3};\\[6pt] 3)\quad a_{i3}=\frac{\partial(\Delta L_{3i}^{(0)})}{\partial B_3} =\frac{\operatorname{tg} A'_{3i}}{V_3^2\cos B_3}, \end{gathered} \tag{12} \]

where, in computing the coefficients \(a_{i3}\), we set \(B_3 \simeq B_3^{(0)}\).

For the exact computation of the longitude differences \(\Delta L_{i3}^{(0)}\), several ways may be proposed. In the first general method [1] for solving the direct spheroidal intersection, for this purpose use was made of expansions in powers of the small parameter \(\kappa_{i3}^2 \leqslant e^2\), obtained for the integral

\[ \beta_{i3}\Delta L_{i3}=(\gamma\mu)_{i3} \int_{\varphi_i}^{\varphi_2} \frac{d\varphi}{(1-m_{i3}^2\sin^2\varphi)\sqrt{1-\kappa_{i3}^2\sin^2\varphi}}, \qquad \beta_{i3}=\pm1, \tag{13} \]

where \(\gamma_{i3}\), \(\mu_{i3}\), \(m_{i3}^2\) are parameters depending on \(B_i\) and \(A_{i3}\), while \(\varphi\) is the variable of integration, depending on \(B_i\), \(A_{i3}\), and \(B\). For computing the longitude differences \(\Delta L_{i3}^{(0)}\) in the general case, one may also use the expansions found in (2), in powers of \(e^2\nu_{i3}^2\), of the integral

\[ \beta_{i3}\Delta L_{i3} = \int_{A_{i3}}^{A'_{3i}} \sqrt{\frac{\sin^2 A-e^2\nu_{i3}^2}{\sin^2 A-\nu_{i3}^2}}\,dA . \tag{14} \]

Finally, if the lengths \(s_{i3}\) of the intersecting sides \(i3\) do not exceed certain definite limits, then the differences \(\Delta L_{i3}^{(0)}\) can be computed quite simply and accurately from the closed expression

\[ \operatorname{tg}\frac{1}{2}\Delta L_{i3}^{(0)} = \frac{\sin\frac{1}{2}\bigl(B_3^{(0)}-B_i\bigr)} {\cos\frac{1}{2}\bigl(B_3^{(0)}+B_i\bigr)} \operatorname{tg}\frac{1}{2}\bigl(\Delta A_{i3}^{(0)}-\delta_{i3}\bigr), \tag{15} \]

where \(\Delta A_{i3}^{(0)}=A_{3i}^{\prime(0)}-A_{i3}\), and \(\delta_{i3}\) is a very small correction, not exceeding \(0''.5\) when \(s_{i3}\simeq1500\) km. In the present article, the method of solving the direct spheroidal intersection is considered for a limited length \(s_{i3}\) of the intersecting sides \(i3\), in which expression (15) is used to compute the differences \(\Delta L_{i3}^{(0)}\). The principal content of the article is the derivation of an expression for the very small correction \(\delta_{i3}\) entering into (15). We shall give this derivation.

Map the rectified arc \(\Delta\Gamma_{1.2}\) of the terrestrial spheroid onto the rectified arc \(\Delta\Gamma'_{1.2}\) of a sphere of radius \(a\) in such a way that for any

the pairs of corresponding points \(C, C^\circ\) on the arcs \(\Delta \Gamma_{1.2}, \Delta \Gamma^\circ_{1.2}\) were equal to their spheroidal \(B\) and spherical \(\varphi\) latitudes, while for the arcs \(\Delta \Gamma_{1.2}, \Delta \Gamma^\circ_{1.2}\) as a whole their spheroidal \(\Delta L_{1.2}\) and spherical \(\Delta \lambda_{1.2}\) differences of longitude of points \(1, 2\) and \(1^\circ, 2^\circ\) were equal. Thus we adopt the following law of mapping the spheroid onto the sphere:

\[ 1)\ B=\varphi,\qquad 2)\ \Delta L_{1.2}=\Delta\lambda_{1.2}. \tag{16} \]

If, moreover, \(A\) and \(a\) are the azimuths of the arcs \(\Delta \Gamma_{1.2}, \Delta \Gamma^\circ_{1.2}\) at the corresponding points \(C, C^\circ\), and \(L\) and \(\lambda\) are the longitudes of these points, then in general

\[ 1)\ L\ne \lambda,\qquad 2)\ A\ne a, \tag{17} \]

which follows from the differential relation

\[ \frac{d\lambda}{dL}=\frac{da}{dA} =(1+e'^2\cos B)\frac{\operatorname{ctg} a}{\operatorname{ctg} A} =V^2\frac{\operatorname{ctg} a}{\operatorname{ctg} A}, \tag{18} \]

valid under the mapping law (16).

