ON A UNIFIED FORMULA FOR SOLVING CONNECTING TRIANGLES

A. I. VOLKOV, P. A. MARCHENKO

(Presented by the scientific seminar of the Department of Mine Surveying and Geodesy)

Up to the present time, in mine-surveying literature (textbooks, handbooks, instructions) the question of choosing a formula for solving a connecting triangle has been considered depending on its shape. Such a choice was quite necessary, since each of the recommended formulas has an advantage when solving only triangles of a definite shape. Thus, the technical instruction for carrying out mine-surveying work recommends applying the sine formula for solving triangles in which \(0^\circ < A < 20^\circ\) and \(B > 160^\circ\); in the remaining cases it is proposed to use the formula of sides.

In 1952, when considering the question of connection in mine orientation, engineer A. P. Belokrys, in his article [1], proposed a new formula for solving connecting triangles. A feature of this formula is that it does not contain the side \(c\). This is a very important circumstance, since the side \(c\) (the distance between plumb lines) is measured with less accuracy than the other sides of the triangle.

In 1956 a paper [2] by engineer P. G. Sheverdin was published, in which the author proposes using for solving connecting triangles the same formula as engineer A. P. Belokrys. In addition, this work gives an analysis of the proposed formula and indicates the field of its application in comparison with other formulas.

The tangent formula proposed by the above-mentioned authors has the following form:

\[ \tg A = \frac{a \cdot \sin C}{b - a \cos C} \quad \text{and} \quad \tg B = \frac{b \sin C}{a - b \cos C}. \tag{1} \]

The error of the computed angle \(A\) in this case is equal to:

\[ M_A = \pm \frac{1}{C} \sqrt{(\sin^2 A + \sin^2 B)m_S^2 \rho^2 + a^2 M_C^2 \cos^2 B}, \tag{2} \]

where \(M_C\) is the error of measuring angle \(C\);
\(m_S\) is the error of measuring the sides \(a\) and \(b\).

Analyzing expression [2] and analogous expressions when computing angles by the side and sine formulas, the author of [2] quite rightly comes to the conclusion that the tangent formula, for any shape of the triangle, gives better (in some cases identical) results than the side formula. For an elongated form of the connecting triangle (for values of \(A\) from \(0^\circ\) to \(3\text{–}5^\circ\)) the tangent formula is equivalent to the sine formula; in the remaining cases, however, the sine formula, within the limits recommended by the instruction, in the author’s opinion gives somewhat better results. On the basis of the indicated analysis the author recommends that, when solving triangles of any shape, only the tangent formula be used. One cannot but agree with the author’s conclusion regarding the advantage of the tangent formula in comparison with the side formula; this advantage is indisputable. A certain advantage of the sine formula over the tangent formula, however, is open to doubt.

Engineer A. P. Belokrys correctly notes in [1] that the sine formula includes one of the most roughly measured elements of the connecting triangle—the distance between plumb lines, i.e., side \(c\). In the overwhelming majority of cases this distance is measured between freely swinging plumb lines, whereas the other sides of the triangle are measured between a fixed point and one swinging plumb line. It follows from this that side \(c\) is always measured with lower accuracy than the remaining sides of the triangle.

The question of the accuracy of measuring the sides of the connecting triangle under the above conditions requires special investigations. However, proceeding from the method and conditions of measurement of the indicated quantities, it may be assumed that the error of side \(c\), under the condition of asynchronous swinging of the plumb lines, will be expressed by the formula:

\[ m_c = \pm \sqrt{2m_o^2 + 2m_k^2}, \]

where \(m_o\) is the error due to readings on the tape;

\(m_k\) is the error due to the swinging of the plumb lines,

or, taking \(m_k = m_o\), by the formula

\[ m_c = 2m_o. \]

When measuring sides \(a\) and \(b\), however, it is possible by a method known in practice to eliminate the error due to oscillation of the plumb lines, and then the error of measurement of these sides will be equal to

\[ m_{a,b} = \pm m_o\sqrt{2}. \]

Hence it follows that side \(c\) is measured 1.4 times more roughly than sides \(a\) and \(b\).

In order to reveal the influence of unequal-accuracy measurement of the sides of the connecting triangle on the accuracy of computing the angle, Table 1 is given below, in which the absolute values of the errors of angle \(A\) are given, found for the following initial data: \(a = 6\) m; \(c = 3\) m; \(M_C = \pm 7''\); \(m_a = m_b = m_c = 0.0004\) m; with \(m_c = 1.4\,m_{a,b} = \pm 0.00056\) m.

It follows from Table 1 that, indeed, with equally accurate measurements of the sides of the connecting triangle, i.e., with \(m_a = m_b = m_c\), the sine formula, within the limits of its application recommended by the instruction, has a certain advantage in comparison with the tangent formula. Taking into account the fact that side \(c\), which enters into the sine formula, is measured less accurately than sides \(a\) and \(b\), this advantage, as follows from Table 1 (columns 5 and 6), is completely lost.

Table 1

Thus, the tangent formula is in fact universal, since the scope of its application is not limited to any particular form of the connecting triangle, and it can successfully replace both the side formula and the sine formula.

The merit of this formula, as is seen from Table 1, is that the errors of the angles computed by this formula never tend to infinity, but have a definite limit. Taking into account the measurement accuracy specified in the instructions when connecting to the alignment by means of a connecting triangle, the limiting error of the angles computed by the tangent formula is equal to:

\[ M_{\text{lim.}}=\pm \frac{m_s \rho}{c}\sqrt{2}. \tag{3} \]

For the example given in Table 1, the limiting error of angle \(A\) is \(38''.8\).

Using formula (3) and taking into account the requirements of the technical instructions concerning the accuracy of measurements when connecting by means of a connecting triangle, one can, in advance, for various distances between the plumb lines indicate the limits of variation of the error of the computed angle in de-

depending on the shape of the triangle. Such limits for the value \(m_s = 0.0005\) m are given in Table 2.

On the basis of the data in Table 2, one may conclude that, when the distance between plumb lines is 5 or more meters, any form of connecting triangle provides the necessary accuracy of the connecting survey; therefore, as Eng. P. G. Sheverdin correctly notes, the choice of the instrument station point in this case is of secondary importance.

Table 2

In conclusion, it is necessary once again to note the advantage of the tangent formula in comparison with the side and sine formulas currently recommended, and to consider its prompt introduction into practice necessary.

For unfavorable shapes of connecting triangles, the use of this formula should be mandatory; this will lead to a sharp increase in the accuracy of connecting surveys.

REFERENCES

1. Belokrys A. P. On the theory of connection in mine orientation. Studies on Questions of Mine Surveying, collection XXV, 1952.

2. Sheverdin P. G. A new formula for solving connecting triangles. Mine Surveying (collection of scientific papers of the Department of Mine Surveying and Geodesy of the Donetsk Industrial Institute, issue IV, 1956).

3. Technical Instructions for the Performance of Mine-Surveying Work, 1959.