Graph-Analytical Method for Precomputing the Accuracy of Triangulation Sides

M. V. POSTNIKOV

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

To evaluate the merit of the geometric construction of triangulation chains, as well as of triangulation developed by the method of geodetic intersections, during design and reconnaissance the reciprocal weight of each figure, \(\frac{1}{P_F}\), is determined, and then these quantities are summed over the entire triangulation chain. The resulting value characterizes the accuracy of determining the terminal side of the chain. Such precomputation of the merit of triangulation with respect to the accuracy of transferring side lengths is, to a certain degree, approximate in character. As is known, the reciprocal weight \(\frac{1}{P_F}\) of any element of triangulation (direction angle, side, etc.) can be computed rigorously from the expression:

\[ \frac{1}{P_F} = [f . f . n]. \]

However, the application of this formula involves considerable computations, and in practice, when precomputing the accuracy of triangulation, approximate formulas are usually used. In choosing formulas for this purpose, great importance is attached to their simplicity and convenience of use.

For precomputing the accuracy of designed triangulation determined by the method of intersections, the quantity \(Q\) is computed according to the formula [1]:

\[ Q = 2\delta^2_{A+B} + \delta^2_A - 2\delta_{A+B}\cdot \delta_A, \tag{1} \]

where \(Q\) is the quantity reciprocal to the weight of the triangle;
\(\delta_{A+B}, \delta_A\) are the increments of the logarithms of the sines of the angles of the triangles when these angles are changed by one second, in units of the 6th decimal place of logarithms.

To determine the value of \(Q\), a table was compiled at TsNIIGAiK under the direction of Prof. A. I. Durnev, published in the “Collection of Articles of GUGK, 1944.”

The procedure for determining the value of \(Q\) from this table is as follows:

1. On the basis of the network scheme compiled as a result of design (and/or reconnaissance), the angles \(A\) and \(B\) are measured with a protractor to an accuracy of one degree (Fig. 1).

2. Using the arguments \(A\) and \(B\), for all triangles of each transfer, the values of \(Q\) are obtained from the table by double interpolation. (The tables are compiled with two entries—the angles \(A\) and \(B\)).

For the correct determination of the value \(Q\), the following rule should be adopted for denoting the angles \(A\) and \(B\).

Let the direction in which the length of the line is transferred be indicated by an arrow (Fig. 1). Then, in triangle No. 1, the rear angle at the side of the traverse line \(S\) should be denoted by \(A_1\), and the forward angle by \(B_1\); in triangle No. 2 the rear angle corresponds to \(B_2\), the forward one to \(A_2\), etc.

Fig. 1.

In the horizontal row of the table the value of angle \(B\) is found; in the vertical column the value of angle \(A\) is taken; at the intersection of the lines the value \(Q\) is read. Then, as recommended by A. I. Durnev [1], for each transfer the arithmetic mean values \(Q_{cp}'\) and \(Q_{cp}''\) are computed from the two triangles. At the same time the weight \(P\) for each transfer and \(Q_m\) for both transfers are computed. The weight of each transfer \(P\) and the value \(Q_m\) are computed by the formulas:

\[ Q_{cp}'=\frac{Q_1' + Q_2'}{2}; \qquad Q_{cp}''=\frac{Q_1'' + Q_2''}{2}, \tag{2} \]

\[ Q_m=\frac{Q_{cp}' + Q_{cp}''}{2}, \tag{3} \]

\[ P_1'=\frac{100}{Q_{cp}'}; \qquad P_2''=\frac{100}{Q_{cp}''}. \tag{4} \]

The computations of the value \(Q\) and of the weight of the transfer \(P\) are carried out in a statement of the following form (Table 1):

The indicated rules for computing \(Q_m\) and \(P_m\) are valid only when the angles \(A\) and \(B\) are equal. If the angles \(A\) and \(B\) differ sharply from one another, then the computation of the value \(Q\) should be carried out differently.

A double transfer increases the accuracy (weight) of the result; therefore the weight of the balanced value of the side will be equal to the sum of the weights of both transfers, i.e.

\[ P=P_1+P_2 \]

or

\[ P=\frac{1}{Q'}+\frac{1}{Q''}=\frac{Q'+Q''}{Q'\cdot Q''}; \]

then

\[ Q_m=\frac{Q'\cdot Q''}{Q'+Q''}, \]

where

\[ Q'=Q_1'+Q_2'; \qquad Q''=Q_1''+Q_2''. \]

Table 1.

