Mixed Method for Solving the Direct Plane Resection

B. F. Krutoy

§ 1. Formulation of the Problem and Possible Methods of Its Solution

In geodesy, a direct plane resection is the problem of finding the observed position \(1\) of a point to be determined on a plane by means of two observed adjoining rays \(L_{2\cdot 1}\) and \(L_{3\cdot 1}\), intersecting at point \(1\) and drawn to this point from two fixed adjoining points \(\overline{\Pi}\ni 2, 3\)^[The symbol \(\ni\) indicates that the set \(\overline{\Pi}\) of adjoining points breaks up into the separate points \(2, 3\).] of the plane (Figs. 1.1 and 1.2). The construction of the rays \(L_{2\cdot 1}\), \(L_{3\cdot 1}\) is carried out with the aid of the observed angles \(\gamma_2, \gamma_3\). In the general case (Fig. 1.1) these angles are laid off at the adjoining points \(2, 3\), clockwise, from two fixed directing rays \(L_{2\cdot 4}\) and \(L_{3\cdot 5}\), drawn from the fixed adjoining points \(2, 3\) to the neighboring fixed directing points \(\overline{H}\ni 4, 5\). In the special case (Fig. 1.2), each of the adjoining points \(\overline{\Pi}\) may at the same time be a directing point \(\overline{H}\) for the adjoining point neighboring it. On grounds that will become clear from the subsequent exposition, the plane on which the points \(1, 2, 3, 4, 5\) are situated will henceforth be called the principal plane.

Fig. 1.1

Solving the direct plane resection, i.e., finding the observed position of the point to be determined \(1\) and the adjoining segments \(\overline{\Pi 1}\) on the principal—

Fig. 1.2

plane, can be carried out in two ways: 1) by a graphical method—quickly, but approximately; 2) by a computational method—more complicated, but accurate. However, the computational solution of the direct resection is rather cumbersome, especially if the length of the side \(\overline{2.3}\) is not given to us, and only the coordinates of the points \(\overline{2}, \overline{3}\) are known. Therefore we have developed a third possible method for solving the direct resection—a mixed method, which is a combination of the first two methods and gives results with a prescribed degree of accuracy, while requiring less labor than the second of the indicated methods. The solution by the mixed method is carried out by successive approximations.

§ 2. Mixed method for solving the direct plane resection

First approximation

Using the given coordinates \(\bar{x}_{T}, \bar{y}_{T}\) of the fixed points \(\bar{T} > \bar{\Pi}, \bar{H}^{1)}\) \((m > n,h;\ n \ni 2,3;\ h \ni 4,5)\) of the principal plane and the observed approximate angles \(\gamma_n\), we shall draw on graph paper an accurate working drawing of the resection with a relative reduction \(1:Y^{[0]}\) (Fig. 2.1). We shall choose the reduction \(Y^{[0]}\) so that the lengths of the sides of the resection fall within the range \(50\text{–}150\ \mathrm{mm}\).

Let \(\bar{m}^{[0]}, 1^{[0]}\) denote the exact positions on the working drawing of the images of the fixed \(\bar{T}\) and of the sought point \(1\) of the principal plane, with
\(\bar{m}^{[0]} > \bar{n}^{[0]}, \bar{h}^{[0]}\). By \(\bar{m}^{[0]}, 1^{[0]}\) we shall denote the approximate positions of the points \(\bar{m}^{[0]}, 1^{[0]}\) obtained on the working drawing by construction, which we shall call the constructed images of these points. Let, further, on the principal plane the points \(\bar{m}^{[0]}, 1^{[0]}\) of the working drawing correspond to the points \(\bar{T}^{(0)}, 1^{(0)}\)—the positions of the determined point \(1\) constructed in the first approximation

Fig. 2.1

\(^{1)}\) The sign \(>\) indicates that the total set \(\bar{T}\) of fixed points of the direct resection is decomposed into two partial sets \(\bar{\Pi}, \bar{H}\) of initial and directing fixed points.

and fixed \( \bar T \) points of the resection. Then the error \( \nabla 1^{[0]} \) in the position of point \(1^{[0]}\) on the working drawing will correspond to the error \( \nabla 1^{(0)} \) in the position of point \(1^{(0)}\) on the principal plane, where \( \nabla 1^{[0]} \) is the length of the segment \(\dot 1^{[0]} 1^{[0]}\), and \( \nabla 1^{(0)}\) is the length of the segment \(11^{(0)}\). Obviously,

\[ \nabla 1^{(0)} = y^{[0]}\cdot \nabla 1^{[0]} . \tag{2.1} \]

With careful execution of the working drawing, the error \( \nabla 1^{[0]} \) usually does not exceed \(1\ \mathrm{mm}\).

Let us now measure on the working drawing the lengths \(\delta^{[0]}_{n.1}\) of the adjacent sides \(\bar n^{[0]}1^{[0]}\) constructed in the first approximation. We shall denote the exact values of the lengths of the segments \(\bar n^{[0]}1^{[0]}\) by \(\partial^{[0]}_{n.1}\). Then, between the exact \(\partial^{[0]}_{n.1}\) and the approximate \(\delta^{[0]}_{n.1}\) lengths of the adjacent sides on the working drawing, the following relation will hold:

\[ \delta^{[0]}_{n.1} - \partial^{[0]}_{n.1} = \Delta \delta^{[0]}_{n.1}\quad (n=2,3), \tag{2.2} \]

where \(\Delta \delta^{[0]}_{n.1}\) is the correction to the length \(\partial^{[0]}_{n.1}\), usually not exceeding \(2\ \mathrm{mm}\).