Suppose now that the spheroidal triangle \(P12\), formed by the meridian arcs \(P1, P2\) and the aligned arc \(\Delta \Gamma_{1.2}\), is mapped according to the law (16) onto a sphere of radius \(a\). Then for the corresponding mapped triangle \(P^\circ 1^\circ 2^\circ\) on the sphere, which by assumption will be aligned on this surface, one may write the known relation

\[ \operatorname{tg}\frac{1}{2}\Delta\lambda_{1.2} = \frac{ \sin \frac{1}{2}(\varphi_2-\varphi_1) }{ \cos \frac{1}{2}(\varphi_2+\varphi_1) } \operatorname{tg}\frac{1}{2}\Delta a_{1.2}, \]

where \(\Delta a_{1.2}=a'_{2.1}-a_{1.2}\) is the difference of the forward azimuths at points \(2^\circ\) and \(1^\circ\) of the aligned arc \(\Delta \Gamma^\circ_{1.2}\). On the basis of the equalities (16), this relation can be rewritten in the following form, coinciding with (15):

\[ \operatorname{tg}\frac{1}{2}\Delta L_{1.2} = \frac{ \sin \frac{1}{2}(B_2-B_1) }{ \cos \frac{1}{2}(B_2+B_1) } \operatorname{tg}(\Delta A_{1.2}-\delta_{1.2}), \tag{19} \]

where

\[ 1)\ \Delta A_{1.2}=A'_{2.1}-A_{1.2},\qquad 2)\ \delta_{1.2}=\Delta \alpha_{1.2}-\Delta A_{1.2}. \tag{20} \]

The difference \(\Delta A_{1.2}\) can be computed quite exactly if the latitudes \(B_1, B_2\) of the endpoints \(1,2\) of the arc \(\Delta \Gamma_{1.2}\) and the azimuth \(A_{1.2}\) of this arc at point 1 are known, since

\[ \sin A'_{2.1} = \frac{V_2\cos B_1}{V_1\cos B_2}\sin A_{1.2} = g_{1.2}\sin A_{1.2}. \tag{21} \]

Therefore our main task consists in obtaining a working expression for the correction \(\delta_{1.2}\) and in investigating the magnitude of this correction for various lengths \(s_{1.2}\) of the arc \(\Delta \Gamma_{1.2}\).

To solve the problem posed, let us take into account that Prof. F. A. Sludskii in [2] found an exact relation connecting the azimuth \(\widetilde A_{x\mu}\) of the normal-section line \(\Delta \widetilde\Gamma_{x\mu}\) at point \(x\) of the spheroid

with the azimuth \(\alpha_{x\mu}\) of the rectified arc \(\Delta T^{0}_{x\mu}\) at the corresponding point \(x^{c}\) of the sphere when the spheroid is represented on the sphere according to the law (16):

\[ \operatorname{ctg}\widetilde A_{x\mu} = \operatorname{ctg}\alpha_{x\mu} - \frac{ e^{2}\,[V_x\sin B_\mu - V_\mu\sin B_x]\cos B_x }{ V_x\cos B_\mu\sin\Delta L_{x\mu} }, \qquad \binom{x=1,2}{\mu=2,1}^{*}. \tag{22} \]