Computation of the reciprocal weight of a triangulation transmission

Formulas:

\[ Q=2\delta_{A+B}^{2}+\delta_A^{2}-2\delta_{A+B}\cdot\delta_A . \]

\[ Q_{cp}+\frac{Q_1+Q_2}{2} \qquad Q_m=\frac{Q'_{cp}+Q''_{cp}}{2} \]

\[ P=\frac{100}{Q_{cp}} \]

Hence it is easy to prove that

\[ \frac{Q'\cdot Q''}{Q'+Q''}<\frac{Q'+Q''}{2}. \]

Let us note that such a method of finding the reciprocal weight of a transmission is not entirely rigorous, since

\[ \frac{1}{P_F}=[f.f.n]<\frac{Q'\cdot Q''}{Q'+Q''}. \]

However, this rule for determining the reciprocal weight will give results closer to the values computed by the rigorous formula.

Below is the procedure for computing \(Q_m\) and \(P\).

1. From the values of \(Q\) determined for each triangle of the transmission, the quantities \(Q'\) and \(Q''\) are found:

\[ Q'=Q'_1+Q'_2;\qquad Q''=Q''_1+Q''_2. \tag{5} \]

1. From the found \(Q'\) and \(Q''\), the weight of the unequated value of side \(S_2\) is computed:

\[ P'=\frac{100}{Q'};\qquad P''=\frac{100}{Q''}. \tag{6} \]

1. The weight of the adjusted value of the side, \(P_m\), and \(Q_m\) are computed:

\[ P_m=P'+P''. \tag{7} \]

\[ Q_m=\frac{1}{P_m}=\frac{Q'\cdot Q''}{Q'+Q''}. \tag{8} \]

The computation of the value \(Q\) and of the weight \(P\) is carried out in a schedule of the following form:

Table 2

Computation of the reciprocal weight of a triangulation side

Formulas:

\[ Q' = Q'_1 + Q'_2 \]

\[ P' = \frac{100}{Q'} \]

\[ Q'' = Q''_1 + Q''_2 \]

\[ P'' = \frac{100}{Q''} \]

\[ P_m = P' + P'' \]

\[ Q_m = \frac{1}{P_m} = \frac{Q' \cdot Q''}{Q' + Q''} \]

To accelerate and simplify the obtaining of the quantities \(Q\), one may measure in the triangulation triangles not the angles \(A\) and \(B\), but the lengths of the projections \(a\) and \(b\) adjacent to these angles (Fig. 2), for a definite value of the height of the triangle (for example: \(h=40\) mm), and then determine the quantities \(Q\) from a special table using the arguments \(a\) and \(b\).

Fig. 2.

The theoretical basis of this method is as follows: in triangle \(ABC\) (Fig. 2), drop the perpendicular \(h=CD\) from vertex \(C\) to side \(AB\), which divides side \(AB\) into the segments \(AD=a\) and \(DB=b\).

Obviously, the ratio \(\dfrac{a}{h}\) will express the cotangent of angle \(A\), and the ratio \(\dfrac{b}{h}\)—the cotangent of angle \(B\).

Draw the line \(A'B'\) parallel to the side \(AB\) at a distance from the vertex of the triangle \(C\) equal to the prescribed value \(CD'=h'\); then:

\[ \frac{a'}{h'}=\frac{a}{h}=\operatorname{ctg} A . \tag{9} \]

\[ \frac{b'}{h'}=\frac{b}{h}=\operatorname{ctg} B . \tag{10} \]

Formula (1) may be represented in another form, passing from \(\delta\) to the cotangents of the angles:

\[ Q=\left[\operatorname{ctg}^{2} A+2\operatorname{ctg}^{2}(A+B)-2\operatorname{ctg}(A+B)\cdot \operatorname{ctg} A\right]\left(\frac{10^{6}\psi}{\rho}\right)^{2}. \tag{11} \]

Expressing the cotangents of the angles through the elements of the triangle \(a\), \(b\), and \(h\), formula (11) will be written as follows:

\[ Q=\left\{ \left(\frac{a}{h}\right)^{2} +2\left[ \frac{\frac{a}{h}\cdot\frac{b}{h}-1}{\frac{a}{h}+\frac{b}{h}} \right]^{2} -2\left(\frac{a}{h}\right) \left[ \frac{\frac{a}{h}\cdot\frac{b}{h}-1}{\frac{a}{h}+\frac{b}{h}} \right] \right\} \left(\frac{10^{6}\psi}{\rho}\right)^{2} \tag{12} \]

or, after the corresponding transformations:

\[ Q= \left\{ \frac{(ab-h^{2})^{2}+(a^{2}+h^{2})^{2}}{h^{2}(a+b)^{2}} \right\} \left(\frac{10^{6}\psi}{\rho}\right)^{2}. \tag{13} \]

For measuring the segments \(a\) and \(b\), with the value \(h\) constant, one can construct a grid on a transparent base (Fig. 3).

