On solving intersections and Hansen's problem using formulas recommended by D. N. Ogloblin in the course Orientation of Underground Surveying through a Single Vertical Shaft
Abstract Generated abstract
This paper critiques arithmometric formulas recommended by D. N. Ogloblin for solving intersection problems and Hansen’s problem in mine surveying. Volkov analyzes the proposed principal and check formulas, arguing that they verify only arithmetic consistency and not the correctness of direction angles, tangents, or the solution as a whole. Through a numerical example using Ogloblin’s own data, the paper shows that erroneous coordinates can pass the stated checks, creating a risk of production errors. It recommends supplementing the methods with independent checks based on recomputing measured quantities from coordinates, especially by deriving the tangent of the direction angle between computed points from their coordinates.
Full Text
ON THE SOLUTION OF INTERSECTION PROBLEMS AND HANSEN’S PROBLEM USING THE FORMULAS RECOMMENDED BY D. N. OGLOBLIN IN THE COURSE “ORIENTATION OF UNDERGROUND SURVEYING THROUGH A SINGLE VERTICAL SHAFT”1
A. I. Volkov.
The question of arithmometric methods for solving mine-surveying and geodetic problems, in connection with the widespread introduction of calculating machines into computational practice, is acquiring exceptionally urgent significance. The great practical importance of this question also explains the attention paid to it in the pages of periodical literature.
At present, a whole series of methods has already been developed for the arithmometric solution of mine-surveying and geodetic problems. Naturally, for production one should recommend that method of solution which gives the greatest effect with respect to reliability, simplicity, and speed of computation; moreover, the first requirement is decisive in choosing the method. A method that does not provide reliable control of the solution of the problem as a whole has no practical value and, consequently, cannot be recommended for production.
Considering from this point of view the arithmometric methods for solving intersection problems and Hansen’s problem recommended by D. N. Ogloblin in his work Orientation of Underground Surveying through a Single Vertical Shaft, it is easy to arrive at the conclusion that they do not satisfy this basic requirement.
On pages 122–125 of the book Orientation of Underground Surveying through a Single Vertical Shaft, two methods are given for the arithmometric solution of an intersection problem.
In solving by the first method (p. 123), the following formulas are proposed (the form and example are given on p. 121)2:
\[ x_A - x_R = \frac{y_S - y_R - (x_S - x_R)\tg\psi}{\tg\varphi - \tg\psi}, \tag{1} \]
\[ x_A - x_S = \frac{y_S - y_R - (x_S - x_R)\tg\varphi}{\tg\varphi - \tg\psi}, \tag{2} \]
\[ y_A-y_R=(x_A-x_R)\cdot \operatorname{tg}\varphi, \tag{3} \]
\[ y_A-y_S=(x_A-x_S)\cdot \operatorname{tg}\psi \tag{4} \]
and it is said that “formulas (1)–(4) are the working formulas by which all computations are performed.”
No other remarks are made by D. N. Ogloblin concerning the formulas cited above. This indicates that D. N. Ogloblin evidently assumes that, since when formulas (1)–(4) are applied the result is obtained twice, the checking of the solution of the problem is thereby fully ensured.
Analyzing formulas (1)–(4), it is not difficult to conclude that they do not provide an exhaustive check on the correctness of the solution of the problem as a whole. The presence of these formulas provides only a check on the correctness of carrying out the arithmetic operations indicated in the formulas and on the computation of \(y_S-y_R\) and \(x_S-x_R\), without revealing an error in the computation of the direction angles \(\varphi\) and \(\psi\), or of their tangents, since for any values of \(\operatorname{tg}\varphi\) and \(\operatorname{tg}\psi\), except \(\operatorname{tg}\varphi=\operatorname{tg}\psi\), there exists only one point of intersection of the straight lines and, consequently, only one (even if erroneous) solution. This latter circumstance is precisely the weak point of formulas (1)–(4), depriving them of practical value.^1
For solving the intersection by the second method (p. 123), the following formulas are proposed (the form and an example are given on p. 200):
\[ x_A-x_R= \frac{(x_S-x_R)\cdot \operatorname{ctg}l+(y_S-y_R)} {\operatorname{ctg}l+\operatorname{ctg}r}, \tag{5} \]
\[ y_A-y_R= \frac{(y_S-y_R)\cdot \operatorname{ctg}l-(x_S-x_R)} {\operatorname{ctg}l+\operatorname{ctg}r}, \tag{6} \]
\[ x_A-x_S= \frac{-(x_S-x_R)\cdot \operatorname{ctg}r+(y_S-y_R)} {\operatorname{ctg}l+\operatorname{ctg}r}, \tag{7} \]
\[ y_A-y_S= \frac{-(y_S-y_R)\cdot \operatorname{ctg}r+(x_S-x_R)} {\operatorname{ctg}l+\operatorname{ctg}r}. \tag{8} \]
Formulas (5) and (6) are adopted as the principal ones; formulas (7) and (8) are given for checking.