The azimuth \(\widetilde A_{x\mu}\) of the straight normal section \(\Delta\widetilde T_{x\mu}\) at point \(x\) of the spheroid, for distances \(\widetilde s_{x\mu}\leq 1000\) km, differs very little from the azimuth \(A_{x\mu}\) of the rectified arc \(\Delta R_{x\mu}\) at the same point \(x\) of the spheroid. This is seen from the expression obtained in [3] for the difference \(\delta A_{x\mu}=\widetilde A_{x\mu}-A_{x\mu}=\eta_{x\mu}\) of the indicated azimuths:

\[ \left. \begin{aligned} \delta A''_{x\mu} &= (\widetilde A_{x\mu}-A_{x\mu})'' = \rho''\,\frac{(e')^{2}}{6}\,\bar\sigma^{\,2}_{x\mu}\cos^{2}B_x\sin A_{x\mu} \Biggl[ \cos A_{x\mu} \\ &\quad -\frac{\bar\sigma_{x\mu}}{4}\tg B_x +\frac{\bar\sigma^{\,2}_{x\mu}}{6}\cos A_{x\mu} -(e')^{2}\cos^{3}A_{x\mu} \Biggr] \\ &= \rho''\,\frac{(e')^{2}}{12}\,\bar\sigma^{\,2}_{x\mu} \left( 1-\frac{\bar\sigma^{\,2}_{x\mu}}{6} \right) \cos^{2}B_x\sin 2A_{x\mu} -\rho''\,\frac{(e')^{2}}{48}\,\bar\sigma^{\,3}_{x\mu}\sin 2B_x\sin A_{x\mu} \\ &\quad -\rho''\,\frac{(e')^{4}}{12}\,\bar\sigma^{\,2}_{x\mu}\cos^{2}B_x\sin 2A_{x\mu}\cos^{2}A_{x\mu} = \eta''_{x\mu}, \end{aligned} \right\} \tag{23} \]

where

\[ 1)\ \binom{x=1,2}{\mu=2,1},\qquad 2)\ \bar\sigma_{x\mu}=\frac{s_{x\mu}}{N_x}\,\bar\sigma_{\mu x},\qquad 3)\ N_x^{2}=\frac{a^{2}(1-e'^{2})}{1+e'^{2}\cos^{2}B_x}, \tag{24} \]

\[ e'^{2}=\frac{a^{2}-b^{2}}{b^{2}}; \]

\(a, b\) are the major and minor semiaxes of the terrestrial spheroid.

From the considerations given, it follows that the desired correction \(\delta_{1.2}\) should expediently be represented in the following transformed form:

\[ \delta_{1.2}=\Delta A_{1.2}-\Delta\alpha_{1.2} = (\Delta\widetilde A_{1.2}-\Delta\alpha_{1.2}) - (\Delta\widetilde A_{1.2}-\Delta A_{1.2}), \]

where, on the basis of (21),

\[ \Delta\widetilde A_{1.2}-\Delta A_{1.2} = (\widetilde A_{2.1}-\widetilde A_{1.2}\pm 180^\circ) - (A_{2.1}-A_{1.2}\pm 180^\circ) = \]

\[ = (\widetilde A_{2.1}-A_{2.1}) - (\widetilde A_{1.2}-A_{1.2}) = \delta A_{2.1}-\delta A_{1.2} = \eta_{2.1}-\eta_{1.2}. \tag{25} \]

Hence, for the correction \(\delta_{1.2}\), we obtain the refined expression

\[ \delta_{1.2} = (\Delta A_{1.2}-\Delta\alpha_{1.2}) - (\delta A_{2.1}-\delta A_{1.2}) = \varepsilon_{1.2}-\Delta\eta_{1.2}, \tag{26} \]

which reduces the computation of \(\delta_{1.2}\) to the calculation of two partial corrections \(\varepsilon_{1.2}\) and \(\Delta\eta_{1.2}\).

The correction \(\Delta\eta_{1.2}=(\eta_{2.1}-\eta_{1.2})\) can be found according to (23). The determination of this correction is substantially facilitated by the use of nomograms compiled by A. G. Lesnyak [4]. As for the first partial correction \(\varepsilon_{1.2}\), below we shall obtain a simple expression for its calculation through the corrections \(\eta_{1.2}\) and \(\eta_{2.1}\).

\[ \text{*) Sludsky expressed the difference }(\operatorname{ctg}\alpha_{x\mu}-\operatorname{ctg}\widetilde A_{x\mu}) \text{ through the reduced latitudes }u_x,\ u_\mu. \]

