The Optimal Shape of Triangles in Triangulations
Abstract Generated abstract
This paper reexamines the choice of triangle angles in geodetic triangulation, questioning the usual condition that equates relative errors of connecting and intermediate sides and noting that it does not uniquely imply equality of two angles. It instead minimizes the expression governing the relative error of a connecting side, subject to the trigonometric relation among the cotangents of the angles of a plane triangle. The resulting optimum is a triangle with angles A = B = 65 degrees 54 minutes and C = 48 degrees 12 minutes, for which the computed error factors are compared with equilateral triangles and with a previously used form. The calculations indicate that this shape generally reduces relative errors of both connecting and intermediate sides, allowing a longer series of triangles to achieve accuracy comparable to a shorter equilateral series.
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The Most Advantageous Form of Triangulation Triangles
V. S. Nuverlev
In considering this question in courses of higher geodesy it is indicated that the relative errors of the connecting and intermediate sides of a certain triangle must be equal, i.e. the condition must be satisfied:
\[ \operatorname{ctg}^{2} A+\operatorname{ctg}^{2} B+\operatorname{ctg} A \operatorname{ctg} B = \operatorname{ctg}^{2} B+\operatorname{ctg} B \operatorname{ctg} C \tag{a} \]
Further it is indicated that from condition (a) it follows that:
\[ \operatorname{ctg} A=\operatorname{ctg} C,\quad \text{i.e.}\quad A=C \tag{b} \]
Here we must note that equation (a) can be represented in the form:
\[ \operatorname{ctg}^{2} A-\operatorname{ctg}^{2} C = \operatorname{ctg} B(\operatorname{ctg} C-\operatorname{ctg} A) \]
whence
\[ \operatorname{ctg} A+\operatorname{ctg} B+\operatorname{ctg} C=0 \tag{c} \]
i.e. any values of the angles of a triangle satisfying equation (c) will also satisfy equation (a); in this case the equality \(A=C\) need not be satisfied.
Moreover, the idea arises that, by abandoning condition (a), one can obtain connecting and intermediate sides with a smaller relative error. For this it is necessary to seek such a form of triangle in which the best determination of the connecting sides of the row is ensured. Then the intermediate sides will also have smaller relative errors.
As is known, the relative error of the connecting side of a triangle with number \(n\) will be equal to:
\[ \frac{m^{2}_{a_n}}{a^{2}_{n}} = \frac{m^{2}_{b}}{b^{2}} + \frac{2}{3}\cdot\frac{m''^{2}}{\rho^{2}}\, n\bigl(\operatorname{ctg}^{2} A+\operatorname{ctg}^{2} B+\operatorname{ctg} A \operatorname{ctg} B\bigr) \tag{d} \]
The left-hand side of expression (d) becomes least under the condition:
\[ Q_a=\operatorname{ctg}^{2} A+\operatorname{ctg}^{2} B+\operatorname{ctg} A \operatorname{ctg} B=\text{minimum} \tag{e} \]
in the region \(0<A<\pi;\ 0<B<\pi\). The function (e) inside this region is positive, i.e. its least value must also be positive. On the boundary of the region the function assumes the greatest value; consequently, the least value of the function is assumed inside the region.
The cotangents of the angles of a plane triangle are not independent, but are connected by the condition:
\[ \operatorname{ctg} A \operatorname{ctg} B + \operatorname{ctg} A \operatorname{ctg} C + \operatorname{ctg} B \operatorname{ctg} C = 1 \tag{f} \]
i.e. expression (e) can be represented in the form:
\[ Q_a= \operatorname{ctg}^{2} A+\operatorname{ctg}^{2} B+1 -\operatorname{ctg} A \operatorname{ctg} C -\operatorname{ctg} B \operatorname{ctg} C = \min. \]
Equating the first derivatives of \(Q\) to zero, we find:
\[ \begin{aligned} 2\operatorname{ctg} A-\operatorname{ctg} C&=0\\ 2\operatorname{ctg} B-\operatorname{ctg} C&=0 \end{aligned} \qquad\bigg|\, A=B \]
Then expression \((f)\) will have the form:
\[ 5\operatorname{ctg}^{2}A=1 \]
whence \(A=B=65^\circ54'\); \(C=48^\circ12'\).
Thus, we have found the form of triangles for which the least relative errors of the connecting and (as we shall see below) intermediate sides are ensured. For these angles, a series of 12 triangles is equal in length to a series of 10 equilateral triangles. This circumstance is taken into account in computing the values \(\sqrt{nQ}\).
Table of values of \(\sqrt{nQ_a}\) and \(\sqrt{(n-1)Q_a+Q_c}\)
| No. of triangle | \(A=C=52^\circ46'\) \(B=74^\circ38'\) \(Q_a=Q_c=0.856\) |
\(A=B=C=60^\circ\) \(Q_a=Q_c=1.000\) |
\(A=B=65^\circ54'\) \(C=48^\circ12'\) \(Q_a=0.600\) |
\(A=B=65^\circ54'\) \(C=48^\circ12'\) \(Q_c=1.400\) |
|---|---|---|---|---|
| 1 | 0.931 | 1.000 | 0.775 | 1.181 |
| 2 | 1.316 | 1.414 | 1.095 | 1.414 |
| 3 | 1.612 | 1.732 | 1.342 | 1.612 |
| 4 | 1.861 | 2.000 | 1.549 | 1.789 |
| 5 | 2.081 | 2.236 | 1.732 | 1.949 |
| 6 | 2.280 | 2.449 | 1.897 | 2.098 |
| 7 | 2.462 | 2.646 | 2.049 | 2.236 |
| 8 | 2.632 | 2.828 | 2.191 | 2.366 |
| 9 | 2.792 | 3.000 | 2.324 | 2.490 |
| 10 | 2.943 | 3.162 | 2.450 | 2.608 |
| 11 | — | — | 2.569 | 2.720 |
| 12 | — | — | 2.683 | 2.828 |
The results of the computations show that the relative errors of the connecting and intermediate sides of the triangles of the series, with the angles introduced by us, receive smaller values than with angles \(A=C=52^\circ46'\) and \(A=B=C=60^\circ\). The only exception is the intermediate side of the first triangle. If the error of a measured angle for triangulations of the 1st class is taken as equal to \(\pm 0''.7\), then, for the case of equilateral triangles, \(a_{10}\) and \(c_{10}\), computed according to the equations for the figure conditions of the angles, have relative errors (without taking into account the error of the initial side)
\[ \frac{m_{a10}}{a_{10}}=\frac{m_{c10}}{c_{10}}=1:114000; \]
under the same conditions, if \(A=B=65^\circ54'\),
\[ \frac{m_{a12}}{a_{12}}=1:135000,\qquad \frac{m_{c12}}{c_{12}}=1:128000. \]
From this we must conclude that the form of triangles proposed by us significantly increases the accuracy of the computed sides.