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In
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
, triangulation is the process of determining the location of a point by measuring only
angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
s to it from known points at either end of a fixed baseline by using
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, rather than measuring distances to the point directly as in
trilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth (geopositioning).
When more than three distances are involved, it may be called multilateration, for e ...
. 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
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
of systems of very large triangles, called triangulation networks. This followed from the work of
Willebrord Snell
Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, Snell. His name is usually associated with the law of refraction of light known as Snell's law.
The lunar crater Sne ...
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 system
A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows satellite navigation devices to determine their location (longitude, latitude, and altitude/elevation) to high pr ...
s 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
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle.
Angles formed by two ...
''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
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
are strictly correct only for a flat surface. If the curvature of the Earth must be allowed for, then
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
must be used.
Calculation
With
being the distance between ''A'' and ''B'' we have:
:
Using the
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
tan α = sin α / cos α and sin(α + β) = sin α cos β + cos α sin β, this is equivalent to:
:
:
therefore:
:
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
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
,
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
,
metrology
Metrology is the scientific study of measurement. It establishes a common understanding of units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to standardise units in Fran ...
,
astrometry
Astrometry is a branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. It provides the kinematics and physical origin of the Solar System and this galaxy, the Milky Way.
His ...
,
binocular vision
In biology, binocular vision is a type of vision in which an animal has two eyes capable of facing the same direction to perceive a single three-dimensional image of its surroundings. Binocular vision does not typically refer to vision where an ...
,
model rocketry
A model rocket are small rockets designed to reach low altitudes (e.g., for model) and be recovered by a variety of means.
According to the United States National Association of Rocketry (NAR) Safety Code, model rockets are constructed of ...
and gun direction of
weapon
A weapon, arm or armament is any implement or device that can be used to deter, threaten, inflict physical damage, harm, or kill. Weapons are used to increase the efficacy and efficiency of activities such as hunting, crime, law enforcement, s ...
s.
In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the ''
agrimensores
''Gromatici'' (from Latin '' groma'' or ''gruma'', a surveyor's pole) or ''agrimensores'' was the name for land surveyors amongst the ancient Romans. The "gromatic writers" were technical writers who codified their techniques of surveying, most ...
''; but were introduced into medieval Spain through
Arabic treatises on the
astrolabe, such as that by
Ibn al-Saffar
Abu al‐Qasim Ahmad ibn Abd Allah ibn Umar al‐Ghafiqī ibn as-Saffar al‐Andalusi (born in Cordoba, died in the year 1035 at Denia), also known as Ibn as-Saffar (, literally: son of the brass worker), was a Spanish-Arab astronomer in Al-Anda ...
(d. 1035).
Donald Routledge Hill
Donald Routledge Hill (6 August 1922 – 30 May 1994)D. A. King, “In Memoriam: Donald Routledge Hill (1922-1994)”, ''Arabic Sciences and Philosophy,'' Volume 5 / Issue 02 / September 1995, pp 297-302 was a British engineer and historian of sc ...
(1984), ''A History of Engineering in Classical and Medieval Times'', London: Croom Helm & La Salle, Illinois: Open Court. . pp. 119–122 Abu Rayhan Biruni
Abu Rayhan Muhammad ibn Ahmad al-Biruni (973 – after 1050) commonly known as al-Biruni, was a Khwarazmian Iranian in scholar and polymath during the Islamic Golden Age. He has been called variously the "founder of Indology", "Father of Co ...
(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
The term Jacob's staff is used to refer to several things, also known as cross-staff, a ballastella, a fore-staff, a ballestilla, or a balestilha. In its most basic form, a Jacob's staff is a stick or pole with length markings; most staffs ar ...
, used specifically for measuring angles, which dates from about 1300; and the appearance of accurately surveyed coastlines in the
Portolan charts
Portolan charts are nautical charts, first made in the 13th century in the Mediterranean basin and later expanded to include other regions. The word ''portolan'' comes from the Italian ''portulano'', meaning "related to ports or harbors", and whi ...
, the earliest of which that survives is dated 1296.
