TWO-GROUP ADJUSTMENT OF A GEODETIC PENTAGON

B. F. Krutoi

In adjusting a geodetic pentagon by the method of least squares, the most labor-intensive part is the formation and simultaneous solution of 9 normal equations. This stage of the computational work can be shortened or, at least, simplified if two conditions are met:

1) the simultaneous adjustment of all 9 condition equations must be replaced by separate adjustment by the so-called two-group method (Urmaev. On the successive adjustment of trigonometric networks);

2) the system of condition equations of the figures must be composed in such a way that the resulting system of normal equations admits a simple solution. Hence, as a consequence, it would become possible to give general formulas for the correlates, expressing them through the misclosures.

Having established the basic principles for rationalizing adjustment computations, we shall briefly set out the two-group method of adjustment. Its essence is that all equations are divided into 2 groups, with the first group including equations of the simplest form, preferably with coefficients $\pm 1$. Solving the equations of the first group gives the primary corrections. From the angles corrected by the primary corrections, the condition equations of the 2nd group are formed. In order to make them independent of the equations of the 1st group, the coefficients of the equations of the 2nd group are transformed by formulas analogous to those for the primary corrections. Solving the transformed equations gives secondary corrections. As theory shows, the sum of the primary and secondary corrections is exactly equal to the corrections obtained under simultaneous adjustment. Consequently, the two-group method of adjustment is equivalent, in its final results, to the ordinary Gauss method.

Fig. 1

Proceeding to the derivation of formulas for the geodetic pentagon, let us first note that the angles should be numbered in the order indicated in the drawing, since the formulas have been developed with reference to this numbering.

In accordance with the above, we divide all condition equations into 2 groups, including in the 1st group 6 equations of figures, as the simplest, and in the 2nd group 3 pole equations. To attain the greatest simplicity of formulas, we write the condition equations of the figures in a somewhat peculiar way.

namely: 5 equations are obtained from the equality of the vertical angles at the 5 interior points of intersection of the diagonals; the 6th equation is obtained from the sum of the interior angles of the pentagon. The pole equations, however, are written from 3 geodetic quadrilaterals having the common pole \(A\) and the common side \(AE\): \(\Delta\Delta ABCE\), \(ACDE\), and \(ABDE\) (although the combination formula \(C_4^3=4\) indicates the possibility of obtaining still a 4th quadrilateral—\(ABCD\), we discard it as being a consequence of the first 3). The general form of the conditional equations of the figures is given below; the compilation of the pole equations we regard as generally known, and therefore do not include them. Moreover, the method of compiling them is clear from the attached form. Having compiled the conditional equations of group 1, we find the general formulas of the primary corrections. The further computations are arranged in the form of schemes and corresponding formulas.

Scheme of the conditional equations of group 1

Conditional equations of group 1

a) 5 equations of the form

\[ (3i-2)+(3i-1)-(3i+3)-- \]

\[ --(3i+6)+w_i=0, \tag{1} \]

where

\[ w_i=\varepsilon_{2i-1}-\varepsilon_{2i} \]

(see the form);

b) the 6th equation of the form

\[ \sum_{s=1}^{15}(i_s)+w_6=0, \]

where

\[ w_6=\sum_{r=1}^{11}\delta_r-540^\circ . \tag{2} \]

Correlates of group 1

\[ k_1=\frac{-19w_1-w_2+5w_3+5w_4-w_5}{66} \]

\[ k_2=\frac{-w_1-19w_2-w_3+5w_4+5w_5}{66} \]

\[ k_3=\frac{+5w_1-w_2-19w_3-w_4+5w_5}{66} \]

\[ k_4=\frac{+5w_1+5w_2-w_3-19w_4-w_5}{66} \]

\[ k_5=\frac{+w_1+5w_2+5w_3-w_4-19w_5}{66} \]

\[ k_6=-\frac{w_6}{15} \]

(For the compilation and solution of the normal equations of group 1, see overleaf).

