Triangulation

# Triangulation

Overview
In trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, triangulation is the process of determining the location of a point by measuring angles
Angles
The Angles is a modern English term for a Germanic people who took their name from the ancestral cultural region of Angeln, a district located in Schleswig-Holstein, Germany...

to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly (trilateration
Trilateration
In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...

). The point can then be fixed as the third point of a triangle with one known side and two known angles.
Discussion

Encyclopedia
In trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

and geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, triangulation is the process of determining the location of a point by measuring angles
Angles
The Angles is a modern English term for a Germanic people who took their name from the ancestral cultural region of Angeln, a district located in Schleswig-Holstein, Germany...

to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly (trilateration
Trilateration
In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...

). 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
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

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
Resection (orientation)
Resection is a method for determining a position using a compass and topographic map .-Resection versus intersection:...

. 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 SAT NAV system is a system of satellites that provide autonomous geo-spatial positioning with global coverage. It allows small electronic receivers to determine their location to within a few metres using time signals transmitted along a line-of-sight by radio from...

s in the 1980s.

## Applications

Optical 3d measuring systems use this principle as well in order to determine the spatial dimensions and the geometry of an item. Basically, the configuration consists of two sensors observing the item. One of the sensors is typically a digital camera device, and the other one can also be a camera or a light projector. The projection centers of the sensors and the considered point on the object’s surface define a (spatial) triangle. Within this triangle, the distance between the sensors is the base b and must be known. By determining the angles between the projection rays of the sensors and the basis, the intersection point, and thus the 3d coordinate, is calculated from the triangular relations.

## Distance to a point by measuring two fixed angles

The coordinates and distance to a point can be found by calculating the length of one side of a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

, given measurements of angles and sides of the triangle formed by that point and two other known reference points.

The following formulas apply in flat or Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

. They become inaccurate if distances become appreciable compared to the curvature of the Earth, but can be replaced with more complicated results derived using spherical trigonometry
Spherical trigonometry
Spherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...

.

### Calculation

Therefore

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:

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
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

Navigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...

, metrology
Metrology
Metrology is the science of measurement. Metrology includes all theoretical and practical aspects of measurement. The word comes from Greek μέτρον , "measure" + "λόγος" , amongst others meaning "speech, oration, discourse, quote, study, calculation, reason"...

, astrometry
Astrometry
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...

, binocular vision
Binocular vision
Binocular vision is vision in which both eyes are used together. The word binocular comes from two Latin roots, bini for double, and oculus for eye. Having two eyes confers at least four advantages over having one. First, it gives a creature a spare eye in case one is damaged. Second, it gives a...

, model rocketry and gun direction of weapon
Weapon
A weapon, arm, or armament is a tool or instrument used with the aim of causing damage or harm to living beings or artificial structures or systems...

s.

The use of triangles to estimate distances goes back to antiquity. In the 6th century BC the Greek philosopher Thales
Thales
Thales of Miletus was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition...

is recorded as using similar triangles to estimate the height of the pyramids by measuring the length of their shadows at the moment when his own shadow was equal to his height; and to have estimated the distances to ships at sea as seen from a clifftop, by measuring the horizontal distance traversed by the line-of-sight for a known fall, and scaling up to the height of the whole cliff. Such techniques would have been familiar to the ancient Egyptians. Problem 57 of the Rhind papyrus, a thousand years earlier, defines the seqt or seked as the ratio of the run to the rise of a slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

, i.e. the reciprocal of gradients as measured today. The slopes and angles were measured using a sighting rod that the Greeks called a dioptra
Dioptra
A dioptra is a classical astronomical and surveying instrument, dating from the 3rd century BCE. The dioptra was a sighting tube or, alternatively, a rod with a sight at both ends, attached to a stand...

, the forerunner of the Arabic alidade. A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria
Hero of Alexandria
Hero of Alexandria was an ancient Greek mathematician and engineerEnc. Britannica 2007, "Heron of Alexandria" who was active in his native city of Alexandria, Roman Egypt...

(c. 10–70 AD), which survived in Arabic translation; but the knowledge became lost in Europe. In China, Pei Xiu
Pei Xiu
Pei Xiu , style name Jiyan , was a minister, geographer, and cartographer of the state of Cao Wei during the Three Kingdoms period of Chinese history, as well as the subsequent Jin Dynasty. Pei Xiu was very much trusted by Sima Zhao, and participated in the suppression of Zhuge Dan's coup...

(224–271) identified "measuring right angles and acute angles" as the fifth of his six principles for accurate map-making, necessary to accurately establish distances; while Liu Hui
Liu Hui
Liu Hui was a mathematician of the state of Cao Wei during the Three Kingdoms period of Chinese history. In 263, he edited and published a book with solutions to mathematical problems presented in the famous Chinese book of mathematic known as The Nine Chapters on the Mathematical Art .He was a...

