Talk:Geodesics on an ellipsoid/Archive 2

CLEANUP. This article is a mess
I intend to clean this article to where it can be deveoped to GA status.

Compare this to other Mathematics GA's like Directed acyclic graph or Final stellation of the icosahedron Inlinetext (talk) 14:18, 17 January 2017 (UTC)


 * Inlinetext, I assume that you are still in the process of revising this article; so I'll hold off on making changes for now. However, I recommend restoring the figures showing geodesics on biaxial and triaxial ellipsoids.  These are essential to give readers new to the subject a good idea of what the article is about.  Also, you've dismissed the section "Equations for a geodesic" with the statement "Blatant copy violation combining plagiarisms from Rapp (1992) and Borre/Strang (2012)".  I was the editor responsible for this section; I'm curious to see the examples you found of copying and plagiarism. cffk (talk) 15:05, 18 January 2017 (UTC)


 * Considering this is an existing article, I am still evaluating aspects like the audience to be addressed (with particular emphasis on the general audience), the structure, layout and voice required by WP:MSM specifically MOS:MATH. As previous reviewers have noted, the parent article on Geodesy takes a different approach (treatment) which is pretty concise and effective, and does not specifically touch the biaxial or triaxial elliposid issues. Hence, can I know the source of the figures and equations which were deleted ? As there are so many approaches to realising equations for a geodesic, the alternatives would imply that your approach / results are either original research or a hoax. Inlinetext (talk) 16:29, 18 January 2017 (UTC)


 * Inlinetext, it might be a good idea to refrain from using terms like "plagiarism" and "hoax" unless you have some concrete evidence that they are warranted. The material from "Equations for a geodesic" follows the derivation of Bessel (1825) fairly closely (I was one of the translators of the English version of this paper).  I generated the figures.  If you click on a figure you will find its provenance described; in the case of figures showing geodesics, this specifies the ellipsoid parameters, the starting point and length of the geodesic, and the viewing direction. cffk (talk) 18:23, 18 January 2017 (UTC)


 * Considering that you are essentially the author of this article, see WP:WPM Identifying pseudo-mathematics, hoaxing, and other unattributable material. Most Wikipedians, naturally enough, don't feel qualified to pronounce on articles purporting to be mathematical, so it is up to us to identify crankery, hoaxes, and material with dubious or unverifiable sources. I bring this up, reluctantly, since I (as have others) notice a consistent pattern of your treating this article as some kind of research paper / research brief. I put it to you directly that none or few of the connected articles in the infobox are as verbose as this one. Most Wikipedia articles on Ellipsoids, Spheriods and Geodesics are within 20,000 bytes in text size. Of course the equations are now much improved since Bessel's times wouldn't you say, but nowadays rendered increasingly inrrelevant by numerical / alternative computational methods?  Perhaps Vincenty (A) and (B) would be a better starting point than Bessel considering the standard approach's well known shortcomings for certain situations. That neither of Vincenty's seminal papers contain figures is an advantage. Inlinetext (talk) 19:20, 18 January 2017 (UTC)


 * You will note that I have not objected to your shortening the article. I merely expressed concern that you were tossing out too much. No, Bessel's approach is still the standard one.  It's not irrelevant in the age of modern computers; indeed, extending the series the Bessel gives, it allows geodesics to be computed rapidly to full precision.  Vincenty adopts Bessel's method (via Helmert and Rainsford) but leaves out all the derivations.  Vincenty's paper is just a quick cook-book recipe providing little understanding of the problem; this quality of the paper is illustrated by http://www.movable-type.co.uk/scripts/latlong-vincenty.html "I discovered to my surprise that while the mathematics is utterly beyond me, it is actually quite simple to program."  The Wikipedia article needs to do better than this; and, as an aid to understanding, figures are crucial. cffk (talk) 20:44, 18 January 2017 (UTC)


 * Actually the link cited by you shows that a person relatively unfamiliar with mathematics could understand Vincenty's formulae directly (without diagrams) and program it simply. This implies that Vincenty's approch is the correct one for Wikipedia's audience and only the bare formulae are sufficient. Where your version of the article fails is the attempt to derive formulae based on your personal assumptions, approaches and diagrams. This is strongly deprecated However, an article that "speaks" to the reader runs counter to the ideal encyclopedic tone of most Wikipedia articles. Article authors should avoid referring to "we" or addressing the reader directly. Since this article is way too long for a Good Article, it will first have to be trimmed to around 20,000 bytes, sections can be moved later into subpages. The References section will also be vastly reduced. I suggest Spherical trigonometry as the model article we use for this purpose. Inlinetext (talk) 06:31, 19 January 2017 (UTC)


 * Just to be clear... my original post here was to find out if there was any substance to the claim of plagiarism. I still don't know!  You asked what the section in question was based on, and I replied Bessel.  Then you went off on a tangent about whether Bessel is the best resource for deriving the equations for a geodesic.  Please feel free to substitute a better derivation!  However, if the first sentence of "solution of the direct problem" refers to the "auxiliary sphere", readers should know what this means. cffk (talk) 21:29, 18 January 2017 (UTC)


 * Since you subsequently informed me on this page that "Cffk" is Karney CFF (which was not known to me at the time I removed that text), it becomes obvious that your derivations are not strictly Bessel's original equations and include apparently your original researches, eg. Algorithms for geodesics heavily based on Rapp's equations (or via Helmert), and therefore not suitable for inclusion on Wikipedia until they are widely accepted and validated by your peers. Your text curiously did not explicitly cite your own work as the source but instead misleadingly implied these being derived in Rapp 1993, Borre, Strang 2012 et al which are presumably covered by copyright. Had you explcitly cited your own work(s) the claim of plagiarism would obviously not have been made by me. IMO the derivations can be avoided here and the interested reader referred to accepted works, including those which take the (simpler) Reimennian approach for their derivations. I have not attempted to tamper with your text, so when large sections of text are removed in the editorial process a few sentences to maintain continuity would need to be inserted back, as would also inline equation numbers being updated. I suggest that our common approach to improving the article should be to further trim it to half its present size ie. between 20K to 30K bytes Inlinetext (talk) 06:31, 19 January 2017 (UTC)

I see that the article has been reverted to its state prior to Inlinetext's deletions. Over the next week or so, I will look into trimming some of the dead wood in the article. One of Inlinetext's deletions that I may adopt is the removal of the portraits. These were introduced to satisfy an earlier reviewer. However, I agree with Inlinetext, that they are distracting. Comments? cffk (talk) 13:13, 30 January 2017 (UTC)


