NOMOGRAMS FOR CALCULATING THE CORRECTION FOR THE DEVIATION OF A GEODESIC LINE FROM A STRAIGHT NORMAL SECTION

A. G. LESNYAK

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

The formula for the correction for the deviation of a geodesic line from a straight normal section, in the form in which it was given by F. N. Krasovskii, is as follows:

\[ \delta''= \frac{\eta^2 \rho''}{6} \left(\frac{S}{N}\right)^2 \sin A_{1.2} \left[ \cos A_{1.2} - \frac{S}{4N}\tg B_1 + \frac{S^2}{6N^2}\cos A_{1.2} + \eta^2\cos^3 A_{1.2} \right], \tag{1} \]

where \(\eta^2=e'^2\cos^2 B_1,\quad N=\dfrac{a}{\sqrt{1-e^2\sin^2 B_1}},\quad S\) is the distance, \(\rho''=206264.8\),

\(A_{1.2}\) is the azimuth of the geodesic line at the point with latitude \(B_1\).

This formula has high accuracy of the order of 1000 km.

The calculation of \(\delta''\), as is evident from (1), requires a considerable amount of computation. Nomographing this formula makes it possible to eliminate the computational process completely. We shall carry out the nomographing of (1) in parts, for which we represent formula (1) in the form:

\[ \delta''= \frac{e'^2\rho''}{12}\sigma^2\sin 2A_{1.2}\cos^2 B_1 - \frac{e'^2\rho''}{48}\sigma^3\sin A_{1.2}\cos 2B_1 + \]

\[ + \frac{e'^2\rho''}{72}\sigma^4\sin 2A_{1.2}\cos^2 B_1 + \frac{e'^4\rho''}{6}\sigma^2\sin A_{1.2}\cos^3 A_{1.2}\cos^4 B_1, \tag{2} \]

where \(\sigma\) is the new variable

\[ \sigma=\frac{S}{N}. \]

Denote

\[ \left. \begin{aligned} (3')\qquad &\frac{e'^2\rho''}{12}\sigma^2\sin 2A_{1.2}\cos^2 B_1=\delta_1,\\[4pt] (3'')\qquad &\frac{e'^2\rho''}{48}\sigma^3\sin A_{1.2}\cos 2B_1=\delta_2,\\[4pt] (3''')\qquad &\frac{e'^2\rho''}{72}\sigma^4\sin 2A_{1.2}\cos^2 B_1=\delta_3,\\[4pt] (3^{\mathrm{IV}})\qquad &\frac{e'^4\rho''}{6}\sigma^2\sin A_{1.2}\cos^3 A_{1.2}\cos^4 B_1=\delta_4. \end{aligned} \right\} \tag{3} \]

Then the formula takes the form:

\[ \delta''=\delta_1-\delta_2+\delta_3+\delta_4. \]

Each \(\delta_i\) \((i=1,2,3,4)\) contains four variables: \(\sigma,\ A_{1.2},\ B_1,\ \delta_i\).

We shall carry out the nomographing of each of equations (3) by the method of eliminating variables. Let us consider in detail the construction of the nomogram for \((3')\); the construction of nomograms for \((3'')\), \((3''')\), \((3^{IV})\) is performed analogously.

Represent \((3')\) in the form of two equations, introducing a new variable \(\gamma\):

\[ \sin 2A_{1.2}\cos^2 B_1=\gamma \tag{4} \]

\[ \frac{e'^2\rho''}{12}\sigma^2\gamma=\delta_1. \tag{5} \]

Equations (4) and (5) now each contain three variables and are equations of the 3rd nomographic order, the nomographing of which presents no particular difficulties, provided that they are reduced to one of the three canonical forms:

\[ f_1 f_2=f_3, \tag{6} \]

\[ f_1+f_2=f_3, \tag{7} \]

\[ f_1+f_2+f_3=f_1 f_2 f_3. \tag{8} \]

Our equations (4) and (5) are evidently reduced to the first canonical form of type (6).

To combine the two nomograms of equations (4) and (5) into one, it is necessary that the scale \(\gamma\) be common. For this purpose we rewrite (5) in the form:

\[ \delta_1\frac{12}{\sigma^2 e'^2\rho''}=\gamma. \tag{9} \]

To make the nomogram adapted, we transform (9), introducing the multiplier \(\mu_1\)

\[ \mu_1\delta_1\frac{12}{\mu_1\sigma^2 e'^2\rho''}=\gamma, \tag{10} \]

after which we write (4) and (10) in the form of Soreau equations:

\[ \left| \begin{array}{ccc} \cos^4 B_1 & \cos^2 B_1 & 1\\ \sin^2 2A_{1.2} & -\sin 2A_{1.2} & 1\\ \gamma & 0 & 1 \end{array} \right|=0 \]

\[ \left| \begin{array}{ccc} \left(\dfrac{12}{\mu_1\sigma^2 e'^2\rho''}\right)^2 & \dfrac{12}{\mu_1\sigma^2 e'^2\rho''} & 1\\ (\mu_1\delta_1)^2 & -\mu_1\delta_1 & 1\\ \gamma & 0 & 1 \end{array} \right|=0 \]