On the principal plane, equality (2.2) corresponds to the equality:

\[ \dot D_{n.1} - D^{(0)}_{n.1} = \Delta D^{(0)}_{n.1}\quad (n=2,3), \tag{2.3} \]

which is obtained from (2.2) by multiplying both of its sides by \(y^{[0]}\); hence, in expanded form, we obtain:

\[ \dot D_{2.1} - D^{(0)}_{2.1} = \Delta D^{(0)}_{2.1} \]

\[ \dot D_{3.1} - D^{(0)}_{3.1} = \Delta D^{(0)}_{3.1} \tag{2.3.1} \]

Consequently, in (2.3):

\[ \dot D_{n.1} = y^{[0]}\cdot \delta^{[0]}_{n.1} = D_{n.1} \]

\[ D^{(0)}_{n.1} = y^{[0]}\cdot \partial^{[0]}_{n.1} \tag{2.4} \]

\[ \Delta D^{(0)}_{n.1} = y^{[0]}\cdot \Delta \delta^{[0]}_{n.1} \]

Obviously, \(\dot D_{n.1}=D_{n.1}\) is the length of the segment \(\bar n1\), \(D^{[0]}_{n.1}\) is the length of the segment \(\bar n^{(0)}1^{(0)}\), and \(\Delta D^{(0)}_{n.1}\) is the correction to the length \(D^{(0)}_{n.1}\), equal to the length of the segment \(1.1^{(0)}\).

To find the corrections \(\Delta D^{(0)}_{n.1}\), let us first compute the directional (bearing) angles \(t_{n.1}\) of the adjacent rays \(L_{n.1}\).

For the positive direction of measuring the angles \(\gamma_n\) adopted in Figs. 1.1 and 1.2 (clockwise), we shall have:

a) in the direct resection of general form (Fig. 1.1):

\[ t_{2.1} = \bar t_{2.4} + \gamma_2 \]

\[ t_{1.3} = \bar t_{5.3} + \gamma_3 \tag{2.5.1} \]

b) in the direct resection of a special form (Fig. 1.1)

\[ t_{2.1}=\bar t_{2.3}+r_2 \]

\[ t_{1.3}=\bar t_{2.3}+r_3 \tag{2.5.2} \]

We next multiply the first of equations (2.3.1) by \(\cos t_{1.3}\), and the second by \(\cos t_{2.1}\), and add the equalities obtained. After this we multiply the first of equations (2.3.1) by \(\sin t_{2.1}\), and the second by \(\sin t_{1.3}\), and again add the equalities obtained. As a result we arrive at the following two equalities:

\[ D_{2.1}\cos t_{2.1}+D_{3.1}\cos t_{1.3} - D^{(0)}_{2.1}\cos t_{2.1} - D^{(0)}_{3.1}\cos t_{1.3} = \]

\[ =\Delta D^{(0)}_{2.1}\cos t_{2.1} + \Delta D^{(0)}_{3.1}\cos t_{1.3} \tag{C} \]

\[ D_{2.1}\sin t_{2.1}+D_{3.1}\sin t_{1.3} - D^{(0)}_{2.1}\sin t_{2.1} - D^{(0)}_{3.1}\sin t_{1.3} = \]

\[ =\Delta D^{(0)}_{2.1}\sin t_{2.1} + \Delta D^{(0)}_{3.1}\sin t_{1.3} \]

But

\[ D_{2.1}\cos t_{2.1}+D_{3.1}\cos t_{1.3} = \bar x_3-\bar x_2 = \bar \xi_{2.3} \]

\[ D_{2.1}\sin t_{2.1}+D_{3.1}\sin t_{1.3} = \bar y_3-\bar y_2 = \bar \eta_{2.3}, \tag{2.6.1} \]

where \(\bar x_2,\ \bar y_2,\ \bar x_3,\ \bar y_3\) are the given coordinates of the fixed control points \(2,\ 3\). In addition, let us introduce the notation:

\[ D^{(0)}_{2.1}\cos t_{2.1} + D^{(0)}_{3.1}\cos t_{1.3} = \xi^{(0)}_{2.1}+\xi^{(0)}_{1.3} = \xi^{(0)}_{2.3} \]

\[ D^{(0)}_{2.1}\sin t_{2.1} + D^{(0)}_{3.1}\sin t_{1.3} = \eta^{(0)}_{2.1}+\eta^{(0)}_{1.3} = \eta^{(0)}_{2.3}, \tag{2.6.2} \]

and also the notation:

\[ \Delta D^{(0)}_{2.1}\cos t_{2.1} + \Delta D^{(0)}_{3.1}\cos t_{1.3} = \Delta \xi^{(0)}_{2.1}+\Delta \xi^{(0)}_{1.3} = \Delta \xi^{(0)}_{2.3} \]

\[ \Delta D^{(0)}_{2.1}\sin t_{2.1} + \Delta D^{(0)}_{3.1}\sin t_{1.3} = \Delta \eta^{(0)}_{2.1}+\Delta \eta^{(0)}_{1.3} = \Delta \eta^{(0)}_{2.3} \tag{2.6.3} \]

Then the original equalities \((C)\), in the adopted notation, can be written more concisely as follows:

\[ (\bar x_3-\bar x_2)-\xi^{(0)}_{2.3} = \Delta \xi^{(0)}_{2.3} \]

\[ (\bar y_3-\bar y_2)-\eta^{(0)}_{2.3} = \Delta \eta^{(0)}_{2.3} \tag{2.7} \]

Let us examine the geometric meaning of equalities (2.7). For this purpose, on the survey plane, along the rays \(L_{2.1},\ L_{3.1}\) from the control points \(2,\ 3\) in the direction of the point \(1\) to be determined, lay off the segments \(\overline{2\,1^{(0)}_2},\ \overline{3\,1^{(0)}_3}\), whose lengths are respectively \(D^{(0)}_{2.1},\ D^{(0)}_{3.1}\) (Fig. 2.2). Then Fig. 2.2 and formulas (2.6.3) show that the segments \(1^{(0)}_2 C^{(0)}_1\) and \(C^{(0)}_1 1^{(0)}_3\), parallel to the coordinate axes, are the sums \(\Delta \xi^{(0)}_{2.3}, \Delta \eta^{(0)}_{3.2}\) of the projections of the corrections