Gemma Frisius
On land, the cartographer
Gemma Frisius
Gemma Frisius (; born Jemme Reinerszoon; December 9, 1508 – May 25, 1555) was a Frisian physician, mathematician, cartographer, philosopher, and instrument maker. He created important globes, improved the mathematical instruments of his d ...
proposed using triangulation to accurately position far-away places for map-making in his 1533 pamphlet ''Libellus de Locorum describendorum ratione'' (''Booklet concerning a way of describing places''), which he bound in as an appendix in a new edition of
Peter Apian
Petrus Apianus (April 16, 1495 – April 21, 1552), also known as Peter Apian, Peter Bennewitz, and Peter Bienewitz, was a German humanist, known for his works in mathematics, astronomy and cartography. His work on "cosmography", the field that d ...
's best-selling 1524 ''Cosmographica''. This became very influential, and the technique spread across Germany, Austria and the Netherlands. The astronomer
Tycho Brahe
Tycho Brahe ( ; born Tyge Ottesen Brahe; generally called Tycho (14 December 154624 October 1601) was a Danish astronomer, known for his comprehensive astronomical observations, generally considered to be the most accurate of his time. He was k ...
applied the method in Scandinavia, completing a detailed triangulation in 1579 of the island of
Hven
Ven ( da, Hven, older Swedish spelling Hven) is a small Swedish island in the Øresund strait, between Scania and Zealand (Denmark). It is part of Landskrona Municipality, Scania County. The island has 371 inhabitants and an area of . During th ...
, where his observatory was based, with reference to key landmarks on both sides of the
Øresund
Øresund or Öresund (, ; da, Øresund ; sv, Öresund ), commonly known in English as the Sound, is a strait which forms the Danish–Swedish border, separating Zealand (Denmark) from Scania (Sweden). The strait has a length of ; its width v ...
, producing an estate plan of the island in 1584. In England Frisius's method was included in the growing number of books on surveying which appeared from the middle of the century onwards, including
William Cuningham
William Cuningham, also known as Kenningham, was a 16th-century English physician, astrologer, and engraver. He practised at Norwich around 1559. Cunningham published his work ''The Cosmographical Glasse'' that year. It contains many woodc ...
's ''Cosmographical Glasse'' (1559), Valentine Leigh's ''Treatise of Measuring All Kinds of Lands'' (1562),
William Bourne's ''Rules of Navigation'' (1571),
Thomas Digges
Thomas Digges (; c. 1546 – 24 August 1595) was an English mathematician and astronomer. He was the first to expound the Copernican system in English but discarded the notion of a fixed shell of immoveable stars to postulate infinitely many s ...
's ''Geometrical Practise named Pantometria'' (1571), and
John Norden
John Norden (1625) was an English cartographer, chorographer and antiquary. He planned (but did not complete) a series of county maps and accompanying county histories of England, the ''Speculum Britanniae''. He was also a prolific writer ...
's ''Surveyor's Dialogue'' (1607). It has been suggested that
Christopher Saxton
Christopher Saxton (c. 1540 – c. 1610) was an English cartographer who produced the first county maps of England and Wales.
Life and family
Saxton was probably born in Sowood, Ossett in the parish of Dewsbury, in the West Riding of Yorkshire ...
may have used rough-and-ready triangulation to place features in his county maps of the 1570s; but others suppose that, having obtained rough bearings to features from key vantage points, he may have estimated the distances to them simply by guesswork.
[Martin and Jean Norgate (2003)]
Saxton's Hampshire: Surveying
University of Portsmouth
Willebrord Snell
The modern systematic use of triangulation networks stems from the work of the Dutch mathematician
Willebrord Snell
Willebrord Snellius (born Willebrord Snel van Royen) (13 June 158030 October 1626) was a Dutch astronomer and mathematician, Snell. His name is usually associated with the law of refraction of light known as Snell's law.
The lunar crater Sne ...
, who in 1615 surveyed the distance from
Alkmaar
Alkmaar () is a city and municipality in the Netherlands, located in the province of North Holland, about 30 km north of Amsterdam. Alkmaar is well known for its traditional cheese market. For tourists, it is a popular cultural destination. The ...
to
Breda
Breda () is a city and municipality in the southern part of the Netherlands, located in the province of North Brabant. The name derived from ''brede Aa'' ('wide Aa' or 'broad Aa') and refers to the confluence of the rivers Mark and Aa. Breda has ...
, approximately 72 miles (116 kilometres), using a chain of quadrangles containing 33 triangles in all. Snell underestimated the distance by 3.5%. The two towns were separated by one degree on the
meridian
Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to
Science
* Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon
* ...
