Triangulation (surveying)

In surveying, triangulation is the process of determining the location of a point by measuring only angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly as in trilateration. The point can then be fixed as the third point of a triangle with one known side and two known angles.

Triangulation can also refer to the accurate surveying of systems of very large triangles, called triangulation networks. This followed from the work of Willebrord Snell in 1615–17, who showed how a point could be located from the angles subtended from three known points, but measured at the new unknown point rather than the previously fixed points, a problem called resectioning. Surveying error is minimized if a mesh of triangles at the largest appropriate scale is established first. Points inside the triangles can all then be accurately located with reference to it. Such triangulation methods were used for accurate large-scale land surveying until the rise of global navigation satellite systems in the 1980s.

Triangulation may be used to find the position of the ship when the positions of A and B are known. An observer at A measures the angle α, while the observer at B measures β .

The position of any vertex of a triangle can be calculated if the position of one side, and two angles, are known. The following formulae are strictly correct only for a flat surface. If the curvature of the Earth must be allowed for, then spherical trigonometry must be used.

Take L as the distance between A and B.

It can readily be seen that d = L * (tan α * tan β) / (tan α + tan β) which is the result required.

However, this simple rule is only useful when a manually observed result is to be manually calculated. In that case, the anomaly when α is a right angle can be handled by simply saying d = L*(tan β) when the case is encountered. The procedure below is perfectly general, at a higher cost (negligible to a computer).

The cost is:
  Simple: Lookup, lookup, divide, divide.
  General: Lookup, lookup, multiply, add, lookup, add, lookup, divide, multiply.

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