To find the indicated correction $\varepsilon_{1,2}$, we first of all reduce the initial expression (22) to the form

\[ \sin(\widetilde A_{x\mu}-\alpha_{x\mu}) = \frac{e^2\cos B_x \sin \widetilde A_{x\mu}}{V_x\sin\sigma_{x\mu}} \left(V_x\sin B_\mu - V_\mu\sin B_x\right), \tag{27} \]

where it has been taken into account that, if $\sigma_{x\mu}$ is the angular measure of the arc $\Delta\Gamma^\circ_{x\mu}$, then

\[ \frac{\sin\alpha_{x\mu}}{\cos B_\mu\sin\Delta L_{x\mu}} = \operatorname{cosec}\sigma_{x\mu}. \tag{28} \]

Now replacing in the right-hand side of equality (27) the quantity $\widetilde A_{x\mu}$ by $A_{x\mu}$ according to (21):

\[ \widetilde A_{x\mu}=A_{x\mu}+\delta A_{x\mu}, \]

we expand $\sin\widetilde A_{x\mu}$ in a Taylor series. We obtain:

\[ \sin\widetilde A_{x\mu} = \sin A_{x\mu} +\cos A_{x\mu}\,\delta A_{x\mu} -\frac{1}{2}\sin A_{x\mu}(\delta A_{x\mu})^2-\ldots \tag{29} \]

In addition, we shall take into account that, by the basic property of the rectified curve $\Gamma_{x\mu}$ on the spheroid of revolution,

\[ \frac{\cos B_x}{V_x}\sin A_{x\mu} = \frac{\cos B_\mu}{V_\mu}\sin A'_{\mu x} = -\frac{\cos B_\mu}{V_\mu}\sin A_{\mu x}. \tag{30} \]

Then, using the chain relation (30) and the equality $\sigma_{x\mu}=\sigma_{\mu x}$, we introduce in (27) the abbreviated notations $Q_{x\mu}$, $Q_{\mu x}$, defining them by the relations:

\[ Q_{x\mu}\sin A_{x\mu} = \left[ \frac{e^2\cos B_x}{V_x\sin\sigma_{x\mu}} \left(V_x\sin B_\mu - V_\mu\sin B_x\right) \right]\sin A_{x\mu} = \]

\[ = \left[ \frac{e^2\cos B_\mu}{V_\mu\sin\sigma_{\mu x}} \left(V_\mu\sin B_x - V_x\sin B_\mu\right) \right]\sin A_{\mu x} = Q_{\mu x}\sin A_{\mu x}. \tag{31} \]

Finally, expanding in the left-hand side of equality (27) the sine of the small arc $(\widetilde A_{x\mu}-\alpha_{x\mu})$ in a series, we represent (27) in the following expanded form:

\[ (\widetilde A_{x\mu}-\alpha_{x\mu}) -\frac{1}{6}(\widetilde A_{x\mu}-\alpha_{x\mu})^3 = \]

\[ = Q_{x\mu} \left[ \sin A_{x\mu} +\cos A_{x\mu}\,\delta A_{x\mu} -\frac{1}{2}\sin A_{x\mu}(\delta A_{x\mu})^2 \right]. \tag{27.1} \]

In order now to derive the expression we need for the correction

\[ \varepsilon_{1,2}=\Delta\widetilde A_{1,2}-\Delta\alpha_{1,2}, \]

entering into (26), let us put in (27.1) first $x=1$, $\mu=2$, and then take $x=2$, $\mu=1$. Subtracting then from the second value of equality (27.1) its first value and neglecting very small differences of order higher than $e^6$:

\[ 1)\quad \frac{1}{6} \left[ (\widetilde A_{2.1}-\alpha_{2.1})^3 - (\widetilde A_{1.2}-\alpha_{1.2})^3 \right], \]

\[ 2)\quad -\frac{1}{2} \left[ Q_{2.1}\cos A_{2.1}(\delta A_{2.1})^2 - Q_{1.2}\cos A_{1.2}(\delta A_{1.2})^2 \right], \]

we obtain, taking into account the notation (31), the following compressed expression for the difference
\((\tilde A_{2.1}-\alpha_{2.1})-(\tilde A_{1.2}-\alpha_{1.2})=(\tilde A_{2.1}-\tilde A_{1.2}\pm180^\circ)-(\alpha_{2.1}-\alpha_{1.2}\pm180^\circ)\Delta\tilde A_{1.2}-\Delta\alpha_{1.2}=\varepsilon_{1.2}\) through the corrections \(\delta A_{1.2}, \delta A_{2.1}\):

\[ \varepsilon_{1.2}=\Delta\tilde A_{1.2}-\Delta\alpha_{1.2} =\left(Q_{2.1}\cos A_{2.1}\right)\delta A_{2.1} -\left(Q_{1.2}\cos A_{1.2}\right)\delta A_{1.2}. \tag{32} \]

Since the quantities \(Q_{\mu}, \delta A_{\mu}\), entering equality (32), are small of order not lower than \(e^2\), the quantity \(\varepsilon_{1.2}\) itself, determined by this equality, will be of order not lower than \(e^4\) for distances \(s_{1.2}\) up to 1000 km.

