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In mathematics, a triangulation is not a process, but perhaps the final result of the process. I'm not sure I can quote a good definition at the moment, but the definition on this page is unduly narrow and needs work. Michael Hardy 01:24 Apr 8, 2003 (UTC)

I don't want to argue, but I think that for most non-mathematician triangulation is a process, the end result beeing the lenght, height or area just determinated by ... triangulation. -- looxix 21:03 Apr 8, 2003 (UTC)

The triangulation in topology and geometry are highly related, who can merge the two parts together? wshun 02:06, 19 Dec 2003 (UTC)

I vote not. They are related in a way that the topological notion is a natural generalization of the geometrical one. So, IMO the better idea is to change their order in the article and mention the fact of generalization.... I am doing it right now... Will not be worse anyway :-) Mikkalai 03:08, 19 Dec 2003 (UTC)

Whether 'd' will have physical meaning if alpha(or beta)>90 degrees(because then no perpendicular can be drawn between baseline & the object to be viewed)? — Preceding unsigned comment added by (talk) 12:15, 11 January 2012 (UTC)

I added above question because if we derive this factor 'd' then it requires assumption of existence of right triangles(as shown in the original figure also) which will not hold when alpha(or beta)>90 degrees. — Preceding unsigned comment added by (talk) 13:03, 11 January 2012 (UTC)

It will still all work fine. You'll just find that tan(α) is negative, because the perpendicular lands to the left of A rather than between A and B. Jheald (talk) 16:38, 11 January 2012 (UTC)
Thanks,i convinced myself by deriving the relation when alpha>90 degree,i found it to give same expression for 'd'(i should have done this earlier,i admit),also i convinced myself for the fact that d>0 always,through geometrical proof. — Preceding unsigned comment added by (talk) 05:54, 12 January 2012 (UTC)

geodetic triangulation[edit]

In surveying/geodesy, triangulation is also the process of covering an area with triangles and computing the co-ordinates of the nodal points. This could be made clearer.

I couldn't agree more.--www.doc 12:39, 3 April 2006 (UTC)

serious rewrite needed[edit]

Process or not, this page is obtuse to the point of being an obstruction to learning. How on earth this would qualify as an encyclopaedic entry is beyond me. It makes a hash of a simple concept and its application. Malangthon 10:54 SPT. 05/09/06

I really don't understand triangulation after reading this article. I think it needs some clarification, and I think that an example would really help in the explanation.

is this triangulation?[edit]

Hi. Would either of these be considered triangulation:

  1. Using "angular size" (how large an object looks from a point) and distance to determine size of object, or using "angular size" (how large an object looks from a point) and size of object to determine distance,
  2. Using parallax, from either two points or eyes to determine size of object or distance,
  3. Or using three points on a map, etc, to form a triangle, making smaller triangles within it by drawing points on the centres of the sides, and repeating the process to find the centre?

Thanks. AstroHurricane001(Talk+Contribs+Ubx) 01:02, 16 February 2007 (UTC)

Removed elementary from first line[edit]

I removed it because triangulation is usually not taught in elementary. That's why I removed the word.N734LQ 23:58, 29 June 2007 (UTC)

Where is the θ in the image?[edit]

The text refers to the angle θ, but the image on the right does not show where this angle is. Can someone create a new image with that angle? enderminh 16:25, 3 July 2007 (UTC)

The text explains it quite clearly: the third angle of the triangle. Two angles are marked α and β, and the triangle has only three angles. I hope that is enough to find out where the θ angle is.... --CiaPan 06:20, 4 July 2007 (UTC)
I know that. I just think that given that the text refers to it often, it would be nice to have an image that actually has that angle. It's just a suggestion. enderminh 08:07, 7 July 2007 (UTC)

Fair use rationale for Image:Triangulation mip.jpg[edit]

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BetacommandBot 00:28, 26 July 2007 (UTC)

Triangulation in chess[edit]

Out of curiosity, could we put a section about triangulation in chess endgames ? Seigneur101 (talk) 01:47, 26 March 2008 (UTC)

Ah well, never mind... :) Seigneur101 (talk) 12:39, 26 March 2008 (UTC)

History ?[edit]

Resolved: Section added.