Grid

for determining the segments
\(a\) and \(b\)

Fig. 3.

On the line \(CD\), equal segments of 20 mm are laid off, and through their ends straight lines \(A'B'\), \(A''B''\), and \(A'''B'''\) are drawn perpendicular to \(CD\), on which scales with one-millimeter divisions are applied. For the correct setting of any scale with millimeter divisions parallel to the base of the triangle, auxiliary dashed lines are drawn between the straight lines \(A'B'\), \(A''B''\), and \(A'''B'''\) and parallel to them.

The procedure for determining the value \(Q\) by means of linear segments will be as follows:

1. A transparent grid is placed on the diagram of the constructed triangulation network; point \(S\) is made to coincide with vertex \(C\) of triangle \(ABC\) (Fig. 1), and one of the scales is set parallel to the base \(AB\) of the triangle.

When measuring the linear segments \(a\) and \(b\), the following rule should be adopted: segment \(a\) will always be the outer segment of each bundle of intersections, and segment \(b\) the one adjacent to the middle (connecting) side.

The segments \(a\) and \(b\) are measured to an accuracy of one millimeter.

1. From the table, using the arguments \(a\) and \(b\), the value \(Q\) is selected.

Sometimes one of the connecting angles in a triangle may be obtuse, i.e., exceed \(90^\circ\) (Fig. 4).

In this case the segments \(a\) and \(b\), which determine the values of angles \(A\) and \(B\), are measured in the same direction from the vertical line \(CD\), and the segment determining the obtuse angle will be negative. Thus, for case (a) in Fig. 4 the line \(D'A' = a\), \(D'B' = -b\); for case (b), \(D'A' = -b\) and \(D'A'' = a\).

Fig. 4.

The table for determining the values \(Q\) from linear segments must have two entries. In the horizontal row of the argument the segments \(b\) are entered, and in the vertical column—the segments \(a\). At the intersection of the horizontal and vertical lines the value \(Q\) is obtained. It is recommended that the table of values \(Q\) be compiled for \(h = 40\) mm, which corresponds to the average triangle of a fourth-order triangulation plotted on a drawing at a scale of 1:25000.

In this case, when determining the segments \(a\) and \(b\) on the grid, the middle scale should be used.

When compiling the table of values \(Q\), the lengths of the segments \(a\) and \(b\) must be set at different intervals, taking into account the need to ensure the accuracy of determining the values \(Q\) to \(2\)—\(3\)% (i.e., without interpolation).

A table compiled in this way will practically not require interpolation when determining \(Q\).

With a constant triangle height \(h\), the required accuracy of measuring the segments \(a\) and \(b\) is determined by the rate of change of the value \(Q\). In order to obtain, from the linear segments, the value \(Q\) with an accuracy to the unit of the sixth decimal place of the logarithm, the segments \(a\) and \(b\) at angles \(A\) and \(B\) greater than \(90^\circ\) should be measured with an error not exceeding 1 mm; for angles \(A\) from 40 to \(105^\circ\) and angles \(B\) from 20 to \(40^\circ\), the value \(Q\) changes very slowly, and here the segments \(a\) and \(b\) may be measured with an error up to 3–4 mm.

The table for computing the value \(Q\) from linear segments is given in Appendix No. 1. To compile it, a table of values of \(Q\) prepared at TsNIIGAiK was used, but instead of the entries \(A^\circ\) and \(B^\circ\), new entries \(a\) mm and \(b\) mm were computed for \(h = 40\) mm.

This table does not have corresponding integral intervals for the values \(a\) and \(b\) and cannot fully reveal the advantage of the new method of determining \(Q\).

If the triangle is very small, the segments should be measured on the upper scale (at \(h = 20\) mm), but to obtain the value \(Q\) the measured segments must be doubled. If the triangle is of considerable size, the segments are measured on the lower scale (at \(h = 60\) mm).

To obtain the value \(Q\) in this case, the measured segments \(a\) and \(b\) must be multiplied by \(2/3\).

Example 1.

Given: \(a = 63\) mm; \(b = 55\) mm, find \(Q\);
scale: medium
answer: \(Q = 7\).