\(\Delta D_{2.1}^{(0)}, \Delta D_{3.1}^{(0)}\), onto the same axes. In doing so it is important to note that the computation of the sums \(\Delta \xi_{2.3}^{(0)}, \Delta \eta_{2.3}^{(0)}\) by formulas (2.7) does not require preliminary knowledge of the required corrections \(\Delta D_{2.1}^{(0)}, \Delta D_{3.1}^{(0)}\) of the adjoining sides. Hence it follows that, after finding the quantities \(\Delta \xi_{2.3}^{(0)}, \Delta \eta_{2.3}^{(0)}\) by formula (2.7), we can use these quantities for the inverse determination of the corrections

Fig. 2.2

\(\Delta D_{2.1}^{(0)}, \Delta D_{3.1}^{(0)}\), applying for this purpose the simple construction that is clear from consideration of Fig. 2.2. Since the lengths of the corrections \(\Delta D_{2.1}^{(0)}, \Delta D_{3.1}^{(0)}\), according to (2.4), may reach a considerable value, \(2Y^{[0]}\) mm (if one takes \(|\Delta d_{\text{p.1}}^{[0]}|_{\max}=2\) mm), the indicated auxiliary construction can be carried out only under the condition of a corresponding reduction of these corrections to dimensions not exceeding, for example, 100 mm. It is easy to see that this will be the case if the reduction \(Y^{[1]}\) of the auxiliary construction is chosen so that the lengths \(|\Delta \xi_{2.3}^{(0)}:Y^{[1]}|\), \(|\Delta \eta_{2.3}^{(0)}:Y^{[1]}|\) fall within the range 60–80 mm.

To determine the possible value of \(Y^{[1]}\) for a given value of \(Y^{(0)}\), let us compute the magnification \(K^{[1]}\) of the auxiliary construction, which we define by the double equality:

\[ K^{[1]}=\frac{Y^{[0]}}{Y^{[1]}} =\frac{\left|\delta d_{\text{p.1}}^{[1]}\right|_{\max}} {\left|\nabla d_{\text{p.1}}^{[0]}\right|_{\max}}, \tag{2.8} \]

where

\[ \left|\nabla d_{\text{p.1}}^{[0]}\right|_{\max} \]

is the greatest absolute value of the error in the lengths \(d_{\text{p.1}}^{[0]}\) of the adjoining sides measured on the working drawing;

\[ \left|\delta d_{\text{p.1}}^{[1]}\right|_{\max} \]

is the corresponding absolute value of the correction \((\Delta d_{\text{p.1}}^{[0]})_{\max}\) increased \(K^{[1]}\) times, \(= -(\nabla d_{\text{p.1}}^{[0]})_{\max}\).

Assuming \((\nabla d^{[0]}_{\mathrm{p.1}})_{\max}=\pm 2\ \mathrm{mm}\), \((\delta d^{[0]}_{\mathrm{p.1}})_{\max}=\pm 100\ \mathrm{mm}\), from (2.8) we obtain:

\[ K^{[1]}=\frac{y^{[0]}}{y^{[1]}}=\frac{100}{2}=50. \tag{2.8.1} \]

In a number of cases \(K^{[1]}\) may be increased to 100.

The auxiliary construction, performed with a reduction \(y^{[1]}\) and serving to find the corrections \(\Delta D^{(0)}_{\mathrm{p.1}}\) in the first approximation, is carried out as follows. Having determined the reduction \(y^{[1]}\) of the auxiliary construction from the condition that the lengths \(\left|\Delta \xi^{(0)}_{2.3}:y^{[1]}\right|\), \(\left|\Delta \eta^{(0)}_{2.3}:y^{[1]}\right|\) should not exceed 60–80 mm, on the tracing paper from some point \(1^{[1]}_{2}\) we lay off the segment

\[ \Delta \xi^{[1]}_{2.3}=\frac{1}{y^{[1]}}\cdot \Delta \xi^{(0)}_{2.3}=1^{[1]}_{2}\,c^{[1]}_{1} \]

in accordance with the rule:

\[ \begin{array}{ll} \text{upward} & \text{from point } 1^{[1]}_{2}, \text{ if } \Delta \xi^{[1]}_{2.3}>0,\\[4pt] \text{downward} & \text{from point } 1^{[1]}_{2}, \text{ if } \Delta \xi^{[1]}_{2.3}<0. \end{array} \tag{2.9} \]

Next, from the endpoint \(c^{[1]}_{1}\) of the segment \(1^{[1]}_{2}c^{[1]}_{1}\), at a right angle to it, the segment

\[ \Delta \eta^{[1]}_{2.3}=\frac{1}{y^{[1]}}\cdot \Delta \eta^{(0)}_{2.3}=c^{[1]}_{1}\,1^{[1]}_{3} \]

is laid off in accordance with the rule:

\[ \begin{array}{ll} \text{to the right} & \text{from point } c^{[1]}_{1}, \text{ if } \Delta \eta^{[1]}_{2.3}>0,\\[4pt] \text{to the left} & \text{from point } c^{[1]}_{1}, \text{ if } \Delta \eta^{[1]}_{2.3}<0. \end{array} \tag{2.10} \]

The method of construction set forth here is explained in Fig. 2.3. Having drawn on the tracing paper, according to the indicated rules, a pair of segments \(1^{[1]}_{2}c^{[1]}_{1}\) and \(c^{[1]}_{1}1^{[1]}_{3}\), we then place this tracing paper on the working drawing so that

