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Tienstra's Formula Revisited

Mesa Mike
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A few months back there was a thread ("Dolphin Has an Evil Idea") where 3-point resection was discussed. At that time, I created a puzzle geocache ("Help, I've Lost My Cache! [GC1B0Q9]") that required solving a resection problem in order to locate it, although the only two finders so far did it by other than intended methods.


The intended method, of course, was to use Tienstra's Formula, but online solvers for it are elusive -- one of them that is found has a bug and gives wrong results, and another only allows the input of whole degrees, apparently. So at that time I wrote a simple JavaScript solver to check my work. So far, it seems accurate.


It's been a while since all that transpired, and I wanted to hold off sharing it until there was a chance for the various geocaches resulting from that thread to get found fair and square by people who knew what they were doing. It's been long enough, I think.


So... here it is, if you'd like to check it out.

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Hi Mike:


Thanks! Will check it out ASAP.

Of course, I will take this opportunity to plug my "Tienstra Cache" also. It seems to have attracted a few local (and even non-local !) cachers, but interest has waned. I guess there is only a certain "type" that enjoy it. :P;)


Some of my finders have resorted to drawing lines on maps (which can work if done well). A few have found other mathematical ways to solve the problem due to a "hole" I left in my logic. One other found a unique non-Tienstra based solution (and found the cache) that I don't fully understand (yes, he was a math major!), but it works sometimes, but not always (huh? ;) ).

Edited by Klemmer & TeddyBearMama
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I'm having problems with your web calculator. I assume (?) it is designed for decimal degrees of Lat & Long? That does calculate a solution (UTM does not, I get error "NaN"), but the answer is not what I was expecting. It's off by good sized fractions of a degree. I used my cache and just stations LEMON (DX4720), SPRINGS (DX4264) and IRV 22 (DX4263).


I couldn't figure how to do a cut / paste of the results, or I would post them. Saving as an htm file didn't work. But I printed & scanned the result. I think I have your email address at home (in office today), and I'll email you that and also my excel spreadsheet with all the details of my calculations (which work fine, out to 3 decimals of UTM). BTW: I used NGS FORWARD for the BM to BM angles.


Looking at your code, it looks like you are using UTM (Easting amd Northing), so now I'm really confused....

More tonight or tomorrow. I've got to do some work here, darn....

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I assume your calculator is using plane coordinates. Thus you could input UTM values, or State Plane Coordinate values. In either case those are on a projection and for a large figure grid angles are not equal to the geodetic angles you would get if you were actually on the point turning them.


The turned angles can be corrected to grid by means of computing a thing called the T-t correction, however an approximate calculation of the position would have to be made to compute the correction, followed by small changes in the angles.


Anyway using SPC coordinates for the GC1B0Q9 resection problem the solution using plane coordinates is within a few 1/100's of a foot of the correct solution. When considering that the angles are given to 0.00005 of a degree which amounts to 0.2 seconds the solution has inherent error of a few tenths of a foot anyway. Thus the error in using uncorrected grid is pretty small in this case.


- jlw

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It dawned on me that when I cut & pasted the UTM coords from my Excel sheet, it might have took it with commas (not sure, but I know I didn't remove them). I'll re-do it & check tomorrow. I guess you should note on the site that the required format for the stations is UTM. Funny decimal degrees "worked" for me (it calculated, anyway). I guess it took them as if they were UTM, which wouldn't work well for lots of reasons.


BTW: I normally use the NGS UTM conversion program, and keep everything in UTM (not State Plane, if it even matters in a case like this).


So, using NGS UTM and Forward (which I believe yields geodetic angles?), I've had very good results on my excel sheet, with the calculations done in UTM & decimal degrees. I did notice in the process (seems like ages ago) that if I didn't carry the intermediate results out to about (I think) eight decimal points (e.g. BM to BM angles), the solution went astray noticeably.

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I used it and the coords were spot on using a Triangulation Station.

But the angles did not appear upon calculation.



Point A: 432360.586, 4054323.318


Point B: 432362.931, 4054282.837


Point C: 432414.465, 4054252.944


Result GF0907: 432362.931, 4054282.837


If you were to use the azimuths you could easily count the angles on the circle between the points giving you the degree or angle.

This circle could be refrenced to the many map projections.


This is at a short distance as well.

I know over about 156 miles for GPS baselines it says it tends to get distorted.

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It works! Sorry for the confusion. The problem was indeed the commas. Removed the commas, and it works for my cache setup just fine.


Your website results differ from my excel spreadsheet results in the second decimal place of UTM (meters), so we're talking centimeters, which is obviously fine. One or the other must do something slightly differently, which is no problem at all at these short ranges (2100 - 2400 meters, BM to BM). At longer ranges, who knows? I believe my method is completely geodetic (great circle), which might not even be "proper" use of Tienstra (?).


Anyway, I'm good with either method (for purposes of finding a cache or missing BM). For real survey purposes, well, I'll leave that to the professionals :ph34r: .


Great work, Mike. Nice contribution to the "hobby", or maybe even "profession"!

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Some various comments:


Resection is a nice mathematical problem that I'd like to dig into except that I already have several projects I'm working on.


If I were to want to solve this kind of problem I would probably put the data into least squares fit software. I use a Matlab script (just function minimizer search, not matrix solution) that seems to work. The advantage is that if I am taking data and get redundant measurements (repeated measurements at a point, or 4- or 5-point resection), it is easy to throw them into the least squares whereas an algebraic solution doesn't know what to do with the extra data.


I need to figure out Klemmer's geodetic approach. However, using UTM means you are already on a plane projection, so it isn't obvious to me how you incorporate Forward results. Converting back and forth between lat-lon and UTM?


Forward and Inverse work with geodetic azimuths, which will differ greatly from map projection azimuths. However, the angle defined by two lines (difference of two azimuths) will be close enough between map and geodetic, generally within the overall accuracy of UTM or SPC projections.


UTM and State Plane coordinates have many similarities. One difference is that the SPC projections are more accurate because their zones are smaller than UTM zones, resulting in less warping when converting the world to a flat map. This isn't important for finding a cache but explains why construction projects use SPC.

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Actually, I mis-wrote. I used INVERSE (not FORWARD), and that was just to setup the problem (equivalent to shooting the angles from the unknown point). After that, it was all UTM and angles. So it wasn't really geodetic, except the problem setup.


At the short distances I was dealing with, it would be a wash anyway, as Bill said.

Edited by Klemmer & TeddyBearMama
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Ah. We've had seven finds on Triangulation since April. One by a dear, sweet, little old lady using a walker! :unsure: The fact that it's a mystery cache scares off a lot of people. The math bewilders many who attempt it. I'm not sure that anyone has solved it using any complex program. Angles drawn on a map will get you close enough. Good golly! Walk down the Hudson River Walkway until you can see the intersection stations at about the right angle. :sad: The Hudson River Walkway is a public right-of-way built behind the condo complexes that have replaced the old piers from a bygone era. Not a lot of place to hide a cache! Oh, well.

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