 * Removing those portraits would be fine with me. But don't do it because of Inlinetext.  Jrheller1 (talk) 15:49, 30 January 2017 (UTC)


 * I have reverted you. This article has sourcing issues. There are also copyright issues over the derivations which CFFK had discussed with me on my talk page. I also strongly object to your reinstating CFFK's earlier usage of this article for self promotion, which I had removed.Inlinetext (talk) 03:00, 31 January 2017 (UTC)


 * I have removed a lot of material. I think more can be removed. A whole bunch of refs are excessive. Wikiedia is not a collection of links. Yes there are gaps which need to be filled to maintain contunuity, and I have offered to collaborate to improve the article. In my view, much of ths article can be pushed off to other pages. Inlinetext (talk) 03:17, 31 January 2017 (UTC)

Copyright Violations / Plagiarism / Original Research
CFFK claims that the derivations (which I feel can be dropped) are from his co-authored work 10.1002/asna.201011352. Since this work significantly differs from Bessel's and is not a simple translation, the copyright status of CFFK's co-authored work has to be established before these derivatuions can be reproduced. Secondly there are multiple approaches to deriving these equations, and we would need consensus on which approach to adopt, or if these equations should be here in the first place. Also, CFFK has bias (eg. Vincenty) which has crept into his version and which needs to be NPOVed. Inlinetext (talk) 03:34, 31 January 2017 (UTC)


 * Inlinetext, you've put a copyright violation notice on this article without completing the necessary steps for starting a review. Please either add the required notice on Copyright_problems/2017 February 1 (so I can respond) or remove the violation notice.  I must say, I cannot understand why you put the notice there; the links to the "Copyvios report" says Violation Unlikely and only flags the titles of a couple of journal articles. cffk (talk) 12:47, 1 February 2017 (UTC)


 * To concerned Admins, the URL link in the copyvio template is to the reference cited by Cffk and not directly to the infringing work which is behind a paywall. So the dup detector tool will not work here Inlinetext (talk) 19:02, 1 February 2017 (UTC)


 * Inlinetext, I'm still waiting for you to show any evidence of copyright violation. The reference to Bessel (1825) in the article includes a link to a preprint of the translation_arxiv:0908.1824. So the fact that Astronomische Nachricten requires a subscription should not have stopped you from backing up your claims. Incidentally, you seem to have latched onto my statement that I "followed the derivation of Bessel (1825) fairly closely" to mean that I violated the copyright on the translation of that paper.  This isn't the case.  I should hardly have to remark that the equations in a scientific paper would rarely be considered copyrightable since there is little latitude in how an equation is "expressed".  In any case, copyright would not apply to any equations appearing in Bessel's original paper (because of its age). cffk (talk) 17:44, 1 February 2017 (UTC)

I am not saying you violated Bessel's copyrights (which are expired by age) by your translation, but say that from 2009- onwards you created new copyrighted works whose copyrights are claimed as being held by the publishers and therefore cannot be incorporated in Wikipedia without their consent. for eg. kindly look at '1.1 Equations for a geodesic'. You claim on-article that these equations are derived from Bessel(1825) and you link to your own "translation" of Bessel (1825) "Here the equations for a geodesic are developed; these allow the geodesics of any length to be computed accurately. The following derivation closely follows that of Bessel (1825)". Of course Bessel's orginal work does not contain the following equations/assumptiuons which you have included "Consider an ellipsoid of revolution with equatorial radius a and polar semi-axis b. Define the flattening f = (a − b)/a, the eccentricity e2 = f(2 − f), and the second eccentricity e′ = e/(1 − f). (In most applications in geodesy, the ellipsoid is taken to be oblate, a > b; however, the theory applies without change to prolate ellipsoids, a < b, in which case f, e2, and e′2 are negative.)". These are your additions to Bessel under the guise of "The mathematical notation has been updated to conform to current conventions and, in a few places, the equations have been rearranged for clarity. Several errors have been corrected, a figure has been included, and the tables have been recomputed.". Some of these presentations (ie. 'expressions') of the aforementioned assuming equations seem to be from Kai Borre (2001) or his predecessors and would still be under copyright since the derivation you incorporated is not exactly from Bessel's original paper, so perhaps you will clarify where you got these on-article 'expressions' from (assuming of course that these are not your own Original Research) for these derivations. I have linked the copyvio template to the citation you provided for your own work (copyright with the publisher), especially since all the other sources for these derivations you have cited are probably still within copyright (Bagratuni (1962, §15), Gan'shin (1967, Chap. 5), Krakiwsky & Thomson (1974, §4), Rapp (1993, §1.2), and Borre & Strang (2012)). In passing, can you clarify what is meant by "Lee (2011) has compared 17 methods for solving the inverse problem against the method given by Karney (2013)" as there seems to be an anachronism here.Inlinetext (talk) 18:54, 1 February 2017 (UTC)
 * See strikeout above. Because I have taken the stand in the (harassing and frivolous) SPI case that the SPI was filed to pressure my edits to this article, I will wait for the SPI to conclude before either replying to you here or filing the review notice for you to respond. Inlinetext (talk) 19:16, 1 February 2017 (UTC)

Edits in progress 2017-02-04 + 2017-02-05
I'll be doing some house-cleaning on this article over the next two days. I'd appreciate it if other editors could hold off on modifying the article in the meantime. cffk (talk) 16:22, 4 February 2017 (UTC)


 * I don't believe you are the best person for this. Nonetheless, to allow you to fix your own mistakes and errors, I shall respect your request. I observe though that while Helmert mentioned he avoids the methods of Calculus of variations (and Lagrangians which follow), you use it here. I also suggest that you carefully consider the target readers and the principles contained in WP:AS. Inlinetext (talk) 07:54, 5 February 2017 (UTC)


 * Bessel (1825) &sect;2 uses the calculus of variations; you can check his original paper at http://adsabs.harvard.edu/full/1825AN......4..241B so there's no need to rely on my translation. Reading further in Bessel's paper you will see that the derivation given in the article's "Equations for a geodesic" does indeed follow Bessel.  According to Prosesize the "readable prose size" is currently 34 kB.  This is within the range (&lt; 40 kB) given in WP:AS where "length alone does not justify division." cffk (talk) 21:04, 5 February 2017 (UTC)


 * Now that the pruning is taking place in right earnest, perhaps you should evaluate if 1.7 Envelope of geodesics, 1.8 Area of a geodesic polygon and 2.3 Survey of triaxial geodesics, are essential to the present article. I opine these can be safely excised via a sentence or two in text.Inlinetext (talk) 13:29, 5 February 2017 (UTC)

I'm finished with my edit. Thanks to everyone for their patience. I purged the section on projections; this did seem tangential to the subject. In addition I got rid of many discursive remarks. I've addressed some of the Inlinetext's criticisms:
 * 1) The article is now at 34 kB of prose, within the "standard" range for articles (between 1 kB and 40 kB).
 * 2) The references to my own work are purely to my J. Geod.  article.  So this doesn't count as "original research".  The external link to GeographicLib is allowed because this is the implementation advertized in J. Geod.  I include a link a page I wrote on geodesics on a triaxial ellipsoid.  But this was just me organizing my notes on this topic; there's nothing original here.
 * 3) I'm more even handed in my treatment of alternative approaches (Vincenty, numerical integration).
 * 4) I've removed the portraits of the mathematians.