Fig. 2.3

the segment \(1^{[1]}_{2}c^{[1]}_{1}\) is parallel to the \(X\)-axis, and the segment \(c^{[1]}_{1}1^{[1]}_{3}\) is parallel to the \(Y\)-axis. Then, by translating the tracing paper over the drawing, we achieve

so that simultaneously the point \(1^{[1]}_{2}\) coincides with the ray
\(\mathcal{L}^{[0]}_{1}=\uparrow \overline{2^{[0]}1^{[0]}}\),¹ and the point
\(1^{[1]}_{3}\)—with the ray
\(\mathcal{L}^{[0]}_{1}=\uparrow \overline{3^{[0]}1^{[0]}}\) (Figs. 2.4 and 2.5). Having now measured the lengths
\(\delta d^{[1]}_{n.1}\) of the segments \(1^{[1]}_{n}1^{[0]}\) \((n \supset 2,3)\), we shall establish their sign on the basis of

Fig. 2.4

the following rule. If the rays
\(\mathcal{L}^{[0]}_{n.1}=\uparrow \overline{n^{[0]}1^{[0]}}\) and
\(\mathcal{L}^{[1]}_{n.1}=\uparrow \overline{1^{[1]}_{n}1^{[0]}}\):

1) are directed in the same direction, then \(\delta d^{[1]}_{n.1}>0\),

\[ \tag{2.11} \]

2) are directed oppositely, then \(\delta d^{[1]}_{n.1}<0\).

Fig. 2.5

Then the values \(\delta D^{[1]}_{n.1}\) in the first approximation of the sought corrections \(\Delta D^{(0)}_{n.1}\) will be found from the formula:

\[ \delta D^{(1)}_{n.1}=y^{[1]}\,\delta d^{[1]}_{n.1} \tag{2.12} \]

Finally, the values \(D^{(1)}_{n.1}\) in the same approximation of the lengths \(D_{n.1}\) of the adopted sides \(\overline{P}1\) are computed from the formula:

\[ D^{(1)}_{n.1}=D^{(0)}_{n.1}+\delta D^{(1)}_{n.1}. \tag{2.13} \]

The approximate values \(D^{(1)}_{\mathrm{p}.1}\) of the lengths \(D_{\mathrm{p}.1}\) found by the method described above, owing to errors in the construction and in the measurement of the segments \(\delta^{[0]}1^{[1]}_{\mathrm{p}}\), will be erroneous by the amounts \(\Delta D^{(1)}_{\mathrm{p}.1}\), where:

\[ \nabla D^{(1)}_{\mathrm{p}.1} =\delta D^{(1)}_{\mathrm{p}.1}-\Delta D^{(0)}_{\mathrm{p}.1} =y^{[1]}\delta d^{[1]}_{\mathrm{p}.1} -y^{[0]}\Delta d^{[0]}_{\mathrm{p}.1} =y^{[1]}\left(\delta d^{[1]}_{\mathrm{p}.1}-K^{[1]}\Delta d^{[0]}_{\mathrm{p}.1}\right) \tag{2.14} \]

It may be assumed that the possible values of the differences \(\left|\delta d^{[1]}_{\mathrm{p}.1}-K^{[1]}\Delta d^{[1]}_{\mathrm{p}.1}\right|\) are unlikely to exceed \(2\) mm. It follows that

\[ \left|\nabla D^{(1)}_{\mathrm{p}.1}\right|_{\max} = \left|\delta D^{(1)}_{\mathrm{p}.1}-\Delta D^{(0)}_{\mathrm{p}.1}\right|_{\max} \leq 2y^{[1]}\ \text{mm} \tag{2.15} \]

But from (2.4), after replacing corrections by errors, we find:

\[ \left|\nabla D^{(0)}_{\mathrm{p}.1}\right|_{\max} = y^{[0]}\cdot \left|\nabla d^{[0]}_{\mathrm{p}.1}\right|_{\max} = 2y^{[0]}\ \text{mm} = \left|\nabla D^{(0)}_{\mathrm{p}.1}\right|_{\max}. \tag{2.16} \]

The ratio of the quantities \(\left|\nabla D^{(0)}_{\mathrm{p}.1}\right|_{\max}\) and \(\left|\nabla D^{(1)}_{\mathrm{p}.1}\right|_{\max}\) characterizes the degree of reduction \(M^{(1)}\) of the residual errors in the lengths of the adjacent sides as a result of carrying out the first approximation. Formulas (2.15) and (2.16) show that the reduction \(M^{(1)}\)

\[ M^{(1)} = \frac{\left|\nabla D^{(0)}_{\mathrm{p}.1}\right|_{\max}} {\left|\nabla D^{(1)}_{\mathrm{p}.1}\right|_{\max}} = \frac{y^{[0]}}{y^{[1]}} = K^{[1]} = 50 \tag{2.17} \]

is equal to the enlargement \(K^{[1]}\) of the auxiliary construction in the first approximation.

With the computation of the values \(D^{(1)}_{\mathrm{p}.1}\) of the adjacent sides, the first approximation ends. However, if it is intended in general to confine oneself to a single first approximation, then next, for each of the adjacent rays \(D_{\mathrm{p}.1}\), the values \(x^{(1)}_{\mathrm{p}.1}, y^{(1)}_{\mathrm{p}.1}\) in the first approximation of the coordinates \(x_1, y_1\) of point 1 are computed. The coordinates \(x^{(1)}_{\mathrm{p}.1}, y^{(1)}_{\mathrm{p}.1}\) are computed by the formulas:

\[ x^{(1)}_{\mathrm{p}.1} = \bar{x}_{\mathrm{p}} + D^{(1)}_{\mathrm{p}.1}\cos t_{\mathrm{p}.1} = \bar{x}_{\mathrm{p}} + \xi^{(1)}_{\mathrm{p}.1}, \]

\[ y^{(1)}_{\mathrm{p}.1} = \bar{y}_{\mathrm{p}} + D^{(1)}_{\mathrm{p}.1}\sin t_{\mathrm{p}.1} = \bar{y}_{\mathrm{p}} + \eta^{(1)}_{\mathrm{p}.1}, \qquad (\mathrm{p}=2,3) \tag{2.18} \]

and, as final values, the coordinates \(x^{(1)}_1, y^{(1)}_1\) determined by the formulas are taken:

\[ x^{(1)}_1 = \frac{1}{2} \left( x^{(1)}_{2.1}+x^{(1)}_{3.1} \right) \]

\[ y^{(1)}_1 = \frac{1}{2} \left( y^{(1)}_{2.1}+y^{(1)}_{3.1} \right). \tag{2.19} \]