As is evident, these formulas suffer from the same shortcoming as formulas (1)–(4), and therefore, like the latter, cannot be recommended for production use.
From consideration of formulas (1)–(4) and (5)–(8), it is clear that only their joint use can ensure the reliability of the result; i.e., if, for example, formulas (1), (2) and (5), (6) are taken, then the form obtained in this way fully guarantees the correctness of the solution of the problem and, consequently, saves production work from errors.
^1 This shortcoming of formulas 1–4 is also noted by Associate Professor V. S. Nuvar'ev in the 1936 work “Solution of the Pothenot and Hansen Problems on the Plane and in Gauss–Krüger Coordinates.”
[[unclear: beginning of sentence]] can be carried out quickly and reliably by the following formulas1:
\[ \Delta^{A}_{R}x=a.\operatorname{cosec}(e+r).\sin r.\cos\varphi, \]
\[ \Delta^{A}_{R}y=a.\operatorname{cosec}(e+r).\sin r.\sin\varphi, \]
\[ \Delta^{A}_{S}x=a.\operatorname{cosec}(e+r).\sin l.\cos\psi, \]
\[ \Delta^{A}_{S}y=a.\operatorname{cosec}(e+r).\sin l.\sin\psi. \]
For the arithmetical solution of Hansen’s problem, D. N. Ogloblin recommends the following set of formulas (the form and an example are given on p. 200):
\[ \operatorname{tg}(p_1p_2)= \frac{ \Delta^{B}_{A}x\,[\operatorname{tg}\alpha_1.\operatorname{tg}\alpha_2(\operatorname{tg}\beta_2-\operatorname{tg}\beta_1)+\operatorname{tg}\beta_1.\operatorname{tg}\beta_2(\operatorname{tg}\alpha_1-\operatorname{tg}\alpha_2)] +\Delta^{B}_{A}y\,[\operatorname{tg}\alpha_2.\operatorname{tg}\beta_1-\operatorname{tg}\alpha_1.\operatorname{tg}\beta_2] }{ \Delta^{B}_{A}x\,[\operatorname{tg}\alpha_2.\operatorname{tg}\beta_1-\operatorname{tg}\alpha_1.\operatorname{tg}\beta_2] -\Delta^{B}_{A}y\,[\operatorname{tg}\alpha_1.\operatorname{tg}\alpha_2(\operatorname{tg}\beta_2-\operatorname{tg}\beta_1)+\operatorname{tg}\beta_1.\operatorname{tg}\beta_2(\operatorname{tg}\alpha_1-\operatorname{tg}\alpha_2)] } \tag{15} \]
\[ \Delta^{P_1}_{A}x= \frac{ \Delta^{B}_{A}x.\operatorname{tg}[(p_1p_2)+\beta_1]-\Delta^{B}_{A}y }{ \operatorname{tg}[(p_1p_2)+\beta_1]-\operatorname{tg}[(p_1p_2)+\alpha_1] } \tag{13} \]
\[ \Delta^{P_1}_{A}y=\Delta^{P_1}_{A}x.\operatorname{tg}[(p_1p_2)+\alpha_1] \tag{3} \]
\[ \Delta^{P_2}_{A}x= \frac{ \Delta^{B}_{A}x.\operatorname{tg}[(p_1p_2)+\beta_2]-\Delta^{B}_{A}y }{ \operatorname{tg}[(p_1p_2)+\beta_2]-\operatorname{tg}[(p_1p_2)+\alpha_2] } \tag{16} \]
\[ \Delta^{P_2}_{A}x=\Delta^{P_2}_{A}x.\operatorname{tg}[(p_1p_2)+\alpha_2] \tag{17} \]
\[ \Delta^{P_1}_{B}x= \frac{ \Delta^{B}_{A}x.\operatorname{tg}[(p_1p_2)+\alpha_1]-\Delta^{B}_{A}y }{ \operatorname{tg}[(p_1p_2)+\beta_1]-\operatorname{tg}[(p_1p_2)+\alpha_1] } \tag{18} \]
\[ \Delta^{P_1}_{B}y=\Delta^{P_1}_{B}x.\operatorname{tg}[(p_1p_2)+\beta_1] \tag{19} \]
\[ \Delta^{P_2}_{B}x= \frac{ \Delta^{B}_{A}x.\operatorname{tg}[(p_1p_2)+\alpha_2]-\Delta^{B}_{A}y }{ \operatorname{tg}[(p_1p_2)+\beta_2]-\operatorname{tg}[(p_1p_2)+\alpha_2] } \tag{20} \]
\[ \Delta^{P_2}_{B}y=\Delta^{P_2}_{B}x.\operatorname{tg}[(p_1p_2)+\beta_2] \tag{21} \]
Here, formulas (15), (13), (3), (16), (17) are the principal ones, while (18), (19), (20), and (21) are check formulas.