To avoid computations connected with converting readings made on the upper or lower scales to the medium scale, for which the table of values of \(Q\) has been compiled, one may make a template with radial rays (Fig. 5).

Template for determining segments \(a\) and \(b\)

Fig. 5.

Figure 5 shows a template with a bundle of lines diverging from the vertex of the template. The inscriptions 10; 20; 30… 80 at the ends of the lines denote the segments \(a\) and \(b\) in mm, reduced to the medium scale, for which the table has been compiled. When using the template indicated in Fig. 5, the segments \(a\) and \(b\) should be measured along the base of the triangle.

The graph-analytical method for precomputing the accuracy of a network may also be applied to triangulation constructed from solid triangles.

In designing and reconnaissance of triangulation rows, an evaluation is usually made of the merit of their geometric construction by computing, for each figure of the row, the reciprocal weight and summing these quantities over the entire row.

To determine the reciprocal weight of a figure, the quantity \(R\) is computed by the formula:

\[ R = (\delta_A^2 + \delta_B^2 + \delta_A \cdot \delta_B), \tag{14} \]

where \(\delta_A\) and \(\delta_B\) are the increments of the logarithms of the sines of the connecting angles when these angles are changed by one second (in units of the sixth decimal place of the logarithms).

Formula (14) can be represented in another form, passing from \(\delta\) to the cotangents of the angles,

\[ R = (\operatorname{ctg}^2 A + \operatorname{ctg}^2 B + \operatorname{ctg} A \cdot \operatorname{ctg} B) \left(\frac{10^6 \mu}{\rho}\right)^2 . \tag{15} \]

Expressing the cotangents of the angles through the elements of the triangle \(a\), \(b\), and \(h\) (Fig. 2), we write formula (15) as follows:

\[ R = \left( \frac{a^2 + b^2 + a \cdot b}{h^2} \right) \left(\frac{10^6 \mu}{\rho}\right)^2 . \tag{16} \]

The segments \(a\) and \(b\) are measured with the template (Figs. 3 and 5) in the same order as was indicated above.

To obtain the value \(R\) from the linear segments, a table may be compiled, analogous to the table of values of \(Q\), for which in the existing tables

instead of the entries \(A^c\) and \(B^p\), two new entries, \(a\) mm and \(b\) mm, must be computed through the corresponding integer intervals (Appendix No. 2). To obtain the value \(R\) from line segments, nomograms can be constructed [3] (Appendices Nos. 3 and 4). If the value of \(R\) is less than 20 units of the sixth decimal place of the logarithm, it is obtained from nomogram 2.

The use of the nomograms is as follows:

1. With the aid of a transparent grid, the segments \(a\) and \(b\) are measured to an accuracy of 1 mm.
2. Along the curve \(ab\) of the nomogram, two points corresponding to the measured segments \(a\) and \(b\) are found, and they are joined by a straight line to its intersection with the scale \(R\), on which the answer is read. If the values of the segments \(a\) and \(b\) are the same, then at the corresponding point of the curve \(ab\) a tangent is drawn to its intersection with the scale.

Example 2.

Given \(a = 110.6\) mm, \(b = 14.3\) mm. Find \(R\).

Solution. The problem is solved with the aid of nomogram No. 1.

Answer:

\(R = 39.0\) units of the sixth decimal place of the logarithm.

Example 3.

Given: \(a = b = 72\) mm. Find \(R\).

We solve the problem by nomogram No. 1. We draw a tangent to the curve \(ab\) at the point of the nomogram 72 mm. At the intersection of the tangent with the scale we find the answer \(R = 43\).

It should be noted that, from the magnitude of \(R\), one may judge the comparative merits of individual triangles.

Neglecting the error of the initial side, for a chain consisting of \(n\) triangles, the root-mean-square error of the logarithm of the connecting side of the \(n\)-th triangle may be expressed as:

\[ m_{\lg a_n} = m'' \sqrt{\frac{2}{3}\sum_{1}^{n}\left(\delta_A^2 + \delta_B^2 + \delta_A \cdot \delta_B\right)} \tag{17} \]

or

\[ m_{\lg a_n}^{2} = m''^{2}\frac{1}{P_{a_n}}, \tag{18} \]

where

\(m''\)—the root-mean-square error of a measured angle;

\[ \frac{1}{P_{a_n}} \]

—the sum of the values

\[ \frac{1}{P}, \]

computed for each triangle by the formula:

\[ \frac{1}{P}=\frac{2}{3}\left(\delta_A^2+\delta_B^2+\delta_A\delta_B\right). \tag{19} \]

Formula (19), as is known, is completely rigorous when adjusting a triangulation chain by angles under the figure conditions.