Hence we obtain the scale equations:

1) for the variable \(B_1\):

\[ x=\cos^4 B_1, \]

\[ y=\cos^2 B_1; \]

2) for the variable \(A_{1.2}\):

\[ x=\sin^2 2A_{1.2}, \]

\[ y=-\sin 2A_{1.2}; \]

3) for the variable \(\gamma\):

\[ x=\gamma, \]

\[ y=0; \]

4) for the variable \(\sigma\):

\[ x=\left(\frac{12}{\sigma^2 e'^2\rho''\mu_1}\right)^2, \]

\[ y=\frac{12}{\sigma^2 e'^2\rho''\mu_1}; \]

5) for the variable \(\delta_1\):

\[ x=\delta_1^2\mu_1^2, \]

\[ y=-\delta_1\mu_1. \]

For each nomogram there is its own \(\mu_l\) \((l=1,\ldots,11)\). The equation of the supports for the scales of the variables \(A_{1.2}, B_1, \sigma, \delta_l\) coincide and lie on a curve of the second order (a parabola), whose equation is

\[ x = y^2 . \]

The scales of the variables \(B_1\) and \(\sigma\) lie on one branch of the parabola, with the scale \(B_1\) lying on the inner side, while the scales of the variables \(A_{1.2}\) and \(\delta_l\) lie on the other branch of the parabola, with the scale \(A_{1.2}\) lying on the inner side. The scale \(\gamma\) is blank and therefore requires no graduation.

But it is not possible to depict the entire nomographic domain on one drawing for \(\delta_i\), since the limits of variation of the distance \(S\) are large:

\[ 0 \leqslant S \leqslant 1300\ \text{km}. \]

Therefore, for each \(\delta_i\) a set of nomograms has been prepared, divided according to the argument \(\sigma\). The method of using the nomogram is very simple: we join by a straight line the given values \(A_{1.2}\) and \(B_1\) and find the point \(\gamma_1\) at which this straight line intersects the scale \(\gamma\). We join by a straight line the given value \(\sigma_1\) and the point \(\gamma_1\); \(\delta_1\) is read on the scale \(\delta_1\) at the point where the latter straight line intersects the branch of the parabola.

Nomogram No. 1 ensures an accuracy in determining the correction \(\delta_1\) of \(0''.01\)—\(0''.02\); nomograms Nos. 2, 3, 4—with an accuracy up to \(0''.005\)—\(0''.01\); nomogram No. 5—with an accuracy up to \(0''.005\); nomograms Nos. 6, 7, 8, 9, 10, 11—with an accuracy up to \(0''.001\).

For small distances of the order of 20–150 km this type of nomogram becomes unsuitable. For \(\delta_1, \delta_2, \delta_3\) one nomogram each has been made for small distances. The equations of the scales are straight lines. There are altogether 5 scales, corresponding to the number of variables. The scale \(\gamma\) is blank. The method of using such nomograms is as follows: we join by a straight line the given values \(\sigma\) and \(A_{1.2}\) (in nomogram 7—\(\sigma\) and \(B_1\)). We find the point \(\gamma_1\) at which this straight line intersects the scale \(\gamma\). We join the point \(\gamma_1\) with the given value \(B_1\) (in nomogram No. 7, with the given value \(A_{1.2}\)). The intersection of this straight line with the scale \(\delta_i\) gives the answer.

For the parameter \(\sigma\) a table with two entries has been compiled. The determination of \(\sigma\) from the table is carried out with an accuracy of 3 or 4 decimal places according to the arguments \(S\) and \(B_1\), where \(S\) is the distance, and \(B_1\) is the geodetic latitude of the initial point.

Below are given the limits of variation of \(\sigma\) in each nomogram, as well as the values of \(\mu_l\):

nomogram No. 1 for \(\delta_1\)

\[ \mu_1 = 0.600 \qquad 0{,}122 \leqslant \sigma \leqslant 0.157, \]

nomogram No. 2 for \(\delta_1\)

\[ \mu_2 = 1.000 \qquad 0{,}071 \leqslant \sigma \leqslant 0{,}122, \]

nomogram No. 3 for \(\delta_1\)

\[ \mu_3 = 2{,}854 \qquad 0{,}042 \leqslant \sigma \leqslant 0{,}071, \]

nomogram No. 4 for \(\delta_1\)

\[ 0{,}004 \leqslant \sigma \leqslant 0{,}042, \]

nomogram No. 5 for \(\delta_2\)

\[ \mu_5 = 6.907 \qquad 0.143 \leqslant \sigma \leqslant 0{,}204, \]

nomogram No. 6 for \(\delta_2\)

\[ \mu_6 = 20.721 \qquad 0{,}100 \leqslant \sigma \leqslant 0{,}143, \]