I still don't know what to make of Inlinetext's accusations of plagiarism and copyright violation. Inlinetext, perhaps you could evaluate the article in its current state and bring any such problems that you see to my attention. However, I can't do anything unless you provide specific instances!

I've retained
 * 1) The figures showing geodesics and the section "survey of triaxial geodesics".  These aren't original research.  Even though they take some skill and patience to prepare, they are obtained merely by computing a geodesic and projecting the result orthographically.  They are crucial to understanding the problem.
 * 2) The section "envelope of geodesics".  This gives important properties of geodesics, e.g., the number of geodesics paths, the stability of geodesics.
 * 3) The section "area of a geodesic polygon".  This is important in GIS applications.

Let me know of any inconsistencies that have crept into the article (unreferenced citations, etc.). cffk (talk) 23:27, 5 February 2017 (UTC)

Continuing discussion on the scope

 * I wish to record that your present edit does not have my Consensus, I disagree with the "Bloat" this article has due to an article ownership block. I still disagree with the approaches selectively chosen by you to stay within your comfort zone. I disagree with the text-booky / lecturing approach you have adopted and the consequent over-emphasis on formulae along with images you have generated and which images constitute 85% of this page's load content. Now, I strongly urge you to use a webpage analysis tool like pingdom to isolate why your version of the article is especially horrendous to navigate on a mobile device compared to say Spherical trigonometry (which itself is not a Good Article). After extensive analysis based on the several relevant principles of WP:AS (not only the prose size) I suggest the following to be done immediately for this page. (1) Remove 1.1 Equations for a geodesic, 1.2 Behavior of geodesics, 1.3 Evaluation of the integrals (2) Remove 1.6 Differential properties of geodesics, 1.7 Envelope of geodesics, 1.8 Area of a geodesic polygon (3) Remove 2.3 Survey of triaxial geodesics. (4) Trim the lead section by 50% (5) Remove many of the older references and add in some more recent ones (eg. Pittman's method, geodesics in Minkowski). After this 'rough cut' we can mutually review the article and insert the minimal bridging text, formulae and images to cement this article together. Inlinetext (talk) 05:36, 6 February 2017 (UTC)


 * Just to refresh you. WP:CONACHIEVE states A consensus decision takes into account all of the proper concerns raised. Ideally, it arrives with an absence of objections, Inlinetext (talk) 06:43, 7 February 2017 (UTC)

Some points in response to Inlinetext's suggestions: cffk (talk) 15:46, 7 February 2017 (UTC)
 * 1) I tried pingdom and it didn't tell me that anything was particularly awry.  I also borrowed a smart phone and tried loading the article on this.  It looked fine; "horrendous" was not a word that sprang to my mind.
 * 2) I'm not sure what you're referring to with my "comfort zone".  I do have a notion of the comfort zone for potential readers.  For that reason some explanation of the problem and how it is solved is important.  In addition...
 * 3) The figures are crucial.  To the extent that they impact load times on mobile devices, it would be best for the problem to be addressed in a systematic way by Wikipedia (e.g., by allowing figures to be loaded only on demand).
 * 4) If you believe that the emphasis of the article should be narrowed to focus on Vincenty's algorithm, perhaps you should be seeking to improve Vincenty's formulae.
 * 5) However this article is not focused exclusively on geodetic applications (this is not how this article started).  For example, it addresses a problem posed by a scientist at the European Molecular Biology Laboratory of finding geodesic distances on a molecule modeled by a prolate ellipsoid with equatorial radius 90 and polar semi-axis 254 (I'm not sure about the units here, probably nm).  Similarly, the envelope of geodesics gives the geometry of caustics formed by the propagation of signals on the surface of an ellipsoid.  And so on.
 * 6) It's clear that our starting points are completely different.  Yours seems to be that the article is irredeemibly broken.  Mine is that article is basically OK.  Asides from the contentious GA review three years ago (by a reviewer who was subsequently banned), there have been no major complaints registered on the talk page until you came along with "This article is a mess".
 * 7) For this reason, I don't think a strategy that begins by deleting more than half the article makes sense.


 * I object to all of the above. Since this disagreement may need to be taken to a larger lay audience, ie. the Wikipedia commuity, can you unambigously clarify the following statement in the article text. Here the equations for a geodesic are developed; these allow the geodesics of any length to be computed accurately. The following derivation closely follows that of Bessel (1825) with respect to a) Who has developed these equations ? b) Who has evaluated that these equations developed allow the geodesics of any length to be computed ? c) Who has evaluated that the computations based on these equations are "accurate" for any length ? d) How do you say that these derivation closely follows Bessel's 1825 paper ? I am compelled to be cautious considering your first edit to this page sometime after this 2011 rebuke - "Cffk; It's best not to promote your own work this way; let's take it to talk page". Inlinetext (talk) 01:42, 8 February 2017 (UTC)

Inlinetext, in answer to your question about the passage "Here the equations for a geodesic are developed; these allow the geodesics of any length to be computed accurately. The following derivation closely follows that of Bessel (1825)."

(a) Who has developed these equations? Legendre, Oriani, Bessel, Helmert during the course of the 19th century. See also the citations I give to textbooks in the article.

(b) Who has evaluated that these equations developed allow the geodesics of any length to be computed? It would have been obvious to all of the authors above (it results from the basic convergence properties of the series). Bessel makes it explicit in the last paragraph of his 1825 paper.

(c) Who has evaluated that the computations based on these equations are "accurate" for any length? Again Bessel addresses this issue from a computational perspective. But the result is a consequence of elementary numerical analysis. If you want to get hard numbers then you can use GeographicLib compiled with MPFR and specify, for example, 2000 bits of precision, to obtain accurate results for distances up to 10100 meters and compare these to the results using double precision. The result is as expected from the analysis: the error in the position is a few times eps * max(a, s), where eps = 2&minus;52 for double precision.