(c. 263) gives a version of the calculation above, for measuring perpendicular distances to inaccessible places.

In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the agromensores; but were introduced into medieval Spain through Arabic treatises on the astrolabe
Astrolabe
An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and longitude, surveying, triangulation, and to...

, 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
Jacob's staff
The Jacob's staff, also called a cross-staff, a ballastella, a fore-staff, or a balestilha is used to refer to several things. This can lead to considerable confusion unless one clarifies the purpose for the object so named...

, 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.

### Gemma Frisius and triangulation for mapmaking

On land, the Dutch cartographer Gemma Frisius
Gemma Frisius
Gemma Frisius , was a physician, mathematician, cartographer, philosopher, and instrument maker...

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'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
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...

applied the method in Scandinavia, completing a detailed triangulation in 1579 of the island of Hven, where his observatory was based, with reference to key landmarks on both sides of the Øresund, 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 Cunningham's Cosmographical Glasse (1559), Valentine Leigh's Treatise of Measuring All Kinds of Lands (1562), William Bourne
William Bourne (mathematician)
William Bourne was an English mathematician, innkeeper and former Royal Navy gunner who invented the first navigable submarine and wrote important navigational manuals...

's Rules of Navigation (1571), Thomas Digges
Thomas Digges
Sir Thomas Digges 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 stars at varying distances; he was also first to postulate the "dark night sky...

's Geometrical Practise named Pantometria (1571), and John Norden
John Norden
John Norden was an English cartographer, chorographer and antiquary. He planned a series of county maps and accompanying county histories of England, the Speculum Britanniae...

's Surveyor's Dialogue (1607). It has been suggested that Christopher Saxton
Christopher Saxton
Christopher Saxton was an English cartographer, probably born in the parish of Dewsbury, in the West Riding of Yorkshire, England around 1540....

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.

### Willebrord Snell and modern triangulation networks

The modern systematic use of triangulation networks stems from the work of the Dutch mathematician Willebrord Snell, who in 1615 surveyed the distance from Alkmaar
Alkmaar
Alkmaar is a municipality and a city in the Netherlands, in the province of Noord Holland. Alkmaar is well known for its traditional cheese market. For tourists, it is a popular cultural destination.-History:...

to Bergen op Zoom
Bergen op Zoom
Bergen op Zoom is a municipality and a city located in the south of the Netherlands.-History:Bergen op Zoom was granted city status probably in 1266. In 1287 the city and its surroundings became a lordship as it was separated from the lordship of Breda. The lordship was elevated to a margraviate...

, approximately 70 miles (110 kilometres), using a chain of quadrangles containing 33 triangles in all. The two towns were separated by one degree on the meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

, 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
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...

), 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
Resection (orientation)
Resection is a method for determining a position using a compass and topographic map .-Resection versus intersection:...

, 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.

Snell's methods were taken up by Jean Picard
Jean Picard
Jean-Felix Picard was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand. He was the first person to measure the size of the Earth to a reasonable degree of accuracy in a survey conducted in 1669–70, for which he is honored with a...

who in 1669–70 surveyed one degree of latitude along the Paris Meridian
Paris Meridian
The Paris Meridian is a meridian line running through the Paris Observatory in Paris, France—now longitude 2°20′14.025″ east. It was a long-standing rival to Greenwich as the prime meridian of the world, as was the Meridian of Antwerp in Antwerp, Belgium....

using a chain of thirteen triangles stretching north from Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

to the clocktower of Sourdon
Sourdon
Sourdon is a commune in the Somme department in the Picardie region of France.-Population:-External links:*...

, near Amiens
Amiens
Amiens is a city and commune in northern France, north of Paris and south-west of Lille. It is the capital of the Somme department in Picardy...

. 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
Giovanni Domenico Cassini

and his son Jacques Cassini
Jacques Cassini
Jacques Cassini 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 Academy of Sciences, he was elected in 1696 a fellow of the Royal Society of London, and...

surveyed the whole of the Paris meridian from Dunkirk to Perpignan
Perpignan
-Sport:Perpignan is a rugby stronghold: their rugby union side, USA Perpignan, is a regular competitor in the Heineken Cup and seven times champion of the Top 14 , while their rugby league side plays in the engage Super League under the name Catalans Dragons.-Culture:Since 2004, every year in the...

; and between 1733 and 1740 Jacques and his son César Cassini
César-François Cassini de Thury
César-François Cassini de Thury , also called Cassini III or Cassini de Thury, was a French astronomer and cartographer.- Biography :...

undertook the first triangulation of the whole country, including a re-surveying of the meridian arc
Meridian arc
In geodesy, a meridian arc measurement is a highly accurate determination of the distance between two points with the same longitude. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This...