§ 3. The second and subsequent approximations

The second and subsequent \(e\)-th approximations are carried out essentially in the same sequence and by the same methods as the first approximation. Only here, the quantities obtained in the preceding approximations are taken as the initial ones. Hence the following general...

the procedure for the mixed solution of the direct resection at the \(e\)-th stage of approximation.

Denoting by \(D_{\mathrm{p}.1}^{(e-1)}\) the length of the side \(\overline{\Pi\,1_{\mathrm{p}}^{(e-1)}}\), obtained from the \((e-1)\)-st approximation, and by \(\Delta D_{\mathrm{p}.1}^{(e-1)}\) the sought correction to the value \(D_{\mathrm{p}.1}^{(e-1)}\), we shall have for the \((e-1)\)-st approximation the following initial equality, similar to equality (2.4) in the first approximation:

\[ D_{\mathrm{p}.1}-D_{\mathrm{p}.1}^{(e-1)}=\Delta D_{\mathrm{p}.1}^{(e-1)} \tag{3.1} \]

But in accordance with (2.17) we may write:

\[ D_{\mathrm{p}.1}^{(e-1)}=D_{\mathrm{p}.1}^{(e-2)}+\delta D_{\mathrm{p}.1}^{(e-1)}, \tag{3.2} \]

where \(\delta D_{\mathrm{p}.1}^{(e-1)}\) is the correction to the length \(D_{\mathrm{p}.1}^{(e-2)}\), found from the \((e-1)\)-st approximation. Therefore (3.1) can be reduced to the form:

\[ D_{\mathrm{p}.1}-D_{\mathrm{p}.1}^{(e-2)}-\delta D_{\mathrm{p}.1}^{(e-1)} =\Delta D_{\mathrm{p}.1}^{(e-1)} \quad (\mathrm{p}=2,3) \tag{3.1.1} \]

or, in more expanded form:

\[ D_{2.1}-D_{2.1}^{(e-2)}-\delta D_{2.1}^{(e-1)} =\Delta D_{2.1}^{(e-1)}, \]

\[ D_{3.1}-D_{3.1}^{(e-2)}-\delta D_{3.1}^{(e-1)} =\Delta D_{3.1}^{(e-1)}. \tag{3.1.2} \]

It is important to note that the sought correction \(\Delta D_{\mathrm{p}.1}^{(e-1)}\), whose value still has to be found from the \(e\)-th approximation, will be one order of smallness higher than the correction \(\delta D_{\mathrm{p}.1}^{(e-1)}\), already found from the \((e-1)\)-st approximation.

We now multiply the first equality (3.1.2) by \(\cos t_{2.1}\), and the second equality (3.1.2) by \(\cos t_{1.3}\), and add the expressions obtained. Then we multiply the first of equalities (3.1.2) by \(\sin t_{2.1}\), and the second by \(\sin t_{1.3}\), and add the expressions obtained. Then, introducing the previous notations (2.6.1), as well as notations close to (2.6.2) and (2.6.3):

\[ D_{2.1}^{(e-2)}\cos t_{2.1}+D_{3.1}^{(e-2)}\cos t_{1.3} =\xi_{2.1}^{(e-2)}+\xi_{1.3}^{(e-2)} =\xi_{2.3}^{(e-2)}, \]

\[ D_{2.1}^{(e-2)}\sin t_{2.1}+D_{3.1}^{(e-2)}\sin t_{1.3} =\eta_{2.1}^{(e-2)}+\eta_{1.3}^{(e-2)} =\eta_{2.3}^{(e-2)}. \tag{3.2.1} \]

\[ \delta D_{2.1}^{(e-1)}\cos t_{2.1}+\delta D_{3.1}^{(e-1)}\cos t_{1.3} =\delta\xi_{2.1}^{(e-1)}+\delta\xi_{1.3}^{(e-1)} =\delta\xi_{2.3}^{(e-1)} \]

\[ \delta D_{2.1}^{(e-1)}\sin t_{2.1}+\delta D_{3.1}^{(e-1)}\sin t_{1.3} =\delta\eta_{2.1}^{(e-1)}+\delta\eta_{1.3}^{(e-1)} =\delta\eta_{2.3}^{(e-1)} \tag{3.2.2} \]

\[ \Delta D_{2.1}^{(e-1)}\cos t_{2.1}+\Delta D_{3.1}^{(e-1)}\cos t_{1.3} =\Delta\xi_{2.1}^{(e-1)}+\Delta\xi_{1.3}^{(e-1)} =\Delta\xi_{2.3}^{(e-1)} \]

\[ \Delta D_{2.1}^{(e-1)}\sin t_{2.1}+\Delta D_{3.1}^{(e-1)}\sin t_{1.3} =\Delta\eta_{2.1}^{(e-1)}+\Delta\eta_{1.3}^{(e-1)} =\Delta\eta_{2.3}^{(e-1)}. \tag{3.2.3} \]

we obtain:

\[ (\bar{x}_3-\bar{x}_2)-\xi_{2.3}^{(e-2)}-\delta\xi_{2.3}^{(e-1)} =\Delta\xi_{2.3}^{(e-1)} \]

\[ (\bar{x}_3-\bar{x}_2)-\eta_{2.3}^{(e-2)}-\delta\eta_{2.3}^{(e-1)} =\Delta\eta_{2.3}^{(e-1)} \]

