1.22.2. Principle of triangulation

In triangulation, the entire area to be surveyed is divided into a framework of triangles. Vertices of the individual triangles are known as triangulation stations. Triangulation is more accurate than the theodolite traverse, as there is less accumulation of error than that in theodolite traverse.

Triangulation is a surveying method where a geographical area is mapped out using a network of triangles. Each triangle has three points, called vertices, which represent triangulation stations. The reason triangulation is preferred over using a theodolite traverse is that it reduces the number of errors that can accumulate when measuring angles incrementally. In contrast to measuring angles one after another in a traverse, triangulation provides a framework that minimizes errors by relying on the fixed positions of angles and distances within each triangle.

Think of triangulation like navigating a large city by using landmarks. If you know the location of three distinct landmarks (vertices), you can determine your own position (the triangulation stations) more accurately than if you tried to navigate step-by-step by counting blocks and streets (like a theodolite traverse).

A series of triangulation may consist of (i) a chain of simple triangles, (ii) braced quadrilaterals, and (iii) centered polygons, as shown in Figures 1.55a, b and c.

Triangulation systems can vary in their configuration. Simple triangles form the basic units, but on their own, they do not allow for checks on measurement accuracy since the data flows in a single direction. This means that if there's an error in one angle or distance, it can affect the entire triangle. A more effective approach is the braced quadrilateral, which adds complexity by incorporating multiple possible calculations of the sides and angles, offering more points of verification. Lastly, centered polygons contain a triangulation station at their center and improve accuracy further by maximizing points of triangulation, but they require more calculations.

If you imagine building a LEGO structure, simple triangles are like single pieces that fit together without support. A braced quadrilateral is like using additional pieces to create a more stable base, while centered polygons represent using a complex LEGO set where you have a center piece that supports additional layers for added strength.

Although the triangle is the smallest figure in triangulation scheme, the braced quadrilateral containing four corner stations is preferred. Braced quadrilaterals system is treated to be the best arrangement, as it provides a means of computing the length of the sides using different combination of the sides, diagonals and angles, and thereby offering more checks on measured/computed data.

Braced quadrilaterals are favored because they allow for various different ways to calculate a triangle's sides through combinations of different triangles that share common vertices. This redundancy means that if one measurement is inaccurate, others can compensate for it, enhancing the overall reliability of the triangulation effort. Centered polygons, while less efficient in terms of the speed of calculations due to their complexity, offer an even higher degree of accuracy because of the numerous intersections and checks available within their structure.

In terms of a construction project, using a braced quadrilateral arrangement is like having multiple support beams in a building. If one beam has a flaw, it's less likely to affect the overall stability of the building because others provide support. Centered polygons are like an architect drawing multiple plans for the same structure, ensuring that if one design has an error, alternatives exist to ensure that the building is sound.

Such figures provide many more checks than a simple quadrilateral, so it offers a more accurate solution, but takes more computational time. The figures containing centered polygons (such as quadrilaterals, pentagons, or hexagons) and centered triangles is known as centered figures.

Although the centered polygon configurations, such as quadrilaterals or hexagons, allow for improved checks on accuracy, they do inherently require more time to compute. The additional vertices in these shapes mean more angles and distances must be calculated and verified, which can slow down the surveying process. As a result, surveyors often need to balance the desire for accuracy with the time available to conduct their work.

This is similar to baking a complex recipe that requires careful timing and measuring versus one that only requires mixing a few ingredients together quickly. While the complex recipe may yield a more delicious final result (greater accuracy), it requires investing more time and effort in preparing and verifying each ingredient (calculating angles and distances).

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