(d) How do you say that these derivation closely follows Bessel's 1825 paper? By reading the paper carefully (but I'm not sure I understand your question).

I'm not sure what your principal concern here is. Surely, you could have answered some of these questions by looking up the citations? Are you particularly worried about the assertion "these allow the geodesics of any length to be computed accurately"? This is true. However in Bessel's case, he was contrasting his method with earlier methods which involved assuming that the distance is small. I don't believe this phrase needs to be in the article (especially since "accurate" isn't defined), so I've removed it.

The "first edit" was in July 2012 when the page was in User:Cffk/sandbox. It was just me figuring out how to make Harvard citations work. The page didn't become public until August 2013. There's nothing to see here!

The February 2011 "rebuke" was more of a slap on the wrist and was the result of me referencing my work before it was published in the Journal of Geodesy (in June 2012). There's nothing much to see here either!

cffk (talk) 23:56, 8 February 2017 (UTC)


 * You are not answering my queries. Let me rephrase them.


 * 1) Kindly cite the source or sources for the exact expressions (without modification or synthesis) you have used in the article text. I would appreciate exact citations (ie. page numbers and equation /line numbers) from authoritative independent sources for every step in your summarisation of existing results.
 * 2) You cannot ask me or any other editor to go and self-verify these allegedly "obvious" results. I am asking you to provide me verifiable independent reliable sources which attest that these equations (ie. the identical equations used in the article) can be used accurately for a geodesic of any length. Bessel cannot be used in support of himself.
 * 3) I have read Bessel's 1825 paper carefully. I cannot see the equations or steps, which you claim are Bessel's, over there.
 * 4) I will not dignify your explanation for the rebuke with a reply, except to mention that your rebuker's user page appears compellingly similar to the Good Article reviewer's in certain details. Inlinetext (talk) 04:43, 9 February 2017 (UTC)

Inlinetext, we seem to be talking past each other. Part of the reason is that I'm never quite sure what's really bugging you. So perhaps you could help me understand the reasons behind your questions about the section "Equations for a geodesic". Looking back over this talk page and the SPI case, I divine a few possible explanations for your concerns:


 * 1) The material is plagiarized.
 * 2) It violates the copyrights of Wiley, Teubner, or the ACIC.
 * 3) It constitutes original research.
 * 4) The final equations for distance and longitude (4) and (5) are wrong.
 * 5) The equations are right but not accurate for all distances.
 * 6) You don't understand the derivation.
 * 7) The derivation is correct, but is unnecessary.
 * 8) I'm a charlatan and everything I've authored is suspect.

It would help me if you repudiate any of these reasons that don't apply and add any that I have missed!

I regard (4) and (5) as well established results, derivations of which are given by authors stretching from Legendre (1806) thru Rapp (1993) and beyond. There are slight but inconsequential variations in how the equations are written. Thus Rapp (1993), Eq. (1.28), gives (4) in pretty much this form; but a substitution is needed to go from Rapp's Eq. (1.46) to (5). The derivation itself is uncontroversial. Here again there are variations in how this is presented and undoubtedly there are better and worse ways of doing this. In your interrogatories, you imply that any deviation from published sources is disallowed. But this is nonsense. Small modifications in a derivation in Wikipedia do not constitute original research and are often needed to ensure consistency with the rest of the article or to simplify the steps. My view is that this derivation is essential to readers: it introduces the auxiliary sphere which is key to solving the problem and reduces the problem to quadrature. There's always room for improvement: perhaps the section could be shortened (for example, by appealing to conservation of angular momentum and thereby starting with the Clairaut equation); possibly some steps are unclear and some readers would be helped by more explanation.

cffk (talk) 15:52, 11 February 2017 (UTC)


 * Comment In the spirit of WP:AGF, I am focussing on the last comment you made, "perhaps this (article) could be shortened". How do you propose to do this so that the present article is reduced to a stable version (fully compliant with Wikipedia's educative information policies) and has a prose content of, say, 15K bytes (which seems to be just about the Wikipedia norm for "good" mathematics 2nd tier article topics like this one)?Inlinetext (talk) 17:51, 11 February 2017 (UTC)


 * I have no plans to undertake such a massive edit to the article (which would represent excising 55% of the text). However, I see that in this ANI page, you now say Geodesics on an ellipsoid is also stable and there have been significant cooperative improvements on the article after extensive talk page discussion.  So perhaps you no longer think that draconian cuts are necessary? cffk (talk) 17:06, 12 February 2017 (UTC)


 * My remark that this article is "stable" is in the context of attacks/slurs raised by 'Jrheller1' at ANI alleging disruptive editing / edit-warring. The article is stll a work-in-progress (mainly because you are unwilling to reveal the source(s) for the derivations and equations). NB: If you are unable to carry out the substantial edits required to this article to render it readable (and also useful) to the average reader (it does have a poor Flesch score), I could easily do most of it for you. Inlinetext (talk) 17:59, 12 February 2017 (UTC)

Suggestions (2)
It seems there is disagreement between two editors, so I thought I'd offer a third opinion. I'll make each statement in a separate line so that you can agree or disagree with specific topics.
 * Thanks for these suggestions. I've interleaved my responses setting them off with italics. cffk (talk) 16:46, 12 February 2017 (UTC)
 * I shall be replying in leisurely and piecemeal fashion to all this.
 * * It seems both of you have seem to have overlooked that Wikipedia is not a platform for scholarly research (or rather the kind of pseudo-research which Charles has indulged in here to promote himself and his algorithms). I recall, vaguely, that Tseng(2015) had something to say about Charle's algorithms (and not very in a very nce way) and I shall have to look it up on Tuesday before responding further. The arguments being raised in this opinion are in the nature of Strawman arguments and designed to distract from the main areas of this dispute (a) Size, audience and comprehension level for this article, (b) Involvement of a highly conflicted editor like 'Cffk' (c) Integration of this article with the rest of the Geodesy series (scheme of things) (d) Proper sourcing (e) Good Article criteria (f) Article Ownership etc. Inlinetext (talk) 19:01, 12 February 2017 (UTC)