, so from his measurement he was able to calculate a value for the circumference of the earth – a feat celebrated in the title of his book ''Eratosthenes Batavus'' (''The Dutch
Eratosthenes
Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
''), published in 1617. Snell calculated how the planar formulae could be corrected to allow for the curvature of the earth. He also showed how to
resection, or calculate, the position of a point inside a triangle using the angles cast between the vertices at the unknown point. These could be measured much more accurately than bearings of the vertices, which depended on a compass. This established the key idea of surveying a large-scale primary network of control points first, and then locating secondary subsidiary points later, within that primary network.
Further developments
Snell's methods were taken up by
Jean Picard who in 1669–70 surveyed one degree of latitude along the
Paris Meridian
The Paris meridian is a meridian line running through the Paris Observatory in Paris, France – now longitude 2°20′14.02500″ East. It was a long-standing rival to the Greenwich meridian as the prime meridian of the world. The "Paris merid ...
using a chain of thirteen triangles stretching north from
Paris
Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. S ...
to the clocktower of
Sourdon
Sourdon (; pcd, Sordon) is a commune in the Somme department in the Hauts-de-France region of France.
Geography
Sourdon is situated south of Amiens
Amiens (English: or ; ; pcd, Anmien, or ) is a city and commune in northern France, ...
, near
Amiens
Amiens (English: or ; ; pcd, Anmien, or ) is a city and commune in northern France, located north of Paris and south-west of Lille. It is the capital of the Somme department in the region of Hauts-de-France. In 2021, the population of ...
. Thanks to improvements in instruments and accuracy, Picard's is rated as the first reasonably accurate measurement of the radius of the earth. Over the next century this work was extended most notably by the Cassini family: between 1683 and 1718
Jean-Dominique Cassini and his son
Jacques Cassini
Jacques Cassini (18 February 1677 – 16 April 1756) was a French astronomer, son of the famous Italian astronomer Giovanni Domenico Cassini.
Cassini was born at the Paris Observatory. Admitted at the age of seventeen to membership of the French ...
surveyed the whole of the Paris meridian from
Dunkirk
Dunkirk (french: Dunkerque ; vls, label=French Flemish, Duunkerke; nl, Duinkerke(n) ; , ;) is a commune in the department of Nord in northern France.[Perpignan
Perpignan (, , ; ca, Perpinyà ; es, Perpiñán ; it, Perpignano ) is the prefecture of the Pyrénées-Orientales department in southern France, in the heart of the plain of Roussillon, at the foot of the Pyrenees a few kilometres from the ...]
; and between 1733 and 1740 Jacques and his son
César Cassini undertook the first triangulation of the whole country, including a re-surveying of the
meridian arc, leading to the publication in 1745 of the first map of France constructed on rigorous principles.
Triangulation methods were by now well established for local mapmaking, but it was only towards the end of the 18th century that other countries began to establish detailed triangulation network surveys to map whole countries. The
Principal Triangulation of Great Britain
The Principal Triangulation of Britain was the first high-precision triangulation survey of the whole of Great Britain (including Ireland), carried out between 1791 and 1853 under the auspices of the Board of Ordnance. The aim of the survey was ...
was begun by the
Ordnance Survey
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, logo_width = 240px
, logo_caption =
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in 1783, though not completed until 1853; and the
Great Trigonometric Survey
The Great Trigonometrical Survey was a project that aimed to survey the entire Indian subcontinent with scientific precision. It was begun in 1802 by the British infantry officer William Lambton, under the auspices of the East India Company.Gil ...
of India, which ultimately named and mapped
Mount Everest
Mount Everest (; Tibetan: ''Chomolungma'' ; ) is Earth's highest mountain above sea level, located in the Mahalangur Himal sub-range of the Himalayas. The China–Nepal border runs across its summit point. Its elevation (snow heig ...
and the other Himalayan peaks, was begun in 1801. For the Napoleonic French state, the French triangulation was extended by
Jean-Joseph Tranchot into the German
Rhineland
The Rhineland (german: Rheinland; french: Rhénanie; nl, Rijnland; ksh, Rhingland; Latinised name: ''Rhenania'') is a loosely defined area of Western Germany along the Rhine, chiefly its middle section.
Term
Historically, the Rhinelands ...
from 1801, subsequently completed after 1815 by the Prussian general
Karl von Müffling. Meanwhile, the famous mathematician
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
was entrusted from 1821 to 1825 with the triangulation of the
kingdom of Hanover
The Kingdom of Hanover (german: Königreich Hannover) was established in October 1814 by the Congress of Vienna, with the restoration of George III to his Hanoverian territories after the Napoleonic era. It succeeded the former Electorate of Han ...