But in accordance with (2.7):

\[ (\bar{x}_3-\bar{x}_2)-\xi_{2.3}^{(e-2)}=\Delta \xi_{2.3}^{(e-2)} \]

\[ (\bar{y}_3-\bar{y}_2)-\eta_{2.3}^{(e-2)}=\Delta \eta_{2.3}^{(e-2)} . \tag{3.3} \]

Therefore the preceding equalities may be written in the following final form:

\[ \Delta \xi_{2.3}^{(e-2)}-\delta \xi_{2.3}^{(e-1)} = \Delta \xi_{2.3}^{(e-1)} \]

\[ \Delta \eta_{2.3}^{(e-2)}-\delta \eta_{2.3}^{(e-1)} = \Delta \eta_{2.3}^{(e-1)} . \tag{3.4} \]

Having computed the total corrections \(\Delta \xi_{2.3}^{(e-1)}\), \(\Delta \eta_{2.3}^{(e-1)}\), we proceed to the construction, with their aid, of segments \(1^{[0]}1_{\mathrm{p}}^{[e]}\), whose directed lengths we shall denote by \(\delta d_{\mathrm{p}.1}^{[e]}\). For this purpose, on tracing paper, just as was described above with respect to the segments \(1_2^{[1]}c_1^{[1]}\), \(c_1^{[1]}1_3^{[1]}\), we construct the segments \(1_2^{[e]}c_1^{[e]}\) and \(c_1^{[e]}1_3^{[e]}\), whose directed lengths \(\Delta \xi_{2.3}^{[e-1]}\) and \(\Delta \eta_{2.3}^{[e-1]}\) are found as follows:

\[ \Delta \xi_{2.3}^{[e-1]}= \frac{1}{y^{[e]}}\cdot \Delta \xi_{2.3}^{(e-1)} \]

\[ \Delta \eta_{2.3}^{[e-1]}= \frac{1}{y^{[e]}}\cdot \Delta \eta_{2.3}^{(e-1)} \tag{3.5} \]

Here the reduction \(y^{[e]}\) is chosen in such a way that the lengths of the segments \(1_2^{[e]}c_1^{[e]}\), \(c_1^{[e]}1_3^{[e]}\) do not exceed \(70\text{–}80\) mm. Then, by moving the tracing paper over the drawing, we achieve the simultaneous coincidence of the points \(1_2^{[e]}\), \(1_3^{[e]}\) of the tracing paper with the rays \(l_2^{[0]}\), \(l_3^{[0]}\) of the working drawing, under the condition that the segments \(1_2^{[e]}c_1^{[e]}\), \(c_1^{[e]}1_3^{[e]}\) are parallel to the corresponding coordinate axes. Then the directed lengths of the segments \(1^{[0]}1_{\mathrm{p}}^{[e]}\) will be equal, in magnitude and sign, to the required corrections \(\delta d_{\mathrm{p}.1}^{[e]}\) of the lengths \(d_{\mathrm{p}.1}^{[e-1]}\) of the adjoining sides. The sign of the correction \(\delta d_{\mathrm{p}.1}^{[e]}\) is found by rule (2.11).

Having found the corrections \(\delta d_{\mathrm{p}.1}^{[e]}\) to the lengths \(d_{\mathrm{p}.1}^{[e-1]}\) of the adjoining sides on the drawing with reduction \(y^{[e]}\), we find the corresponding corrections \(\delta D_{\mathrm{p}.1}^{(e)}\) on the base plane by the formula (cf. 2.12):

\[ \delta D_{\mathrm{p}.1}^{(e)}= y^{[e]}\cdot \delta d_{\mathrm{p}.1}^{[e]} \tag{3.6} \]

Finally, the values \(D_{\mathrm{p}.1}^{(e)}\) of the lengths of the adjoining sides in the \(e\)-th approximation are found by the formula:

\[ D_{\mathrm{p}.1}^{(e)}= D_{\mathrm{p}.1}^{(e-1)}+\delta D_{\mathrm{p}.1}^{(e)} \tag{3.7} \]

This completes the \(e\)-th approximation.

The convergence of the \(e\)-th approximation, determined by the value of the magnification \(K^{[e]}\),

\[ K^{[e]}=\frac{y^{[e-1]}}{y^{[e]}} \tag{3.8} \]

when passing from the \((e-1)\)-st approximation to the \(e\)-th approximation, will be of the same order as for the first approximation. This follows from consideration of the following direct formula for computing the magnification \(K^{[e]}\):

\[ K^{[e]}= \frac{\left|\delta D_{\mathrm{p}.1}^{[e]}\right|_{\max}} {\left|\nabla D_{\mathrm{p}.1}^{[e]}\right|_{\max}} \tag{3.8.1} \]

Here: \(\left|\delta D_{\mathrm{p}.1}^{[e]}\right|_{\max}\) is the largest value, magnified in the \(e\)-th approximation by \(K^{[e]}\) times, of the length error \(\nabla D_{\mathrm{p}.1}^{[e]}\), which we admitted in the \((e-1)\)-st approximation when constructing the segment \(1_{\mathrm{p}}^{[0]}1^{[e-1]}\) and measuring its length \(\delta D_{\mathrm{p}.1}^{[e-1]}\) on the working drawing. If we impose the requirement that \(\left|\delta D_{\mathrm{p}.1}^{[e]}\right|_{\max}\) not exceed \(100\ \mathrm{mm}\), and take into account that \(\left|\nabla D_{\mathrm{p}.1}^{[e]}\right|_{\max}\) is unlikely to exceed \(2\ \mathrm{mm}\), then from (3.8.1) it follows that

\[ K^{[e]}=\frac{100\ \mathrm{mm}}{2\ \mathrm{mm}}=50=K^{[1]}. \]

Thus, for the \(e\)-th approximation we have obtained the same value of the magnification as for the first approximation.