The page could use a section laying out some history of how triangulation became adopted in surveying. Who first used or described the method? What equipment did they use? How did this understanding spread, and who took it up? Were there initial practical difficulties?

Google books found me this, in Donald Routledge Hill, A History of Engineering in Classical and Medieval Times(pp.119-122):

The word dioptra in Greek means any instrument for taking a line of sight. In a treatise on surveying written by Hero of Alexandria in about the middle of the first century AD, there is a description of a dioptra that is very similar to a modern theodolote apart from the absence of a telescope... Although this is a very ingenious instrument with obvious relevance in the history of surveying, it was probably not very robust, and would soon have gone out of alignment if subjected to the wear-and-tear of use on construction sites. For everyday use in measuring angles, a simple rotateable alidade provided with a scale of angles was probably the normal instrument...
A number of Arabic writers, including the great scientist al-Biruni (d. c.1050) and the Spanish Muslim Ibn al-Saffar (d. c.1035), describe the solution of various triangulation problems using the astrolabe...
Methods of triangulation were unknown to the Roman agromensores, and were introduced to Spain by astrolabic treatises such as that of Ibn al-Saffar. Although such methods were often omitted when the treatises were translated into Latin, an exception is the Geomatria incerti auctoris, a compilation of Hispano-Arabic inspiration which Millas Vallicrosa relates to the Arabised scientific corpus of the Monastery of Ripoli. Roman methods existed side by side, in Spain, with triangulation surveying. Simplified Roman procedures seem to have been used by individual farmers, while triangulation was carried out by professional surveyors -- Christian and Muslim -- employed by large landowners. In the later Middle Ages, triangulation must have become a more common procedure.

Also, the medieval Jacob's staff, used specifically for measuring angles, dates from about 1300.

On the other hand, Pei Xiu in China (224-271) appears not to include triangulation in his "six principles of map-making".

Can anyone provide more detailed information on the history of the uptake of triangulation in surveying? Jheald (talk) 10:33, 29 April 2008 (UTC)

More: according to a page at NOAA, "In the 16th and 17th centuries, triangulation started to be used widely" [1]; while according to this piece, "using the properties of triangles for land surveying did not become common until the 17th century AD... The first practical use of triangulation in mapping a country began in 1668, when Jean Picard and Jean Dominique Cassini, two French astronomers and mathematicians, began the first scientific survey of France." [2]. It would be nice to find a WP:RS that treated the whole history in depth. Jheald (talk) 17:50, 29 April 2008 (UTC)
On second thoughts, while this may be the first use of a triangulation network, use of triangulation itself to plot features is surely much older. An episode of the TV series the Map Man portrayed Christopher Saxton as using triangulations to fix points in his survey of Norfolk, circa 1570; and it was surely used much earlier than this, eg in the creation of medieval portolan sea charts. Plus of course Eratosthenes in effect used triangulation to estimate the radius of the earth. Jheald (talk) 09:32, 30 April 2008 (UTC)
"Important developments in cartography occurred in the 16th century in the Netherlands, Germany and Austria, where triangulation began to be applied to the measurement of land distances. (Bagrow 1964, Tooley & Bricker 1968, Crone 1978 [1953])" -- Michael Jones, "Tycho Brahe, Cartography and Landscape in 16th Century Scandinavia", in Hannes Palang (ed), European Rural Landscapes: Persistence and Change in a Globalising Environment (2004), p.210 [3] -- the paper discusses maps in the papers of Tycho Brahe of the Norweigian coast (1590, 1594) "showing remarkable accuracy for their period", and of the island of Hven (1584), "the first detailed local maps in Scandinavia based on systematic survey and triangulation".
"Tycho Brahe was familiar with the principles of trigonometry and triangulation (Dreyer 1963 [1890]; Christianson 2000), and had visited cartographers in Germany who were amongst the first to apply such principles to making maps (Thoren 1990; cf Tooley & Bricker 1965:35, Crone 1978 [1953]):60-61). In 1579 he had undertaken triangulations between the main landmarks of Hven and the surrounding landscape on both sides of the Øresund..."
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
-- Jheald (talk) 10:23, 30 April 2008 (UTC)
The key developments/publications appear to be
  • Gemma Frisius who "described for the first time the method of triangulation still used today in surveying" (1533).
  • Willebrord van Roijen Snell who first described a method for resectioning (1615), allowing a new point to be located given only the angles subtended between known points, rather than a compass bearing or an angle from a control point.
-- Jheald (talk) 12:23, 30 April 2008 (UTC)
[4] reviews the state of play in England in the 1570s. Systematic triangulation (rare) should be distinguished from a quick compass sketch from a single point; and from a more traditional open or closed traverse. Jheald (talk) 16:07, 30 April 2008 (UTC)
Should have gone to the wiki. The article on triangulation in geodesy is nice, with some history; while also has some nice articles including Triangulation, Resolution of a Triangle, and History of the Triangulation of France. Jheald (talk) 18:04, 29 April 2008 (UTC)