When the chain consists not of triangles, but of quadrilaterals and central systems, the reciprocal weight of these figures will be computed by the approximate formula:

\[ \frac{1}{P}=\frac{1}{2}\left(\delta_A^2+\delta_B^2+\delta_A\delta_B\right). \tag{20} \]

Formulas (19) and (20), used for computing the reciprocal weight of the last side of a chain, can be combined into one:

\[ \frac{1}{P_{a_n}} = k \sum_{1}^{n}\left(\delta_A^2+\delta_B^2+\delta_A\cdot\delta_B\right), \tag{21} \]

where for triangles \(\kappa = 2/3\); for geodetic quadrilaterals and central systems \(\kappa = \dfrac{1}{2}\).

If \(m''\) is understood as the mean square error of the measured direction, then the values of the coefficients in formula (21) will be different. For triangles the value is \(\kappa = \dfrac{4}{3}\), and for a geodetic quadrilateral and a central system \(\kappa = 1.0\).

Formula (21) may also be used when the link consists of different figures; in this case the value of the reciprocal weight should be computed for each figure and then the sum of these weights formed for all the figures composing the link.

Conclusions

When designing triangulation by rows or by the method of geodetic intersections, the graph-analytical method of determining the values \(Q\) and \(R\) accelerates and simplifies the work process.

Experience shows that determination, by means of a palette, of the linear segments \(a\) and \(b\), from which the value \(Q\) or \(R\) is found, is carried out approximately twice as fast as measuring the angles \(A^\circ\) and \(B^\circ\) with a protractor.

Use of the table of values of \(R\), compiled by the U.S. Coast and Geodetic Survey for determining \(R\), requires interpolation in both arguments \(A^\circ\) and \(B^\circ\). A table compiled instead by linear segments with the intervals indicated above eliminates this inconvenience.

The nomograms proposed by the author for determining the values of \(R\) from the segments \(a\) and \(b\) likewise solve the problem of obtaining \(R\) practically without interpolation.

Literature

1. Durnev A. I. New Systems for Constructing Geodetic Networks. Geodezizdat, 1952.
2. Krasovskii F. N. and Danilov V. V. Manual of Higher Geodesy. Part 1, issue 1, Geodezizdat, 1939.
3. Modrinskii N. I. Nomograms for Geodetic Computations. ONTI, 1937.
4. Magnitskii V. A. On the question of evaluating the merit of the geometric construction of triangulation. Collection of Articles of GUGK, issue XVIII. Geodezizdat, 1948.
5. Pavlov F. F. Precomputation of the Accuracy of Intersections in Mine-Surveying Triangulations. Ugletekhizdat, 1951.

Appendix 1

TABLE OF VALUES OF \(Q\)

(in units of the 6th decimal place of the logarithm)

\[ Q = 2\delta^2_{A+B}+\delta_A^2-2\delta_{A+B}\delta_A \]

Compiled for two arguments:
1. Values of the angles \(A\) and \(B\)
2. Lengths of the segments \(a\) and \(b\) in mm for \(h=40\) mm

Appendix 2

TABLE OF VALUES OF THE QUANTITY \(R\)

(in units of the sixth decimal place of the logarithm)

\[ R=\delta_A^2+\delta_B^2+\delta_A\delta_B \]

Compiled for two arguments:
1. Lengths of the segments “\(a\)” and “\(b\)” in mm, with \(h=40\) mm
2. Values of the connecting angles \(A^\circ\) and \(B^\circ\)

Appendix 3

Sheet 1

Nomogram for computing the value \(R\) from the line segments \(a\) and \(b\)

\(R\)

\(\alpha\) — in units of the sixth decimal place of logarithms

\(a\) and \(b\)

millimeters

Scheme of use

\(R\)

\(a\) and \(b\)

given

given

answer

Example

Given:
\(a = 110.5\) mm,
\(b = 14.9\) mm

Obtained:
\(R = 39\) units of the sixth decimal place of logarithms.

Appendix 4
Sheet 2

Nomograms for computing the value \(R\) from the linear segments \(a\) and \(b\)

Left scale: \(R\)

Label along the left scale:

\(R\) in units of the 6th decimal place of logarithms

Central scale:

\(a; b\), mm

Label along the central scale:

Segments \(a\) and \(b\), in mm

Diagram for use

\(R\)  \(a : b\)

Answer

Given \(a\)

Given \(b\)

Example

Given: \(a = 35\) mm
\(b = 2\) mm

Answer: \(R = 3.3\)