What is triangulation?

The art of surveyors is to determine distances and corners, not accessible to direct measurement. Measuring distances with a measured baton or measuring tape is not an easy task, it is much easier to measure angles with a theodolite. This will help triangulation. Let's see what it is. Let us imagine that the surveyor needs to determine the distance between the points A and B (Figure 1, a), which can not be measured directly. The surveyor will measure the length of the segment AC and the angles ACB and CAB. Knowing the side AC and two adjacent angles to it, he can determine all the elements of the triangle ABC, including the side length AB. In more detail, the geodesist calculates the side length AB using the sine theorem:

Measuring distances with a measured baton or measuring tape is not an easy task, it is much easier to measure angles with a theodolite. This will help triangulation

: = sin():sin().

The length of the segment AC and the angle ABC are known directly from the measurements, () = 1800 - () (). Having a calculator on hand, a modern surveyor can easily calculate the length of a segment AB.

If you want to measure a large space, then it is covered with triangles (Figure 1, b), in which all angles are measured. The length must be measured only once, but the number of directly measured angles exceeds the number of angles, the value of which is initially unknown.

For example, it may happen that in any triangle the surveyor measures all three angles independently of each other. Their sum, as is known, should be equal to 1800, but in reality, after adding three angles obtained by measuring the angles, we do not, as a rule, get 1800. A special section of the general theory of errors - the theory of equalization teaches how to distribute the error among the angles of the triangle.

The classical problem of geodesy is the so-called reverse intersection: inside the known triangle, it is required to construct a point D from the measured angles, from which the sides of the triangle are visible (Fig. 1, c). Trigonometric formulas that express the solution of this problem have a rather complex form.

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