Triangulation (surveying)

Source: Wikipedia, the free encyclopedia.
For other uses, see Triangulation (disambiguation).
Triangulation of Kodiak Island in Alaska in 1929.

In surveying, triangulation is the process of determining the location of a point by measuring only angles to it from known points at either end of a fixed baseline by using trigonometry, rather than measuring distances to the point directly as in trilateration. 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

global navigation satellite systems
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 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 are strictly correct only for a flat surface. If the curvature of the Earth must be allowed for, then spherical trigonometry
must be used.

Calculation

See also: Geometric mean theorem and inverse Pythagorean theorem

With being the distance between A and B gives:

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:

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

Liu Hui (c. 263), How to measure the height of a sea island. Illustration from an edition of 1726
Gemma Frisius's 1533 proposal to use triangulation for mapmaking
Nineteenth-century triangulation network for the triangulation of Rhineland-Hesse

Triangulation today is used for many purposes, including

model rocketry and gun direction of weapons
.

In the field, triangulation methods were apparently not used by the Roman specialist land surveyors, the

Portolan charts
, the earliest of which that survives is dated 1296.

Gemma Frisius

On land, the cartographer

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.[3] 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'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 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.[4]

Willebrord Snell

Main article:
Snellius' triangulation

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 Breda, 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, 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 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

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 Jean-Dominique 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 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

simultaneous equations
given more real-world measurements than unknowns.

Today, large-scale triangulation networks for positioning have largely been superseded by the

triangulation pillars set up for retriangulation of Great Britain (1936–1962), or the triangulation points set up for the Struve Geodetic Arc (1816–1855), now scheduled as a UNESCO World Heritage Site
.

See also

References

  1. ^
    ISBN 0-87548-422-0
    . pp. 119–122
  2. ^ O'Connor, John J.; Robertson, Edmund F., "Abu Arrayhan Muhammad ibn Ahmad al-Biruni", MacTutor History of Mathematics Archive, University of St Andrews
  3. ^ 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
  4. ^ Martin and Jean Norgate (2003), Saxton's Hampshire: Surveying, University of Portsmouth

Further reading

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