, 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
Principal Triangulation of Great Britain
The Principal Triangulation of Britain was a triangulation project carried out between 1783 and about 1853 at the instigation of the Director of the Ordnance Survey General William Roy ....

was begun by the Ordnance Survey
Ordnance Survey
Ordnance Survey , an executive agency and non-ministerial government department of the Government of the United Kingdom, is the national mapping agency for Great Britain, producing maps of Great Britain , and one of the world's largest producers of maps.The name reflects its creation together with...

in 1783, though not completed until 1853; and the Great Trigonometric Survey
Great Trigonometric Survey
The Great Trigonometric Survey was a project of the Survey of India throughout most of the 19th century. It was piloted in its initial stages by William Lambton, and later by George Everest. Among the many accomplishments of the Survey were the demarcation of the British territories in India and...

of India, which ultimately named and mapped Mount Everest
Mount Everest
Mount Everest is the world's highest mountain, with a peak at above sea level. It is located in the Mahalangur section of the Himalayas. The international boundary runs across the precise summit point...

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
Rhineland
Historically, the Rhinelands refers to a loosely-defined region embracing the land on either bank of the River Rhine in central Europe....

from 1801, subsequently completed after 1815 by the Prussian general Karl von Müffling. Meanwhile, the famous mathematician Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

was entrusted from 1821 to 1825 with the triangulation of the kingdom of Hanover
Kingdom of Hanover
The Kingdom of Hanover 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 Brunswick-Lüneburg , and joined with 38 other sovereign states in the German...

, for which he developed the method of least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...

to find the best fit solution for problems of large systems of simultaneous equations 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 SAT NAV system is a system of satellites that provide autonomous geo-spatial positioning with global coverage. It allows small electronic receivers to determine their location to within a few metres using time signals transmitted along a line-of-sight by radio from...

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
Trig point
A triangulation station, also known as a triangulation pillar, trigonometrical station, trigonometrical point, trig station, trig beacon or trig point, and sometimes informally as a trig, is a fixed surveying station, used in geodetic surveying and other surveying projects in its vicinity...

set up for retriangulation of Great Britain
Retriangulation of Great Britain
The retriangulation of Great Britain was a triangulation project which involved erecting concrete pillars on prominent hilltops throughout Great Britain...

(1936–1962), or the triangulation points set up for the Struve Geodetic Arc
Struve Geodetic Arc
The Struve Geodetic Arc is a chain of survey triangulations stretching from Hammerfest in Norway to the Black Sea, through ten countries and over 2,820 km, which yielded the first accurate measurement of a meridian....

(1816–1855), now scheduled as a UNESCO World Heritage Site
World Heritage Site
A UNESCO World Heritage Site is a place that is listed by the UNESCO as of special cultural or physical significance...

.

• Multilateration
Multilateration
Multilateration is a navigation technique based on the measurement of the difference in distance to two or more stations at known locations that broadcast signals at known times. Unlike measurements of absolute distance or angle, measuring the difference in distance results in an infinite number of...

, where a point is calculated using the time-difference-of-arrival between other known points
• Trilateration
Trilateration
In geometry, trilateration is the process of determinating absolute or relative locations of points by measurement of distances, using the geometry of circles, spheres or triangles. In addition to its interest as a geometric problem, trilateration does have practical applications in surveying and...

, where a point is calculated given its distances from other known points
• GSM localization
• Parallax
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. The term is derived from the Greek παράλλαξις , meaning "alteration"...

• Resection
Resection (orientation)
Resection is a method for determining a position using a compass and topographic map .-Resection versus intersection:...

• SOCET SET
SOCET SET
SOCET SET is a software application that performs functions related to photogrammetry. It is developed and published by BAE Systems. SOCET SET was the first commercial digital photogrammetry software program...

• Stereopsis
Stereopsis
Stereopsis refers to impression of depth that is perceived when a scene is viewed with both eyes by someone with normal binocular vision. Binocular viewing of a scene creates two slightly different images of the scene in the two eyes due the the eyes' different positions on the head...

• Surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...

• Trig point
Trig point
A triangulation station, also known as a triangulation pillar, trigonometrical station, trigonometrical point, trig station, trig beacon or trig point, and sometimes informally as a trig, is a fixed surveying station, used in geodetic surveying and other surveying projects in its vicinity...

• Triangulation (social science)
Triangulation (social science)
In the social sciences, triangulation is often used to indicate that more than two methods are used in a study with a view to double checking results. This is also called "cross examination"....

The application and combination of several research methodologies in the study of the same phenomenon.