, for which he developed the
method of least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
to find the best fit solution for problems of large systems of
simultaneous equation
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single ...
s given more real-world measurements than unknowns.
Today, large-scale triangulation networks for positioning have largely been superseded by the
global navigation satellite system
A satellite navigation or satnav system is a system that uses satellites to provide autonomous geo-spatial positioning. It allows satellite navigation devices to determine their location (longitude, latitude, and altitude/elevation) to high pr ...
s established since the 1980s, but many of the control points for the earlier surveys still survive as valued historical features in the landscape, such as the concrete
triangulation pillars set up for
retriangulation of Great Britain
__NOTOC__
The retriangulation of Great Britain was a triangulation project carried out between 1935 and 1962 that sought to improve the accuracy of maps made of Great Britain. Data gathered from the retriangulation replaced data gathered during ...
(1936–1962), or the triangulation points set up for the
Struve Geodetic Arc (1816–1855), now scheduled as a UNESCO
World Heritage Site
A World Heritage Site is a landmark or area with legal protection by an international convention administered by the United Nations Educational, Scientific and Cultural Organization (UNESCO). World Heritage Sites are designated by UNESCO for h ...
.
See also
*
Anglo-French Survey (1784–1790)
The Anglo-French Survey (1784–1790) was the geodetic survey to measure the relative position of Greenwich Observatory and the Paris Observatory via triangulation. The English operations, executed by William Roy, consisted of the measureme ...
*
Bilby tower
A Bilby tower is a type of steel surveying, survey tower used by the United States Coast and Geodetic Survey from 1927 to 1984. It is named after Jasper S. Bilby who designed it in 1926. In 1927, Herbert Hoover, then the Secretary of Commerce, c ...
*
Great Trigonometrical Survey
The Great Trigonometrical Survey was a project that aimed to survey the entire Indian subcontinent with scientific precision. It was begun in 1802 by the British infantry officer William Lambton, under the auspices of the East India Company.Gi ...
*
Multilateration Trilateration is the use of distances (or "ranges") for determining the unknown position coordinates of a point of interest, often around Earth (geopositioning).
When more than three distances are involved, it may be called multilateration, for e ...
, where a point is calculated using the time-difference-of-arrival between other known points
*
Parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects ...
*
Resection (orientation) Position resection and intersection are methods for determining an unknown geographic position ( position finding) by measuring angles with respect to known positions.
In ''resection'', the one point with unknown coordinates is occupied and sighting ...
*
SOCET SET
*
Spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
*
Stellar triangulation
Stellar triangulation is a method of geodesy and of its subdiscipline space geodesy used to measure Earth's geometric shape. Stars were first used for this purpose by the Finnish astronomer Yrjö Väisälä in 1959, who made astrometric photograp ...
*
Stereopsis
Stereopsis () is the component of depth perception retrieved through binocular vision.
Stereopsis is not the only contributor to depth perception, but it is a major one. Binocular vision happens because each eye receives a different image becaus ...
*
Trig point
A triangulation station, also known as a trigonometrical point, and sometimes informally as a trig, is a fixed surveying station, used in geodetic surveying and other surveying projects in its vicinity. The nomenclature varies regionally: they a ...
References
Further reading
* Bagrow, L. (1964) ''History of Cartography''; revised and enlarged by R.A. Skelton. Harvard University Press.
* Crone, G.R. (1978
953
Year 953 ( CMLIII) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar.
Events
By place
Byzantine Empire
* Battle of Marash: Emir Sayf al-Dawla marches north into the Byzantine Empire an ...
''Maps and their Makers: An Introduction to the History of Cartography'' (5th ed).
* Tooley, R.V. & Bricker, C. (1969) ''A History of Cartography: 2500 Years of Maps and Mapmakers''
* Keay, J. (2000) ''The Great Arc: The Dramatic Tale of How India Was Mapped and Everest Was Named''. London: Harper Collins. .
* Murdin, P. (2009) ''Full Meridian of Glory: Perilous Adventures in the Competition to Measure the Earth''. Springer. .
{{Authority control
Angle
Elementary geometry
Euclidean geometry
Surveying
Geodetic surveys