Right, I've now added something, as a start anyway, based mostly on the sources above. No doubt it could be improved, perhaps based on the "Further reading" sources, which I haven't had a chance to look through. Jheald (talk) 22:43, 21 September 2008 (UTC)

A source is needed for the assertion that Roman surveyors didn't use trigonometry. The latter introduction of trig into medieval Europe in no way indicates whether it was used prior to the collapse of the Empire. As chordal equivalents were already in existence in Greek it seems plausible that the info would have been available. — Preceding unsigned comment added by (talk) 16:53, 7 June 2012 (UTC)

See the quote from the book by Donald Routledge Hill at the top of this talk section, which is cited in the article. Jheald (talk) 18:47, 7 June 2012 (UTC)

Ambiguous Units of Angle in Calculations? (or explaination needed.)[edit]

Pardon my ignorance of math, etc. but when trying to understand the topic I referred to the article on Angle and was confused as to what units of angle were being used in the calculations in this Triangulation article. I assume the calcs here use radians, but after reading that grad or gon is used "mainly in triangulation" I'm now unsure. I think this article needs to explain the units of angle used here as well as angle unit conventions and their context in triangulation, and/or add a link to the Angle article. Thanks Jd4x4 (talk) 07:01, 25 May 2009 (UTC)

You could be using degrees, radians, grads. It doesn't matter. So long as you've told your calculator which units of angle you're using (or in olden days got out the appropriate book of trigonometry tables), the formulas are the same and you will get the same answer, whatever units you choose. Jheald (talk) 10:33, 25 May 2009 (UTC)
Doh! Thanks for not mentioning my ignorance. I was tired & have math-o-phobia?? (talk) 13:31, 25 May 2009 (UTC)

Triangulation to determine your location[edit]

I notice at present the article is very focused on determining the location of a ship, given two observers on the shore. Triangulation works just as well for one observer on the ship, observing two objects (say perhaps the lights from two ports), if he knows how far apart the objects are (perhaps measured from a map).

Given an observer can see two points (A and B) a known distance (D) apart (in a known direction that is not on the same line as the observer). The observer can narrow down their own location on a map to one of two points (C or C') if they can determine which way they are facing (using a compass or similar). If these points are along a coastline - then one of the two points C or C' will (usually) be in the sea and the other on the land.

Determining your location is more difficult if you are not at ground level - so if you happened to be in a hot air balloon (or on a mountain), you would either need a third reference point, or some measure of altitude to accurately determine your location along the line between C and C' (where the earth's surface cuts the circle of possible locations in 3D) —Preceding unsigned comment added by EdwardLane (talkcontribs) 10:17, 13 May 2011 (UTC)

You are right – however that is NOT a triangulation. Tri–angulation means using three angles (in fact two angles suffice, as the third one can be easily calculated). Of course one length is also necessary, because angles alone give us an information about the shape but not a scale of a figure.
The observer on a ship can not determine his position by triangulation, because he can measure only one angle – at the triangle's vertex where his ship is. That gives him a position line in a form of circular arc (that follows from the inscribed angle theorem, see Inscribed angle#Theorem). So the method which you describe requires some external information (eg. some direction measurement, ie. additional angle between a triangle's side and a magnetic meridian, measured by a magnetic compass, or a distance from the observer to A or B point) and this way it's no longer a triangulation. --CiaPan (talk) 08:44, 2 August 2013 (UTC)

Accuracy of the triangulation[edit]

Is accuracy of the triangulation is better than that of the Differential GPS. Can some one comment on this? --உமாபதி (talk) 13:14, 31 July 2013 (UTC)