Normal equations of group 1 and their solution

Expressing the correlates through the free terms, for the primary corrections we obtain the following formulas:

\[ (1)=(2)=k_1+k_6 \qquad (4)=(5)=k_2+k_6 \qquad (7)=(8)=k_3+k_6 \]

\[ (3)=-k_3-k_5+k_6 \qquad (6)=-k_1-k_4+k_6 \qquad (9)=-k_2-k_5+k_6 \]

\[ (10)=(11)=k_4+k_6 \qquad (13)=(14)=k_5+k_6 \]

\[ (12)=-k_1-k_5+k_6 \qquad (15)=-k_2-k_4+k_6 . \]

Two-group adjustment of a geodetic pentagon

By B. F. Krutoy

Here: \((i)\) is the correction of the 1st, or auxiliary increments \(\Delta\alpha,\Delta\beta,\Delta\gamma\); \(k_i\) is a correlate of the 1st group of equations or of the auxiliary systems \(\alpha,\beta,\gamma\). Further: \((i=1,2,\ldots,5)\); \((s=1,2,\ldots,15)\); \((r=1,3,5,\ldots,11)\). In addition:

1) if \((3i+3)\) or \((3i+6)>15\), then the corresponding \((3i-6)\) and \((3i-9)\) must be taken;
2) if the indices \((i+1),\ldots,(i+4)\), and also \((i-1)\) and \((i-2)>5\), then they are reduced by 5;
3) group 2 equations—polygonal—are written from the geodetic quadrilaterals having a common pole \(A\) and a common side \(AE\): \(ABCE\); \(ACDE\); \(ABDE\).

Adjustment process:

1) finding corrections I; 2) drawing up polygonal conditions by the corrected angles; 3) transforming the coefficients of the equations of group 2; 4) drawing up normal equations, solving them and finding corrections II. Final corrections \((i_s)=(i_s)_1+(i_s)_2\), with \([(i_s)^2]=[(i_s)_1^2]+[(i_s)_2^2]=\min\).

In the general form, the preceding formulas for the correlates and primary corrections will be represented as follows (in view of their cyclic character):

\[ \begin{array}{ll} \text{correlates} & \left\{ \begin{aligned} k_i&=\frac{-19w_i-w_{i+1}+5w_{i+2}+5w_{i+3}-w_{i+4}}{66},\\[6pt] k_0&=-\frac{w'_s}{15}; \end{aligned} \right. \end{array} \tag{3} \]

\[ \begin{array}{ll} \text{corrections} & \left\{ \begin{aligned} (3i-2)&=(3i-1)=k_i+k_0,\\ (3i)&=-k_{i-1}-k_{i+2}+k_0. \end{aligned} \right. \end{array} \tag{4} \]

In all these formulas (1)—(4): \(i=1,2,\ldots,5\); \(s=1,2,\ldots,15\); \(r=1,3,5,\ldots,11\). In addition: 1) if in (1) the corrections \((3i+3)\) or \((3i+6)>15\), then one must take, respectively, \((3i-6)\) and \((3i-9)\); 2) if in (3) the indices of \(w\), \(i+1,i+2,\ldots,i+4\), and likewise in (4) the indices of \(k\), \(i-1\) and \(i+2\), turn out to be greater than 5, they are decreased by 5; 3) if in (4) the index of \(k_{i-1}\) becomes 0, it is increased by a whole period, i.e. by 5. The indicated rules follow from the cyclic character of formulas (1), (3), and (4), which have cycles 5 and 15.

Having obtained the general formulas for the correlates and primary corrections, we may regard our problem as finished, since all further calculations—the formation of the pole equations, the transformation of the coefficients of these equations, and the determination of the secondary corrections—are sufficiently well known and are set forth in the general theory of two-group adjustment. We note only that these transformations are carried out by formulas (3), (4), while \(\omega_s\) are obtained by formulas (1), (2), since the coefficients in the equations of the first group are equal to \(\pm 1\).

Below is given a form for the adjustment computations of a geodetic pentagon.