The criterion for the end of the solution of the direct resection by the mixed method will be the equality to zero (within the limits of computational accuracy) of the residuals \(\Delta \xi_{2.3}^{(n-1)}\), \(\Delta \eta_{2.3}^{(n-1)}\) in some \(n\)-th approximation.

Then the final values of the lengths \(D_{\mathrm{p}.1}\) and of the coordinate increments \(\xi_{\mathrm{p}.1}\), \(\eta_{\mathrm{p}.1}\) along the joining sides \(\overline{\Pi 1}\) will be found from the formulas:

\[ D_{\mathrm{p}.1}=D_{\mathrm{p}.1}^{(0)}+\sum_{e=1}^{n-1}\delta D_{\mathrm{p}.1}^{(e)} \]

\[ \xi_{\mathrm{p}.1}=\xi_{\mathrm{p}.1}^{(0)}+\sum_{e=1}^{n-1}\delta \xi_{\mathrm{p}.1}^{(e)} \tag{3.9} \]

\[ \eta_{\mathrm{p}.1}=\eta_{\mathrm{p}.1}^{(0)}+\sum_{e=1}^{n-1}\delta \eta_{\mathrm{p}.1}^{(e)} \]

We shall check the obtained values \(D_{\mathrm{p}.1}\), \(\xi_{\mathrm{p}.1}\), \(\eta_{\mathrm{p}.1}\) by the formulas:

\[ \xi_{\mathrm{p}.1}=D_{\mathrm{p}.1}\cos t_{\mathrm{p}.1} \]

\[ \eta_{\mathrm{p}.1}=D_{\mathrm{p}.1}\sin t_{\mathrm{p}.1} \tag{3.10} \]

Finally, the final coordinates \(x_1, y_1\) of the point 1 being determined are computed twice in the following way:

\[ \begin{aligned} x_1 &= \bar{x}_2+\xi_{2.3}=\bar{x}_3+\xi_{3.1},\\ y_1 &= \bar{y}_2+\eta_{2.1}=\bar{y}_3+\eta_{3.1}. \end{aligned} \tag{3.11} \]

This completes the solution of direct resection by the mixed method. Usually, solving the resection for side lengths up to 10 km requires 2 approximations.

Let us collect together the formulas for solving direct resection by the mixed method.

Summary of formulas

\[ \left. \begin{aligned} 1)\quad & t_{2.1}=\bar{t}_{2.4}+r_2,\\ 2)\quad & t_{1.3}=\bar{t}_{5.3}+r_3 \end{aligned} \right\} \qquad \text{— general case (Fig. 1.1)} \]

or

\[ \left. \begin{aligned} 1)\quad & t_{2.1}=\bar{t}_{2.3}+r_2,\\ 2)\quad & t_{1.3}=\bar{t}_{2.3}+r_3 \end{aligned} \right\} \qquad \text{— special case (Fig. 1.2)} \]

First approximation

\[ \begin{aligned} 3)\quad & \xi_{2.1}^{(0)}=D_{2.1}^{(0)}\cos t_{2.1},\\ & \xi_{1.3}^{(0)}=D_{3.1}^{(0)}\cos t_{1.3}; \end{aligned} \]

\[ \begin{aligned} 4)\quad & \eta_{2.1}^{(0)}=D_{2.1}^{(0)}\sin t_{2.1},\\ & \eta_{1.3}^{(0)}=D_{3.1}^{(0)}\sin t_{1.3}; \end{aligned} \]

\[ \begin{aligned} 5)\quad & \xi_{2.3}^{(0)}=\xi_{2.1}^{(0)}+\xi_{1.3}^{(0)},\\ & \eta_{2.3}^{(0)}=\eta_{2.1}^{(0)}+\eta_{1.3}^{(0)}; \end{aligned} \]

\[ \begin{aligned} 6)\quad & \Delta \xi_{2.3}^{(0)}=(\bar{x}_3-\bar{x}_2)-\xi_{2.3}^{(0)},\\ & \Delta \eta_{2.3}^{(0)}=(\bar{y}_3-\bar{y}_2)-\eta_{2.3}^{(0)}. \end{aligned} \]

\(e\)-th approximation

\[ \begin{aligned} 7)\quad & \delta \xi_{2.1}^{(e-1)}=\delta D_{2.1}^{(e-1)}\cos t_{2.1},\\ & \delta \xi_{3.1}^{(e-1)}=\delta D_{3.1}^{(e-1)}\cos t_{1.3}; \end{aligned} \]

\[ \begin{aligned} 8)\quad & \delta \eta_{2.1}^{(e-1)}=\delta D_{2.1}^{(e-1)}\sin t_{2.1},\\ & \delta \eta_{1.3}^{(e-1)}=\delta D_{3.1}^{(e-1)}\sin t_{1.3}; \end{aligned} \]

\[ \begin{aligned} 9)\quad & \delta \xi_{2.3}^{(e-1)}=\delta \xi_{2.1}^{(e-1)}+\delta \xi_{1.3}^{(e-1)},\\ & \delta \eta_{2.3}^{(e-1)}=\delta \eta_{2.1}^{(e-1)}+\delta \eta_{1.3}^{(e-1)}; \end{aligned} \]

\[ \begin{aligned} 10)\quad & \Delta \xi_{2.3}^{(e-1)}=\Delta \xi_{2.3}^{(e-2)}-\delta \xi_{2.3}^{(e-1)},\\ & \Delta \eta_{2.3}^{(e-1)}=\Delta \eta_{2.3}^{(e-2)}-\delta \eta_{2.3}^{(e-1)}. \end{aligned} \]