 * Vincenty's approach is still widely used in undergraduate teaching in surveying engineering and geomatics engineering.
 * Yes Vincenty is still widely used in teaching. However, the trend is away from covering the subject in much depth at all.  I believe that Jekeli now holds the same position that Rapp did at Ohio State University.  But whereas Rapp (1993) gives a thorough treatment of Vincenty's approach, Jekeli (2012) does not mention him at all.  The reason for this trend is that in geodetic applications, the ability of computers to do full 3d adjustments (a conceptually simpler but nevertheless larger problem) obviates the need to solve the 2d ellipsoidal problem; and Vincenty spearheaded this change (Vincenty and Bowring, 1978)!
 * I wil respond to this later.Inlinetext (talk)
 * I agree that horizontal or planimetric geodetic network adjustment has been phased out in favor of 3D Cartesian coordinate networks (thanks to GPS). But these historical methods still deserve a more detailed description in this article. fgnievinski (talk) 02:22, 14 February 2017 (UTC)
 * Vincenty (1975) remains the single most cited reference on the topic so it deserves a thorough description here. fgnievinski (talk) 02:22, 14 February 2017 (UTC)


 * Karney's approach seems superior but its superiority has not been widely attested by third-party sources, likely because it's still relatively new.
 * The claims of "superiority" made in the article are (1) that it is more accurate, and (2) that the solution to the inverse inverse problem is always found. This is backed up by the tests documented in Karney (2015).  These rely on an extensive test set that I generated and published.  The algorithms have stood up to the test of time since I made the algorithms available (in 2009).  According to Google Scholar, Karney (2013) now has 58 citations; none has questioned this conclusion.  Lee (2011) uses my algorithms against which to test the others.  The U.S. National Geodetic Survey offers a quasi-endorsement for it as a possible replacement for Vincenty here.  They have been incorporated into PROJ.4, a widely used library within the GIS community, and exposed by the geod utility.
 * "Stood the test of time" ??? Vincenty's algorithms have stood the test of time being validated even after 30 years by Thomas/Featherstone. Pitmann's algorithms are still validated after 25 years by 'Deakin/Hunter'. who also validated and derived Bessel's equations in a somewhat different way than Charles does here. What does Karney have to show for himself except his own Karney(2015). The UNGS does not endorse Karney. It recommends Vincenty and is based on his formulae. Tseng(2015) had deprecated Karney (I will cite it after I access it again). In view of the statements made just above mine by Charles, I strongly urge him to back away from this article due to competence being required and a sheer inability to recognise his own bias inasmuch as what is good for Charles Karney is not good for this project. ..COI editing is strongly discouraged on Wikipedia. It undermines public confidence, and it risks causing public embarrassment to the individuals being promoted. Editors with a COI cannot know whether or how much it has influenced their editing.. Inlinetext (talk) 19:01, 12 February 2017 (UTC)


 * COMMENT (Tseng (2015a)- An Algorithm for the Inverse Solution of Geodesic Sailing without Auxiliary Sphere) -"Karney (2013) uses Newton's method, requires a good starting guess and uses more complex procedures which even involves firstly calculating the reduced length of the geodesic and solving a fourth-order polynomial merely to find a good starting point, and may involve a high computational cost". Notably, (Tseng(2015b)) - THE GEOMETRIC ALGORITHM OF INVERSE AND DIRECT PROBLEMS WITH AN AREA SOLUTION  FOR THE GREAT ELLIPTIC ARCS does not reference Karney in its citation listing. Inlinetext (talk) 03:59, 13 February 2017 (UTC)
 * I think there are additional implicit claims of "superiority": (3) that it is simpler or faster; and (4) that it is widely used. fgnievinski (talk) 02:22, 14 February 2017 (UTC)
 * So we have one reference (Lee, 2011) supporting claim #1 and another reference (Tseng, 2015) opposing claim #3. Both should be cited. Claim #2 is concerned with pathological cases which might not be commonly found in practice. Claim #4 is difficult to quantify and should not be made in this article without evidence. fgnievinski (talk) 02:22, 14 February 2017 (UTC)
 * Comment : This is an article about 'Geodesics of ellipsoids' and not the Karney-2011/13/15 algorithms.Inlinetext (talk) 08:50, 14 February 2017 (UTC)


 * In fact, it is uncited in recent editions of classic textbooks such as Leick et al. (2015; sec. 4.5.2, "Direct and Inverse Solutions on the Ellipsoid" and App. B, "The Ellipsoid"), not to mention those published just prior to Karney (2013) but still current, e.g., Torge (2012; sec. 6.3.3, "Computations on the ellipsoid") and Jekeli (2012; sec. 2.1.3.3, "Geodesics", and 2.1.4, "Direct / Inverse Problems").
 * I am not sure what this is in reference to.Inlinetext (talk) 19:01, 12 February 2017 (UTC)
 * I was saying that Karney is not widely endorsed in the current geodesy and surveying teaching literature. fgnievinski (talk) 02:22,

14 February 2017 (UTC)
 * I agree. Inlinetext (talk) 08:50, 14 February 2017 (UTC)


 * This article emphasizes the rigorous solution in the general case of very long geodesics, when in fact the special case of short geodesics is both more common (specially in triangulation networks) and warrants simplified expressions (which remain sufficiently accurate for practical purposes); these are mentioned only in passing at the end of Geographical distance and would deserve a section with equations in this article.
 * I debated adding a section on approximate methods (including those dealing specifically with short lines). I think including this material is a mistake because: (1) It's too specific to the needs of one community.  (2) Some methods are already covered (especially, Bowring's method) in the article on geographical distance.  (3) Even though these methods may still be taught, such methods must be going out of style in professional geodesy applications (replaced by 3d methods); in GIS applications, where surface distances need to be calculated, the preference is to use a method which is not limited as to the length of the line.  (4) It would add to the length of the article.
 * Well, (1) geodesy is the main community dealing with this topic, so it deserves more attention. (2) Those methods should be moved out of geographical distance. (3) This issue has already been discussed above. (4) I do appreciate considering non-geodetic applications, but we might have to split the more common topic Geodesics on an ellipsoidal Earth to cover the small flattening case without undue weight on the more esoteric general case. fgnievinski (talk) 02:22, 14 February 2017 (UTC)

Now on to a more controversial issue:


 * The article quality is impressive.
 * This is an incorrect POV. If it was so impressive, it would be a Good Article. For examples of good maths articles listed in seriatim see Addition, Ars Conjectandi, Binary logarithm, Book embedding, Commutative property ... whose text prose sizes average out ca. 13K. In comparision this article is a Whale filled with Blubber.Inlinetext (talk) 05:23, 13 February 2017 (UTC)
 * OK, let me rephrase it: the article is dense. Section #Equations, for example, needs subsections. Section #Triaxial ellipsoid certainly deserve to be split into a separate article. The first section would benefit from being broken into #Definition and #History. The lead is overly detailed and need not define geodesic and need not discuss the figure of the Earth, for example. fgnievinski (talk) 02:22, 14 February 2017 (UTC)
 * A lot of this could be spun off to separate articles. eg Vector space which is long and Good Article developed with collaborative editing (not 1 major contributor).Inlinetext (talk) 08:50, 14 February 2017 (UTC)


 * I don't think it constitutes original research in any substantial way.
 * WP:Original Synthesis is a redirect to No original research.Inlinetext (talk)
 * OK, I meant: it's not original research other than being original synthesis. Thanks for pointing it out. fgnievinski (talk) 02:22, 14 February 2017 (UTC)
 * They overlap, and there are grey areas :-) Inlinetext (talk) 08:50, 14 February 2017 (UTC)


 * I'm afraid it does constitute original synthesis, though; see WP:Original Synthesis.
 * As to WP:original synthesis, my question is "where?" I have assembled information from a number of sources for this article. But where have I combined this information to draw a new conclusion?  In looking over the article, I can see at least one place where, in my modesty, I neglected to add a reference to myself, specifically Helmert (1880) gives the expression for m12, but the formula for M12 is given by Karney (2013) (found from Helmert's result by elementary methods).  Perhaps the last sentences of the section "Evaluation of the integrals", "Fast algorithms for computing elliptic integrals are given by Carlson (1995) in terms of symmetric elliptic integrals. Equation (6) can be inverted using Newton's method.", count as synthesis?  If so, they can be eliminated (under slight protest from me).  I'd be happy to hear of any other possible examples; my guess is that the problems could be easily dealt with by inserting a citation or by revising the text.
 * Some problematic passages:
 * Clairaut (1735) first found this relation,
 * however, these are typically comparable in complexity to the method for the exact solution given above
 * the second form of the longitude integral is preferred
 * The basic strategy for solving the geodesic problems on the ellipsoid...
 * , and most subsequent authors.
 * the NGS (2012) implementation which includes Vincenty's fix still fails to converge in some cases.
 * The solution may be expressed as the sum of two independent solutions
 * This result follows from one of Napier's analogies.


 * In fact, it could serve as an excellent review article in a scholarly journal.


 * The ideal solution in my view would be for Karney to publish this material elsewhere under an open-access license so that its content could be reused here with proper attribution.
 * The longer review you are asking for is Karney (2011). This was rejected to the Journal of Geodesy (like I said, the field has moved away from the classical problem), and Karney (2013) is a slimmed down version of this paper which was accepted.  As it turns out this Wikipedia article no longer references Karney (2011).
 * Well, for me, this very much settles the issue: the Wikipedia article should be marked with Template:Free-content attribution and a citation should be made to Karney (2011) using Template:Cite arXiv. It will resolve all contentions about original synthesis.  Please make sure that your arXiv submission is licensed under anything but CC-BY-NC, as the non-commercial clause is incompatible with incorporation into Wikipedia, see WP:COMPLIC and  fgnievinski (talk) 02:22, 14 February 2017 (UTC)


 * This solution already seems to have been adopted in Geodesics on a triaxial ellipsoid, where there is a footnote stating "This section is adapted from the documentation for GeographicLib (Karney 2015, Geodesics on a triaxial ellipsoid)."


 * So the options for external publication venues would be either the software documentation or a scholarly journal article.


 * The software documentation needs to state its license granting reuse rights.
 * The GeographicLib library comes with an MIT license which lets anyone use it.
 * We should use Template:Free-content attribution a second time and cite it using Template:Cite web. I've also checked and the MIT license is acceptable for incorporation into Wikipedia, see FAQ/Copyright. It wouldn't hurt to mention in the software documentation that the documentation itself is covered under the same license as the source code. fgnievinski (talk) 02:22, 14 February 2017 (UTC)


 * The journal could be fully open-access or partial (hybrid open-access journal).


 * This practice of republishing material on Wikipedia is already in place, see Category:Wikipedia articles published in peer-reviewed literature and external coverage in and.


 * I'd recommend a submission to wikiversity:WikiJournal of Science and could offer assistance to handle it with the editors.
 * Thanks for these pointers &mdash; I was not aware that there had been this blurring of the line between Wikipedia and Journals. However, in the light of my remarks above, I don't know whether this additional form of publication is warranted.  (Plus, I'm somewhat gun-shy following the failed GA review of this article in 2013.)
 * In the light of recent information regarding previous availability of this material elsewhere (arXiv and software documentation), I withdraw the suggestion for a separate publication external to Wikipedia. fgnievinski (talk) 02:22, 14 February 2017 (UTC)

Regards. fgnievinski (talk) 14:56, 10 February 2017 (UTC)


 * Thanks. I have 2 small preliminary observations to contribute here.


 * 1) I would prefer that 'Cffk' kicks off on discussing your suggestions/observations, because I would like to reply to him and not to you.
 * 2) Your observations are very welcome, but would not qualify as an actual third opinion. Cheers. Inlinetext (talk) 15:34, 10 February 2017 (UTC)

Roll-up of responses
I thought it would be best to give my responses to recent comments by fgnievinski and Inlinetext in a contiguous section rather than interspersing them. I'll try to give sufficient context to make it clear what I'm responding to.

Minor matters
Not fair, fgnievinski, pointing to Leick (2015) as not covering Karney (2013), since it doesn't cover Vincenty (1975) either! It only discusses the approximate Gauss mid-latitude method. The whole issue of whether this is covered in surveying courses is a distraction. Surely students need only to know the properties of geodesic and rely on software libraries for calculating them. (Compare with trigonometry courses. The student needs to know the properties of sine but not necessarily an algorithm for computing it.)

Ditto, Inlinetext, for dinging me for the lack of citation by Tseng (2015b). That paper is about great elliptic arcs not geodesics.

Synthesis issues
"Clairaut (1735) first found this relation." I've removed "first" (I supposed that this is the problem?)

"however these are typically comparable in complexity to the method for the exact solution given above" This is already sourced: Jekeli (2012) says. "None of these developments is simpler in essence than the exact (iterative, or series) solution which holds for any length of line." No edits are necessary.

"the second form of the longitude integral is preferred" This is already sourced: Cayley (1870), p 333, says "n is thus indefinitely large and the integral Pi(n, c phi) is not conveniently dealt with. But it may be replaced by an expression depending on Pi(c^2/n, c, phi), where c^2/n is indefinitely small..." No edits are necessary.

