Introduction to Triangulation
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
Calculation
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)
therefore:
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
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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FAQs on Triangulation
1. What is the basic principle of triangulation in geography?
Triangulation is a fundamental surveying method used to determine the position of an unknown point by measuring the angles to it from two known points. These three points form a triangle. By knowing the distance between the two known points (the baseline) and the two angles from the baseline to the unknown point, the location of the new point can be precisely calculated using trigonometry. This process creates a network of interconnected triangles to map large areas.
2. What is the main purpose of using triangulation in surveying and mapping?
The primary purpose of triangulation is to establish a highly accurate framework of control points over a large area. Instead of measuring long distances directly, which is prone to errors, surveyors measure angles, which can be done with high precision. This method is crucial for:
Creating accurate large-scale topographical maps.
Ensuring precision in major engineering projects like the construction of bridges, dams, and tunnels.
Establishing national and regional geodetic control networks which are the foundation for all other surveying and mapping activities.
3. Why are triangles considered the ideal shape for triangulation surveys?
Triangles are the ideal shape for survey networks because they are the simplest and strongest geometric figures. A triangle is the only polygon whose shape cannot be changed without altering the length of its sides. This inherent rigidity ensures that once the angles are measured, the shape and size of the triangle are fixed, providing a stable and reliable basis for calculation. Any errors in measurement can be more easily detected and distributed within a triangular framework compared to less rigid shapes like quadrilaterals.
4. What role does an instrument like the Theodolite play in the triangulation process?
A Theodolite is a precision instrument essential for triangulation. Its primary role is to accurately measure horizontal and vertical angles between points. In a triangulation survey, the theodolite is set up at each known station to precisely sight the other stations and measure the included angles of the triangle. The high accuracy of these angular measurements is critical for the overall precision of the entire triangulation network.
5. How does a network of triangles help minimise survey errors over large areas?
A network of triangles minimises errors through a process of systematic expansion and adjustment. The survey starts from a single, very accurately measured baseline. From there, a network of triangles is built outwards. Because the angles of each triangle must add up to 180 degrees, any small measurement errors can be identified and mathematically adjusted across the network. This prevents the accumulation of significant errors that would occur if long distances were measured sequentially and directly.
6. What is the difference between triangulation and trilateration?
The key difference between triangulation and trilateration lies in what is being measured. In triangulation, the primary measurements are the angles of the triangles, with only one initial baseline distance being measured. In trilateration, the primary measurements are the lengths of all three sides of the triangle, with no angles being measured. Modern technology like GPS and Electronic Distance Measurement (EDM) has made trilateration more common, as it relies on distance measurement, which is the fundamental principle of how GPS determines a location.
7. What are some real-world examples of where triangulation principles are used today?
Beyond traditional map-making, the principle of triangulation is a cornerstone of many modern technologies. A key example is the Global Positioning System (GPS), which uses a similar principle called trilateration (or more accurately, multilateration) with signals from multiple satellites to pinpoint a receiver's exact location on Earth. It is also used in cellular networks to determine a mobile phone's location by measuring signals from multiple cell towers.