Having determined the first correction \(\varepsilon_{1.2}\), entering into (26), we can finally also compute the complete correction \(\delta_{1.2}\):

\[ \delta_{1.2}=\varepsilon_{1.2}-\Delta\eta_{1.2} =\left(Q_{2.1}\cos A_{2.1}-1\right)\delta A_{2.1} -\left(Q_{1.2}\cos A_{1.2}-1\right)\delta A_{1.2}. \tag{33} \]

The expression obtained for the complete correction \(\delta_{1.2}\) shows that, for distances \(s_{1.2}\) up to 1000 km, the quantity \(\delta_{1.2}\) will be of order not lower than \(e^2\). The actual values of the correction \(\delta_{1.2}\) for distances \(s_{1.2}\) up to 400 km will be very small, as is seen from Summary 1. On the basis of the data of this summary one may calculate that even for \(s_{1.2}=400\) km the transverse displacement \(\lambda_{1.2}\) of end 2 for the ray \(\overset{\uparrow}{\odot}L_{1.2}\) due to the correction \(\delta_{1.2}\) will be equal to 2 cm, i.e., negligibly small. Therefore, for distances \(s_{1,2}\) up to 400–500 km one may with a high degree of accuracy take \(\Delta\alpha_{1.2}=\Delta A_{1.2}\). For distances \(500\ \text{km}<s_{1.2}\leq1000\ \text{km}\), it is sufficient to set \(\delta_{1.2}=\delta A_{2.1}-\delta A_{1.2}=\eta_{2.1}-\eta_{1.2}\), and to carry out the calculations of all terms of expansion (21), except the first, by the nomograms of A. G. Lesnyak [4].

Summary 1

How small the correction \(\delta_{1.2}\) is even at very large distances \(s_{1.2}\) is shown by the following example. Let \(B_1=50\), \(A_{1.2}=40^\circ\), \(s_{1.2}=1500\) km. Then \(B_2=59^\circ17'03''\), \(\Delta\alpha_{1.2}=17^\circ03'34''\), \(A'_{2.1}=53^\circ57'00''\), and we find that \(\delta_{1.2}=+0''.513\). In this case the transverse displacement \(\lambda_{1.2}\) of end 2 of the ray \(\overset{\uparrow}{\odot}L_{1.2}\) will reach 3.8 m, which, in some calculations, may likewise be neglected.

It follows, therefore, from what has been said that in solving the spheroidal intersection with lengths \(s_{i3}\) of the intersecting sides up to 1000–1500 km, for an accurate computation of the approximate longitude differences \(\Delta l^{(0)}_{i3}\) it is appropriate to use the very simple expression (15), in which, within the limits indicated above for \(s_{i3}\), one may even take \(\delta_{i3}=0\).

As for the approximate value \(B^{(0)}_3\) of the latitude of point 3, which enters into equality (15), it may be taken directly from the map with an error of \(1'—5'\), or obtained by solving the direct intersection on a sphere, as was done in [1]. Then, in order to find the value \(B_3\) with a sufficient degree of accuracy, 2 approximations will be required. Finally, if the measured angle \(\gamma_3\) at the determined point 3 is known, which occurs in surface ray networks with all measured

angles, then \(B_3^{(0)}\) can be computed with a high degree of accuracy by means of the sequence of computational expressions indicated in my paper [5].

REFERENCES

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

2. F. A. Sludsky. Triangulation without a baseline. Moscow, 1865.

3. F. N. Krasovsky. Manual of Higher Geodesy, part 2, Moscow, 1942.

4. A. G. Lesnyak. Nomograms for computing the correction for the deviation of a geodesic line from a straight normal section. Tomsk, Izvestiya TPI, vol. 154, 1967.

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