Completion of the computations

\[ 11)\quad D_{n.1}=D_{n.1}^{(0)}+\sum_{e=1}^{n-1}\delta D_{n.1}^{(e)} \]

\[ 12)\quad \xi_{n.1}=\xi_{n.1}^{(0)}+\sum_{e=1}^{n-1}\delta \xi_{n.1}^{(e)} =D_{n.1}\cos t_{n.1} \]

\[ \eta_{n.1}=\eta_{n.1}^{(0)}+\sum_{e=1}^{n-1}\delta \eta_{n.1}^{(e)} =D_{n.1}\sin t_{n.1} \]

\[ 13)\quad x_1=\bar{x}_2+\xi_{2.1}=\bar{x}_3+\xi_{3.1} \]

\[ y_1=\bar{y}_2+\eta_{2.1}=\bar{y}_3+\eta_{3.1}. \]

§ 4. Example of solving a direct resection by the mixed method

Practical instructions. Conclusion

We give an example for solving a direct resection by the mixed method for the case shown in Fig. 1.2 (Table 1). The working drawing (Fig. 4.1) was made on graph paper with reduction \(y^{[0]} = 100\,000\). Measurements on the drawing in the principal and auxiliary constructions were perfo—

Reduction \(y^{[0]} = 100\,000\)
” \(y^{[1]} = 1000\)
” \(y^{[2]} = 10\)

Fig. 4.1 (reduced by a factor of 2)

Table 1

\(^{1)}\) Taken from the working drawing of the direct resection.

1st approximation

\[ \text{length } \left(1_2^{(1)} C_1^{(1)}\right)=\Delta \xi_{2.3}^{(0)}=+89.90 \]

\[ \text{length } \left(C_1^{(1)} 1_3^{(1)}\right)=\Delta \eta_{2.3}^{(0)}=-25.24 \]

\[ K^{[1]}=100;\qquad y^{[1]}=1000 \]

\[ y^{[1]}\cdot \text{length } \left(1_2^{[1]}1^{[0]}\right) = y^{[1]}\cdot \delta\partial_{2.1}^{[1]} = \delta D_{2.1}^{(1)}=+79.50 \]

\[ y^{[1]}\cdot \text{length } \left(1_3^{[1]}1^{[0]}\right) = y^{[1]}\cdot \delta\partial_{3.1}^{[1]} = \delta D_{3.1}^{(1)}=+90.70 \]

2nd approximation

\[ \text{length } \left(1_2^{(2)} C_1^{(2)}\right)=\Delta \xi_{2.3}^{(1)}=+0.7 \]

\[ \text{length } \left(C_1^{(2)}1_3^{(2)}\right)=\Delta \eta_{2.3}^{(1)}=+0.97 \]

\[ K^{[2]}=100;\qquad y^{[2]}=10 \]

\[ y^{[2]}\cdot \text{length } \left(1_2^{[2]}1^{[0]}\right) = y^{[2]}\cdot \delta\partial_{2.1}^{[2]} = \delta D_{2.1}^{(2)}=+1.13 \]

\[ y^{[2]}\cdot \text{length } \left(1_3^{[2]}1^{[0]}\right) = y^{[2]}\cdot \delta\partial_{3.1}^{[2]} = \delta D_{3.1}^{(2)}=-0.16 \]

3rd approximation

\[ \text{length } \left(1_2^{(3)} C_1^{(3)}\right)=\Delta \xi_{2.3}^{(2)}=-0.01 \]

\[ \text{length } \left(C_1^{(3)}1_3^{(3)}\right)=\Delta \eta_{2.3}^{(2)}=0.00 \]

…agreed to within 1 mm. The problem was solved in 2 approximations. In the 3rd approximation only the values of the quantities \(\Delta \xi_{2.3}^{(2)}\), \(\Delta \eta_{2.3}^{(2)}\) were computed, whose closeness to zero served as an indication of the end of the solution of the resection. The successive constructions on the working drawing, carried out in the course of solving the resection, are shown in Fig. 4.1. The necessary basic and auxiliary computations are arranged in the form of Table 1. Finally, the values of the segments \(\delta\partial_{\mathrm{p}.1}^{[e]}\) measured on the working drawing and the values of the sums \(\Delta \xi_{2.3}^{(e-1)}\), \(\Delta \eta_{2.3}^{(e-1)}\), as well as their geometric meaning and the values of the reductions \(y^{[e]}\), are placed under Table 1.

In conclusion we point out that the solution of the direct resection may be considered complete when, in some \(e\)-th approximation, we arrive at such values of \(\Delta \xi_{\mathrm{p}.1}^{(n-1)}\), \(\Delta \eta_{\mathrm{p}.1}^{(n-1)}\) that are of the same order of smallness as the required accuracy of the final coordinates \(x_1, y_1\) of point 1. Thus, if an accuracy of these coordinates of 1 m were sufficient, it would be possible to confine oneself to a single first approximation.

This, for example, will occur in the processing of networks of angular (trigonometric) leveling, where one may be satisfied with an accuracy of side lengths of 1 m. For computing preliminary values of the coordinates of points in a triangle (triangulation), however, when adjusting it by the method of indirect observations, two approximations will have to be made.

In conclusion we note that an experimental comparison of various methods of complete solution of the direct resection, i.e., with the computation of sides and coordinates, showed a reduction in the volume of work by up to 30% for the mixed method in comparison with the generally accepted ones.

LITERATURE

1. Krasovsky, F. N. Manual of Higher Geodesy, Part I, Issue 2, § 108, 1939.

2. Butler, S. A. Adjustment of Filling Trigonometrical Networks, pp. 63–64, 1936.

1. The sign \(\uparrow\) means that a directed segment is taken. ↩