Triangulation, Multiple Observers
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From: W. eWatson on 8 May 2010 11:29 Google doesn't seem to help on Subject. What math method would solve a triangulation of a point by two, three, or more observers of a point? Assume the observers are on a plane, work from a common x-y coordinate system, and make some error in their angular measurement. Would the same method be used if each observer provided one or more observations rather than only one?
From: Tim Little on 8 May 2010 12:18 On 2010-05-08, W. eWatson <wolftracks(a)invalid.com> wrote: > Google doesn't seem to help on Subject. What math method would solve > a triangulation of a point by two, three, or more observers of a > point? Assume the observers are on a plane, work from a common x-y > coordinate system, and make some error in their angular measurement. One simple method could be to choose the estimated postion so as to minimise some score of error in observations (e.g. sum of squares of angular error). If you want to get a little fancier, you could use a maximum likelihood method based on prior error distributions for each observer. - Tim
From: Greg Neill on 8 May 2010 12:55 W. eWatson wrote: > Google doesn't seem to help on Subject. What math method would solve a > triangulation of a point by two, three, or more observers of a point? > Assume the observers are on a plane, work from a common x-y coordinate > system, and make some error in their angular measurement. Would the same > method be used if each observer provided one or more observations rather > than only one? It's an exercise in trigonometry, and since there are measurement errors (and there always are in real life), there will also be an error analysis component to the data reduction. Is the observed point in 3-space or also on the plane? (one presumes you mean a geometric plane and not an airplane!). How you choose to reduce the observational data to a result may depend upon the geometry of the given circumstance: if each observervation post has different measurement error for its measurements then a data weighting scheme may be needed; or it may be that for a given point location some pairs of observers may be better positioned to give an accurate fix; After a preliminary estimate of the location the data may be sorted and weighted accordingly for the final reduction. The concept and basic mathematics of triangulation is readily located by google. For example: http://en.wikipedia.org/wiki/Triangulation How to go about analysing the data for multiple observations is probably a bit trickier to pin down, since much will depend upon the particulars of the setup. You might want to take a look for documents pertaining to surveying methods.
From: spudnik on 8 May 2010 15:49 the problem appears to be, "some observers measure the angle to the marker, relative to the other observers," which would not give you the distance *on a plane*, because of similar trigona. Gauss meaasured the curvature of Earth with his theodolite *and* a chain measure of distance (working for France in Alsace-Lorraine, triangulatin' that contested area .-) thus: notice that no-one bothered with the "proofs" that I've seen, and the statute of limitation is out on that, but, anyway, I think it must have been Scalia, not Kennedy, who changed his little, oligarchical "Federalist Society" mind. thus: sorry; I guess, it was Scalia who'd "mooted" a yea on WS-is-WS, but later came to d'Earl d'O. ... unless it was Breyer, as I may have read in an article about his retirement. > I know of at least three "proofs" that WS was WS, but > I recently found a text that really '"makes the case," > once and for all (but the Oxfordians, Rhodesian Scholars, and > others brainwashed by British Liberal Free Trade, > capNtrade e.g.). > what ever it says, Shapiro's last book is just a polemic; > his real "proof" is _1599_; > the fans of de Vere are hopelessly stuck-up -- > especially if they went to Harry Potter PS#1. > http://www.google.com/url?sa=D&q=http://entertainment.timesonline.co.... --Light: A History! http://wlym.com --Waxman's capNtrade#2 [*]: "Let the arbitrageurs raise the cost of your energy as much as They can ?!?" * His first such bill was in '91 under HW on NOx & SO2 viz acid rain; so?
From: W. eWatson on 8 May 2010 16:06
On 5/8/2010 9:55 AM, Greg Neill wrote: > W. eWatson wrote: >> Google doesn't seem to help on Subject. What math method would solve a >> triangulation of a point by two, three, or more observers of a point? >> Assume the observers are on a plane, work from a common x-y coordinate >> system, and make some error in their angular measurement. Would the same >> method be used if each observer provided one or more observations rather >> than only one? > > It's an exercise in trigonometry, and since there are > measurement errors (and there always are in real life), > there will also be an error analysis component to the > data reduction. > > Is the observed point in 3-space or also on the plane? > (one presumes you mean a geometric plane and not an > airplane!). > > How you choose to reduce the observational data to a > result may depend upon the geometry of the given > circumstance: if each observation post has different > measurement error for its measurements then a data > weighting scheme may be needed; or it may be that > for a given point location some pairs of observers may > be better positioned to give an accurate fix; After a > preliminary estimate of the location the data may be > sorted and weighted accordingly for the final reduction. > > The concept and basic mathematics of triangulation is > readily located by google. For example: > > http://en.wikipedia.org/wiki/Triangulation > > How to go about analysing the data for multiple > observations is probably a bit trickier to pin down, > since much will depend upon the particulars of the > setup. You might want to take a look for documents > pertaining to surveying methods. > > 2-D in this case (geometric), but, actually, the 3D case is more interesting to me, since it has applicability. I figured I'd start simply. I saw the link you provided before I posted, but it's a bit short on details. I would have guessed this problem is common among surveyors. As far as I can tell, there are no historic guides to the problem. Although Google shows surveying and robotic imaging (vision) as possible sources. Possibly, there's something in rocketry that applies. |