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WikiProject Mathematics (Rated Start-class, Mid-importance)
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Mathematics rating:
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 Field: Geometry


No citation for the proposed method for triangulation is given. Is there an evaluation that the proposed method is "good" or even "optimal" in terms of computational efficiency and/or accuracy? Is there an evaluation of the numerical stability of the proposed algorithm? Are there any comparisons with other algorithms for triangulation? — Preceding unsigned comment added by (talk) 11:54, 16 February 2016 (UTC)


Just a minor point, shouldn't rather


be substituted in the third equation? (talk) 10:47, 19 November 2012 (UTC)

Since there were no objections, I will change the respective line. (talk) 13:50, 28 November 2012 (UTC)

Broken link?[edit]

The link "Efficient Solution and Performance Analysis of 3-D Position Estimation by Trilateration" in the External Links section gives me a 404 error. daviddoria (talk) 14:34, 12 February 2013 (UTC)

As of 30 May 2014 the link points to a file hoster that shows porn when cklicking the download button. The link should probably be removed.--Jmknaup (talk) 11:28, 30 May 2014 (UTC) link (reference number 3) is dead, someone should replace it with something suitable. Alenrajsp (talk) 12:31, 22 July 2016 (UTC)

Wrong generalisation that 3/4 spheres are enough?[edit]

The article states that:

"If it is known that the point lies on the surface of a fourth sphere then knowledge of this sphere's center along with its radius is sufficient to determine the one unique location."

I'm fairly sure that this need not always be true. If you have found 2 points, that all lie on the first three spheres, then there still is an infinite number of other spheres, that also have both points on their surface. Knowing the position and radius of one of these spheres would not help to discriminate between one of the two points. To visualize: take any 4th sphere and rotate it around the axis that is defined by the two points. All spheres that are thus generated do not provide any further information. So, a 4th sphere CAN help, but it need NOT ALWAYS be enough. Or, even: there is no number N such that N spheres are guaranteed to define a single point in space via the intersection of their surfaces. One needs additional constraints as to how the spheres are positioned relative to each other. At the very least the sentence in question should be weakened to something like "is usually sufficient", if that is the case in real-world-scenarios (e.g. GPS). — Preceding unsigned comment added by (talk) 15:21, 18 June 2013 (UTC)

When I think about it, even the statement about 3 spheres narrowing down the position to 2 points in 3D, is not correct:

"In three-dimensional geometry, when it is known that a point lies on three surfaces such as the surfaces of three spheres then the centers of the three spheres along with their radii provide sufficient information to narrow the possible locations down to no more than two."

There is actually an infinity number of different spheres in 3D that share a common "boundary-intersection-circle" (their centers lie on a straight line). I will delete the whole paragraph, because it is factually wrong and I'm not sure what it is supposed to say. -- (talk) 12:41, 15 July 2013 (UTC)

The generalization holds as long as no more than 2 reference points are colinear and no more than 3 are coplanar.--Jmknaup (talk) 11:55, 30 May 2014 (UTC)

Statistical Lateration[edit]

This has probably been addressed before, but I think it is worth reconsidering: the statistical (i.e. in presence of noise) use of (tri)lateration should be emphasized and I think it will greatly improve the article. The current "multilateration" page is essentially a TDOA article (a bit too much, in fact). But the problem of lateration is slightly more abstract (it doesn't concern itself from where the measurements come from, in fact it is also heuristically used in RSS-based localization). I am no expert on the field, thus I can't write about it, but I think the general problem should (at least) be cited. In its statistical usage, "tri" (in the word "trilateration") is omitted, since you usually have nodes (centers of the spheres), with sufficiently high, and the radii are known within a plus or minus "delta" (note that, inside this delta, there are also included the uncertainties in the coordinates of the centers, leaving the centers error-free; in other words, all the errors are put in the radii's uncertainties, simplifying the notation). Note that the , with , are generally different (it cannot be assumed for all ). The "spheres" (spherical shells: , ) typically intersects in a region (the "grey zone" or something like that) and then, for example, the centroid (center of mass) is taken. Obviously, the algorithm discards the "spheres" which do not intersect at all. Can someone with more expertize write about this or is there already an article (except multilateration/TDOA) regarding these topics? — Preceding unsigned comment added by (talk) 13:57, 2 April 2014 (UTC)