"The basic strategy for solving the geodesic problems on the ellipsoid." Bessel (1825) says "Diese Gleichung enthaelt die Relation zwischen zwei Seiten eines sphaerischen Dreiecks 90 - u' und 90 - u und den ihnen gegen ueberstehenden Winkeln 360 - alpha und alpha'. Die dritte Seite desselben sphaerischen Dreiecks und der ihr gegenueberstehende Winkel, welche ich durch sigma und omega bezeichnen werde, geben, wenn man sie in die Rechnung einfuehrt, elegante Ausdruecke fuer die zusammengehoerigen Veraenderungen von s, u und w. [This equation relates two sides of a spherical triangle, 90 - u' and 90 - u, and their opposite angles, 360 - alpha and alpha'. The third side sigma and its opposite angle omega will appear in the following calculations giving elegant expressions for the joint variations of s, u and w.] I've changed this to "The strategy described by Bessel (1825) and Helmert (1880)"

"and most subsequent authors" I've removed the whole sentence.

"the NGS (2012) implementation which includes Vincenty's fix still fails to converge in some cases." This NGS page says "In very rare instances, the results can fail to iterate properly when the two points are close to being anti-podal." (There are also examples of failure cases earlier on this talk page.) No edits are necessary

"The solution may be expressed as the sum of two independent solutions" I've prepended "As a second order, linear, homogeneous differential equation, its solution ..."

"This result follows from one of Napier's analogies." I've removed this.

Validation
The validation question: Vincenty states clearly the limitations of his method: (a) accurate to (about) 0.1 mm for terrestrial ellipsoids and (b) convergence of the inverse method only if the points are not nearly antipodal. Thomas and Featherstone (2005) confirm this. Vincenty's paper was an important milestone.

Karney (2013) presents an "improved" method and claims: (a) accuracy close to double precision round-off for |f| < 0.02, (b) convergence of the inverse method in all cases. (The paper makes other claims but these are not repeated in the Wikipedia article.)

fgnievinski, I can't find the "implicit claims" you say that you detect: simpler, faster, widely used. I therefore don't see the need to establish these within the article. Please tell me if I'm wrong.

There are now two questions: (1) are the claims correct? and (2) does anyone care?

Are the claims correct?
On "are the claims correct?". These were carefully tested and the tests documented in the paper. The paper was accepted after peer review for publication in the Journal of Geodesy. This should suffice for the Wikipedia standard of verifiability. However, it's reasonable to ascertain that the claim have held up over time.

The paper first appeared in mid 2012 and since then has been cited by 58 other papers. On a parallel track, an implementation of the algorithm was made available in 2009 and advertised on the proj.4 mailing list. It's been added to the standard installs for major Linux distributions (Fedora, Debian, etc.) It's been incorporated into proj.4 and thereby been picked up by several GIS packages. Altogether this embodies a large "user base" operating over the span of 8 years. None of this have turned up evidence to overturn the claims.

By all reasonable measures, claims (a) and (b) hold up.

My arguments above don't depend on the following but I include it to put the NGS quasi-endorsement into context. The last person to modify the NGS version of the Vincenty inverse solution was "dgm" implementing Vincenty (1975b) in May 2011. dgm is Dennis Milbert who is now retired from the NGS. Here an exerpt on an E-mail he sent to Dru Smith, the chief geodesist at NGS on July 3, 2013: I can unreservedly recommend Karney's code for adoption. It is well documented by the Journal of Geodesy paper. And, since the series were rederived to a higher limit of accuracy (and since Karney is an adept programmer/analyst), his code will function at higher accuracy. It checks against my secant 4.0 version, as well as a historical Vincenty direct subroutine. I have included a Zip file with sample versions of inverse and direct that invoke his 1.31 release stashed in geodesic.for. This is code that he had sent to you a while back -- I'm just resending it with the latest core. What I would recommend is that if you go with his stuff, you kick the numbering forward to 4.0, since the inverse (and direct) subroutines constitute completely new engines. And, of course, you will want to do some of your own testing, too, if or when resources permit.

In my chats with Karney I did mention the evolution of geometric geodesy from development methods into projective methods, and the associated decline of use of the inverse. So, there is no formal requirement. And, you guys do not have free resources to pursue this category of problems. I know you could make no promises, but it is gracious of you to consider looking at this. I hope this material can help. As it turns out NGS didn't pick up my algorithm probably for the reason Milbert alludes to in the second paragraph above.

Does anyone care?
I come from the school "if a job is worth doing it's worth doing right". So for me, the geodesics represented a well defined mathematical problem for which there should be a robust solution. I know that 0.1 mm sure seems like it's accurate enough. But you're doing an adjustment with many ill-conditioned triangles or you're doing numerical differentiation either of which may magnify the errors to something you care about. Do you really want to stop and analyze each such problem to verify that 0.1 mm is "small enough"? I don't! (Furthermore, I frequently make use of the arbitrary precision version of the algorithms to verify my software implementation, for example, to detect problems where I've accidentally lost precision.)

Reading Vincenty's obiturary, it's clear that he was from the same school. An accuracy of 0.1 mm was all he could achieve with the memory limitation of programmable calculators of his era. And it's clear that Vincenty was bugged by the antipodal failures in his algorithm -- see his 1975b paper.

By the way, fgnievinski, you can't say that this represents a pathological case. For highly eccentric ellipsoids, the antipodal region may encompass most of the ellipsoid. Don't forget that molecular biologist who wanted to solve the inverse problem on an ellipsoid with b/a = 2.8. Should not Wikipedia address his needs? I remind you that this article is "Geodesics on an ellipsoid" and not "Geodesics in geodetic applications". Also you inserted the Geodesy banner in July 2014 nearly a year after the page went live. While geodetic applications are undeniably important, they shouldn't overshadow other applications.

Even sticking with geodetic applications, I can cite Bernhard Bauer-Marschallinger of the Vienna University of Technology was able to code up the azimuthal equidistant projection using Karney (2013). This replaces series approximations (from Snyder probably) which were valid only for small distances. This is now incorporated into proj.4. This change was a "no-brainer" only because the geodesics were computed to (nearly) full double precision accuracy and because the inverse method always converges. As a bonus the resulting code is simpler and easier to understand. This isn't an application which would have made sense with Vincenty's method.

