Triangulation is the method of calculating a dot's direction from either end of a fixed baseline, by only measuring angles to the dot, rather than measuring distances explicitly as in the trilateration process. And the point may be set as the third point of a triangle with two known angles and one known side. That being said, Triangulation methods are mostly used for measuring the scale of the earth and the distances between different sites.
History Associated with Triangulation Trigonometry
Current triangulation is used for many applications, including surveys, navigation, metrology, astrometry, binocular vision, missile modelling, and arms guidance.
In the region, triangulation techniques were not appreciably employed in medieval Spain by Arabic astrolabe treaties like that of Ibne Al-Saffar by Roman land surveyors, agrimensores (d. 1035). Triangulation methods for measuring the scale of the earth and the distances between different sites were also used by Abu Rayhan Biruni (d.1048).
Simplified Roman methods then seem to have coexisted with more advanced techniques used by skilled inspectors. But it was seldom to translate these practices into Latin (the geometry textbook, the uncertain Geomatria Auctoris of the eleventh century is a notable exception), and these approaches seem only slowly to have been percolated into the rest of Europe.
The medieval Jacob's staff, primarily used for measurement angles, dating from about 1300, and the presence of precisely monitored shorelines in the Portolan maps, the earliest of which is dated 1296, can demonstrate increased knowledge and usage of such techniques in Spain.
Theory of Triangulation Trigonometry by Gemma Frisius
In a new version of the best-selling 1524 Cosmographica by Peter Apian, on-site, the cartographer Gemma Frisius suggested the precise use of triangulation to position far-away plazas for maps in his booklet Libellus de Locorumratione in 1533 (booklet for a describing place). The technology distributed through Germany, Austria, and the Netherlands has been very influential. The Scandinavian astronomer Tycho Brahe used the technique to complement the extensive triangulation, in 1579, of the island of Hven, where he was located on his observatory, which produced a property plan for the island in 1584 concerning the main sites on both sides of Øresund.
Theory of Triangulation Trigonometry by Willebrord Snell
The Dutch mathematician's work, Willebrord Snell, examined the distance from Alkmaar to Breda in 1615 by a total of approximately 116 kilometers (72 miles) using a chain of 33 triangles is based on the current method of the use of triangulation networks. The gap was quickly underestimated by 3.5%. The two villages had been divided by a degree on the meridian so that he could derive a value from his calculation of the earth's diameter - an achievement praised by his book Eratosthenes batavus, published in 1617. Snell measured how to correct the planar formula to account for the earth's curvature.
He also demonstrated how the location of the point within a triangle can be resected or calculated using the angels cast between the unknown spots. These may be much more precisely determined than vertical bearings depending on a compass. This led to the fundamental idea to first survey and subsequently locate secondary subsidiary points within a vast primary network of control points.
Principle of Triangulation Trigonometry
We have l being the distance from A to B:
l = d/tanα + d/tanβ
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(cosα/sinα + cosβ/sinβ)
l = d (sin(α+β)/sinαsinβ)
d = l (sinαsinβ/sin(α+β))
The distance of the unknown point from any point of observation, the north/south, east/west offsets of the point of observation, and the full coordinates of the point are also simple to calculate.
Theodolite, the essential instrument used to calculate horizontal and vertical angles, dates back to Leonard Digges, an English mathematician from the 16th century. It consists, in its present form, of a horizontal and vertical fixed telescope. The levelling is achieved using a spiritual degree, crosshairs in the telescope allow for precise synchronization with the sighted object. The corresponding two measurements, vertical and horizontal, are read while the telescope is precisely calibrated.
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