Table of values of \(b\)

Continuation of the table

nomogram No. 7 for \(\delta_2\)
\[ 0.004 \leqslant \sigma \leqslant 0.100, \]

nomogram No. 8 for \(\delta_3\)
\[ \mu_8 = 51.791 \]
\[ 0.155 \leqslant \sigma \leqslant 0.204, \]

nomogram No. 9 for \(\delta_3\)
\[ 0.050 \leqslant \sigma \leqslant 0.155, \]

nomogram No. 10 for \(\delta_4\)
\[ \mu_{10} = 51.250 \]
\[ 0.100 \leqslant \sigma \leqslant 0.190, \]

nomogram No. 11 for \(\delta_4\)
\[ \mu_{11} = 200 \]
\[ 0.043 \leqslant \sigma \leqslant 0.100. \]

For the values of \(\mu_l\) indicated above, the dimensions of the drawing are \(300 \times 340\) mm. In nomograms No. 10 and No. 11, the scale of the variable \(A_{1.2}\) is located both on the inner side of the parabola from \(0^\circ\) to \(32^\circ\), and on the outer side from \(34^\circ\) to \(90^\circ\); in this case no superposition with the scale \(\sigma\) occurs.

References

1. N. M. Gersevanov. Theory and Construction of Engineering Nomograms. Moscow–Leningrad, ONTI, 1937.
2. N. A. Glagolev. Course of Nomography. Moscow, Vysshaya shkola, 1961.
3. M. V. Pentkovskii. Skeletons of Nomograms of Equations of the Third Nomographic Order. Moscow, Publishing House of the Academy of Sciences of the USSR, 1953.

Fig. 1.

No. 2

\[ 0.071 < \sigma' < 0.122 \]

Fig. 2.

Fig. 3.

Visible labels in the nomogram: \(A\), \(B\), \(C\), \(D\), \(R\), \(N3\), \(\delta\), \(0.012 \leq G \leq 0.071\).

Figure 4.

Fig. 5.

Fig. 6.

Visible inscriptions on the nomogram include: \(x\), \(y\), \(S\), \(N6\), \(g^2\), \(0.100 \le g^2 \le 0.143\), and scale markings such as \(0.00\), \(0.05\), \(0.10\), \(0.15\), \(0.20\), \(0.25\), \(0.30\), \(0.35\), \(0.40\), \(0.005\), \(0.010\), \(0.015\), \(0.020\), \(0.025\), \(0.030\), \(0.035\), \(0.040\), \(0.045\), \(0.050\), \(0.055\), \(0.060\), \(0.065\), \(0.070\), \(0.075\), \(0.080\), and degree-like markings \(0, 10, 20, 30, 40, 50, 60, 70, 80\).

Fig. 7.

Fig. 8.

\[ N_8 \qquad \delta_3, \]

\[ 0{,}155 < G < 0{,}204 \]

\[ \alpha,\qquad B_1,\qquad G \]

No. 9* \(\delta_3\)

\[ 0.050 \le \sigma \le 0.160 \]

Figure labels and scales shown:

• \(\sigma\): \(0.160,\ 0.150,\ 0.140,\ 0.130,\ 0.120,\ 0.110,\ 0.100,\ 0.095,\ 0.080,\ 0.070,\ 0.060,\ 0.050\)
• \(\gamma\)
• \(\delta_3''\): \(0.015,\ 0.010,\ 0.005,\ 0.001,\ 0.0005\)
• \(A_{1,2}\): \(45,\ 45;\ 35,\ 55;\ 30,\ 60;\ 25,\ 65;\ 20,\ 70;\ 15,\ 75;\ 10,\ 80;\ 5,\ 85\)
• \(B_\alpha\): \(0,\ 15,\ 25,\ 35,\ 45,\ 55,\ 65,\ 75,\ 85\)

Fig. 9.

* The distance between the \(\sigma\) and \(\gamma\) scales must be 33 mm.

No. 10 \(\delta_4\)

\[ 0.095 < \delta < 0.190 \]

Fig. 10.

Fig. 11.

Visible labels on the nomogram:

\[ 0^\circ \le \varphi \le 45^\circ \]

\[ 500 \le S \le 50000 \]

Scale labels include:

\[ 0.095,\ 0.090,\ 0.085,\ 0.080,\ 0.075,\ 0.070,\ 0.065,\ 0.060,\ 0.055,\ 0.050,\ 0.045,\ 0.044,\ 0.043 \]

\[ 80^\circ,\ 70^\circ,\ 60^\circ,\ 50^\circ,\ 40^\circ,\ 30^\circ,\ 20^\circ,\ 10^\circ,\ 0^\circ \]

\[ 50000,\ 10000,\ 2000,\ 1000,\ 500,\ 400,\ 300,\ 200,\ 100 \]

Other visible symbols:

\[ \varphi,\quad S,\quad \gamma,\quad y \]