In
trigonometry and
geometry,
triangulation is the
process of determining the location of a point by measuring
angles to it from known points at either end of a fixed
baseline, rather than measuring distances to the point directly.
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
161517, 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 minimised if a mesh of triangles at the largest
appropriate scale is established first, that points inside the
triangles can all then be accurately located with reference to.
Such triangulation methods dominated accurate largescale land
surveying until the rise of
Global navigation satellite
systems 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, 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. 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.
Calculation
 l = \frac{d}{\tan \alpha} + \frac{d}{\tan \beta}
Therefore
 d = l \, / \, (\tfrac{1}{\tan \alpha} + \tfrac{1}{\tan
\beta})
Alternative calculation
Alternatively, the distance RC can be calculated by using the
law of sines to calculate the lengths
of the sides of the triangle:

\frac{\sin\alpha}{BC}=\frac{\sin\beta}{AC}=\frac{\sin\gamma}{AB}
The distance AB is known, so we can write the lengths of the other
two sides as
 AC=\frac{AB\cdot\sin\beta}{\sin\gamma} \qquad
BC=\frac{AB\cdot\sin\alpha}{\sin\gamma}
RC can now be calculated using either the sine of the angle α, or
the sine of the angle β:
 RC=AC \cdot \sin\alpha \qquad
 \qquad RC=BC \cdot \sin\beta
Either way, this gives the result
 RC=\frac{AB \cdot \sin\alpha \cdot \sin\beta}{\sin\gamma}
We know that γ = 180 − α − β, since
the sum of the three angles in any triangle is known to be 180
degrees; and since sin(
θ) = sin(180 
θ), we can
therefore write sin(γ)=sin(α+β), to give the final formula
 RC=\frac{AB \cdot \sin\alpha \cdot \sin\beta}{\sin(\alpha +
\beta)}
This formula can be shown to be equivalent to the result from the
previous calculation by using the
trigonometric identity sin(α + β) =
sin α cos β + cos α sin β.
Other quantities
Given AC or BC from the second calculation, the
full
coordinates of the unknown point can be calculated by
using the
sine and
cosine
of its
bearing from the
corresponding observation point to calculate its offsets on the
north/south and east/west axes.
The distance
MC from the midpoint of AB to the
unknown point C can be calculated by finding MR and then using the
Pythagorean theorem
 MR=AMRB=\left(\frac{AB}{2}\right)\left(BC \cdot
\cos\beta\right)
 MC=\sqrt{MR^2+RC^2}
History of Triangulation
Nineteenth century triangulation
network for the triangulation of RhinelandHesse
Triangulation today is used for many purposes, including
surveying,
navigation,
metrology,
astrometry,
binocular
vision,
model rocketry and gun
direction of
weapons.
The use of triangles to estimate distances goes back to ancient
times. In the 6th century BC the Greek philosopher
Thales 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 lineofsight 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,
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, 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 (c. 1070 AD), which
survived in Arabic translation; but the knowledge became lost in
Europe. In China,
Pei Xiu (224–271)
identified "measuring right angles and acute angles" as the fifth
of his six principles for accurate mapmaking, necessary to
accurately establish distances; while
Liu
Hui (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, such as that by Ibn alSaffar (d.
1035).
Abū
Rayhān Bīrūnī (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 coexisted 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.
Gemma Frisius and triangulation for mapmaking
On land, the Dutch cartographer
Gemma
Frisius proposed using triangulation to accurately position
faraway places for mapmaking 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
bestselling 1524
Cosmographica. This became very
influential, and the technique spread across Germany, Austria and
the Netherlands.
The astronomer Tycho
Brahe 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's
Rules of Navigation (1571),
Thomas Digges's
Geometrical Practise named
Pantometria (1571), and
John
Norden's
Surveyor's Dialogue (1607). It has been
suggested that
Christopher Saxton
may have used roughandready 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 the 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 to BergenopZoom, 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, 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), 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 largescale
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 who
in 166970 surveyed one degree of latitude along the Paris Meridian using a chain of thirteen triangles stretching
north from Paris to the
clocktower of Sourdon, near
Amiens. 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 JeanDominique
Cassini and his son Jacques
Cassini surveyed the whole of the Paris meridian from Dunkirk to Perpignan; and between 1733 and 1740 Jacques and his son
César Cassini undertook the first
triangulation of the whole country, including a resurveying of the
meridian, 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 was begun by the Ordnance Survey in 1783, though not completed until 1853; and the
Great Trigonometric
Survey of India, which ultimately named and mapped Mount Everest 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 from 1801, subsequently completed after
1815 by the Prussian general
Karl
von Müffling. Meanwhile, the famous mathematician
Carl Friedrich Gauss was entrusted from
1821 to 1825 with the triangulation of the
kingdom of Hannover, for which he
developed the
method of least
squares to find the best fit solution for problems of large
systems of
simultaneous
equations given more realworld measurements than
unknowns.
Today, largescale triangulation networks for positioning have
largely been superseded by the
Global navigation satellite
systems 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 (19361962), or the triangulation points set up for the
Struve Geodetic Arc (18161855),
now scheduled as a UNESCO
World
Heritage Site.
See also
References
 I, 27
 Proclus, In
Euclidem
 Joseph
Needham (1986). Science and Civilization in China: Volume
3, Mathematics and the Sciences of the Heavens and the Earth.
Taipei: Caves Books Ltd. pp. 539540
 Liu Hui,
The Sea Island Mathematical
Manual
 Kurt Vogel (1983; 1997), A Surveying Problem Travels from China to
Paris, in Yvonne DoldSamplonius (ed.), From China to
Paris, Proceedings of a conference held July, 1997,
Mathematisches Forschungsinstitut, Oberwolfach, Germany. ISBN
3515082239.
 Donald Routledge Hill (1984), A
History of Engineering in Classical and Medieval Times,
London: Croom Helm & La Salle, Illinois: Open Court. ISBN
0875484220. pp.119122
 Michael Jones (2004), " Tycho Brahe, Cartography and Landscape in 16th
Century Scandinavia", in Hannes Palang (ed), European Rural
Landscapes: Persistence and Change in a Globalising Environment,
p.210
 Martin and Jean Norgate (2003), Saxton's Hampshire: Surveying, University of
Portsmouth
Further reading
 Bagrow, L. (1964) History of Cartography; revised and
enlarged by R.A. Skelton. Harvard University Press.
 Crone, G.R. (1978 [1953]) 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. ISBN 0002570629.