As shortcomings of this method for solving Hansen’s problem, D. N. Ogloblin (p. 211) notes only that it, i.e., the method, “is based on very cumbersome formulas, which, if they are not at hand, are to be restored by the mine—
is difficult for the mine surveyor*. The shortcoming of this method noted by D. N. Ogloblin is insignificant.
A more substantial shortcoming of the method is, undoubtedly, its complete lack of checkability (not noted by D. N. Ogloblin), since the proposed check formulas do not reveal an error (for the reasons set forth when considering the question of solving the intersection problem) when the latter has been introduced into the computation of \((P_1P_2)\), or into any one of the following (or all simultaneously) quantities: \((AP_1)\), \((AP_2)\), \((BP_1)\), \((BP_2)\), or their tangents. Obtaining from the check formulas, in the presence of errors in the above quantities, the same result as from the basic formulas, the computer, having full confidence in this check, will take the erroneous result as true and thereby commit a production error.
For example, let us compute the problem cited by D. N. Ogloblin on p. 200, taking the direction angle \((P_1P_2)\) equal to \(45^\circ 00'00''\) instead of \(82^\circ 25'29''\) (as a result of which all direction angles change, with the exception of \((AB)\)). As a result of solving the problem with these incorrect data, we obtain the following values of the coordinates of points \(p_1\) and \(p_2\):
By the basic formulas:
\[ x_1=-1228{,}004;\quad y_1=1167{,}812;\quad x_2=-1225{,}255;\quad y_2=1166{,}862; \]
by the check formulas:
\[ x_1=-1228{,}004;\quad y_1=1167{,}812;\quad x_2=-1225{,}255;\quad y_2=+1166{,}862, \]
i.e., we obtain complete agreement of the results, although the problem has obviously been solved incorrectly. The true value of the coordinates of points \(p_1\) and \(p_2\) is equal to \(^{1)}\):
\[ x_1=-1222{,}566;\quad y_1=+1164{,}428;\quad x_2=-1220{,}594;\quad \]
\[ -y_2=+1179{,}256. \]
The erroneousness of the solution in this case could have been established by computing \(tg\,(P_1P_2)\) from the obtained coordinates of points \(P_1, P_2\).
\[ tg\,(P_1P_2)=\frac{y_2-y_1}{x_2-x_1}=\frac{-0{,}950}{+2{,}749}. \]
The fact that the obtained value of \(tg\,(P_1P_2)\) is not equal to unity indicates the erroneousness of the computed coordinates of points \(P_1\) and \(P_2\). However, if, as a result of the errors made, points \(P_1\) and \(P_2\), after being displaced, remain on the direction \((P_1P_2)\) computed (correctly or incorrectly) by formula 15, then this check also will not give the desired result.
From all that has been said above it becomes perfectly clear that the apparatus of formulas proposed by D. N. Ogloblin for the arithmetical solution of Hansen’s problem must be supplemented by a special check guaranteeing the correctness of the solution of the problem as a whole. The latter can be accomplished by com—
\(^{1)}\) Taken from Ogloblin’s book, p. 200.
calculation of directly measured quantities through the tangents of the corresponding direction angles, computing in this case the tangent of the direction angle \((P_1P_2)\) necessarily from the coordinates. Thus, the angle \(\beta_1\) can be computed by the formula:
\[ \operatorname{tg}\beta_1 = \frac{\operatorname{tg}(p_1B)-\operatorname{tg}(p_1p_2)} {1+\operatorname{tg}(p_1B)\cdot \operatorname{tg}(p_1p_2)} \]
The remaining angles may be computed analogously. This check will completely eliminate the need for the check formulas given by D. N. Ogloblin, provided that the tangents of all direction angles entering into the computation of the directly measured quantities are determined from the coordinates.