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 by using trigonometry, 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.
Principle 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 length 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.
Calculation
With ℓ {\displaystyle \ell } being the distance between A and B gives:
ℓ = d tan α + d tan β {\displaystyle \ell ={\frac {d}{\tan \alpha }}+{\frac {d}{\tan \beta }}}
Using the trigonometric identities: tan α = sin α / cos α and sin(α + β) = sin α cos β + cos α sin β, this is equivalent to:
ℓ = d ( cos α sin α + cos β sin β ) {\displaystyle \ell =d\left({\frac {\cos \alpha }{\sin \alpha }}+{\frac {\cos \beta }{\sin \beta }}\right)}
ℓ = d sin ( α + β ) sin α sin β {\displaystyle \ell =d\ {\frac {\sin(\alpha +\beta )}{\sin \alpha \sin \beta }}}
therefore:
d = ℓ sin α sin β sin ( α + β ) {\displaystyle d=\ell \ {\frac {\sin \alpha \sin \beta }{\sin(\alpha +\beta )}}}
From this, it is easy to determine the distance of the unknown point from either observation point, its north/south and east/west offsets from the observation point, and finally its full coordinates.
History
Triangulation today is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons. In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the agrimensores; but were introduced into medieval Spain through Arabic treatises on the astrolabe, such as that by Ibn al-Saffar (d. 1035). Abu Rayhan Biruni (d. 1048) also introduced triangulation techniques to measure the size of the Earth and the distances between various places. Simplified Roman techniques then seem to have co-existed with more sophisticated techniques used by professional surveyors. But it was rare for such methods to be translated into Latin (a manual on geometry, the eleventh century Geomatria incerti auctoris is a rare exception), and such techniques appear to have percolated only slowly into the rest of Europe. Increased awareness and use of such techniques in Spain may be attested by the medieval Jacob's staff, used specifically for measuring angles, which dates from about 1300; and the appearance of accurately surveyed coastlines in the Portolan charts, the earliest of which that survives is dated 1296.
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