Here's another symptom of Vincenty not quite working properly. Kallay (2007) justifies using (misguidedly in my opinion) great ellipses to underpin the Microsoft SQL Server Geography type with this statement Computing points along geodesic curves is notoriously difficult and expensive. An exact differentiable parameterizations is not known, and approximate ones are difficult and expensive to compute. I'm not quite sure how he got this idea, but the fact that Vincenty fails for antipodal points probably contributed to the author's misconception.

Pittman (1986)
Pittman (1986) is indeed an interesting paper. A similar method was proposed by Levallois and Dupuy (1952) (this paper is referred to by Rainsford (1955)). It's not so clear that Pittman's method is practical. Even though he publishes his code (good for him!), his inverse method fails if one of the end points is close to a vertex (not an unusual situation). He hints at this problem in the last paragraph of his section "Inverse Method". The problem was also encountered by Gerald Evenden (the author of PROJ.4) in 2009 and documented here. Unfortunately the "validation" performed by Deakin and Hunter did not extend to checking the numerical algorithm extensively. Here's what Deakin (private communication, 2017) tells me His (Pittman's) inverse method relied on an iterative scheme based on an approximation and in some cases failed to converge. In these cases a divide and conquer method had to be developed and you probably couldn’t say his inverse method was a general solution. I have Matlab code for the inverse case that works; but not in all cases.

Tseng (2014)
If you read Tseng (2014) (not 2015a), you will see that he adopts, practically verbatim, the method of Karney (2013) for solving the inverse problem. (Compare the beginning of Section 4 in Tseng's paper with the beginning of Section 4 in my paper. These are close enough to have drawn the attention of the publisher.) The gripes that the author expresses about my method (Newton's method + the need in some cases to solve a quartic to get a good starting point) are pumped up in an effort to justify his method. Note that he doesn't say that what I did is wrong. He's just claiming he can solve the problem better. However, he does not publish an implementation of his algorithm; so there's no way to evaluate whether his way is, in fact, better. Tellingly, the author's web site includes an admission that he later discovered the need to solve "the similar quartic equation to Karney’s derivation (2013)" after all (see his point 5).

Everyone is making incremental steps
Inlinetext, I don't think I've ever come across anyone who has given me more credit than you.

I and Deakin translate Bessel's paper into English and you regard it as an entirely new scientific endeavor basing this on the statement "The mathematical notation has been updated to conform to current conventions and, in a few places, the equations have been rearranged for clarity. Several errors have been corrected, a figure has been included, and the tables have been recomputed." A quick comparison of the German and English versions would show that I stayed faithful to the original. 100% of the credit still belongs to Bessel.

Similarly you seem to regard everything I say in Karney (2015) as originating with me so that if a similar thought appears in the Wikipedia article I'm touting my approach. The reality is that nearly all scientific papers really heavily on what came before. In particular both Vincenty (1975) and Karney (2013) rely (at roughly the 80% level) on the same body of preceding work.

Part of the difficulty is that Vincenty jumps right into his algorithm with few prefatory remarks, whereas I tried to cover the earlier work so that my paper is more self-contained. So when I cover the earlier work you think I'm indulging in "pseudo-research to promote myself and my algorithms". (Really, Inlinetext, you should watch your language -- it verges on the offensive.)

Let me show how Vincenty's Eq. (6) for $$\Delta\sigma$$ is a mere hop, skip, and jump away from Helmert (1880). (I'm using the "improved" formulas Vincenty gives in the supplement to his 1975 paper.)

Helmert (1880) Eq 5.5.8 reads

s(\sigma) = b \frac{1+\frac14 k_1^2}{1-k_1} [ \sigma - (\tfrac12 k_1 - \tfrac3{16}k_1^3 ) \sin2\sigma -\tfrac1{16}k_1^2\sin4\sigma - \tfrac1{48}k_1^3\sin6\sigma ] + \ldots $$ (accurate to third order in the flattening), where I've substituted $$ \phi = \sigma + \tfrac12\pi $$ to switch from Helmert's convention of measuring from the vertex to Bessel's + Rainsford's + Vincenty's convention of measuring from the node.

Vincenty (1975, supplement) substitutes

\begin{align} A &= \frac{1+\frac14 k_1^2}{1-k_1}, \\ B &= k_1(1-\tfrac38 k_1^2),\quad\text{or} \\ k_1 &= B (1+\tfrac38 B^2 + O(B^4)) \end{align} $$ to give

s(\sigma) = b A [ \sigma - B \sin2\sigma -\tfrac1{16}B^2\sin4\sigma - \tfrac1{48}B^3\sin6\sigma ] + \ldots $$ (also accurate to third order in the flattening). Evaluating

s(\sigma_m+\sigma/2)-s(\sigma_m-\sigma/2) = b A (\sigma - \Delta\sigma) $$ gives, after some manipulation of the trigonometric terms,

\Delta \sigma = B \sin \sigma \Big\{ \cos2 \sigma_m + \tfrac{1}{4} B \big[ \cos \sigma \big(-1+2 \cos^2 2 \sigma_m \big) - \tfrac{1}{6} B \cos 2 \sigma_m (-3+4 \sin^2 \sigma) \big(-3+4 \cos^2 2 \sigma_m\big) \big] \Big\}, $$ which is Vincenty's Eq. (6).

(Aside: You might wonder what Vincenty gains with the substitution of $$B$$. The answer is "not much".  However, in his original 1975 paper, Vincenty starts with Rainsford (1955) Eqs. (18-20) which are expansions in terms of $$u^2$$, and then the substitutions Vincenty Eqs. (3-4) are important in simplifying the result (yielding Eq. (6) once again). Note that because the series in $$u^2$$ converges more slowly than the one in $$k_1$$, Vincenty has to retain fourth order terms in the definitions of $$A$$ and $$B$$ in terms of $$u^2$$.  Using $$k_1$$, Vincenty can work consistently at third order.)

Once you know what Vincenty is trying to do, the jump from Helmert's Eq. (5.5.8) to Vincenty's Eq. (6) is trivial. Similar steps can be taken to obtain Vincenty's longitude equations (10-11).

My paper basically does two things: Extend the Helmert's series to higher order and fix the iterative scheme for the inverse method so that it converges in the antipodal case. These are important contribution but nevertheless just a natural extension of the earlier work

R.E.S.P.E.C.T.
Yesterday was the 50th anniversary of the recording of Aretha Franklin's R.E.S.P.E.C.T!

Inlinetext, I realize that my edits deserve more strutiny than usual because it covers some of my work. However, you need to stop throwing about insults. My paper is a solid piece of scholarship; please treat it as such.

cffk (talk) 03:39, 16 February 2017 (UTC)