# Talk:Geodesics on an ellipsoid

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Field:  Geometry

## Archives

• GA review ended with a fail at my request. See /GA1. cffk (talk) 05:35, 13 December 2013 (UTC)
• Archiving talk page /Archive1. cffk (talk) 19:31, 12 March 2017 (UTC)
• Archiving discussions mainly involving Inlinetext /Archive2. cffk (talk) 17:53, 4 April 2017 (UTC)

## An alternative proposal

I'd like to propose an alternative approach to shortening this article taking a leaf out of Jekeli's course notes (see his p. 2-30). Why does the reader need to know much about the details of the solutions of the direct and inverse problems? Much better that she understands the properties of geodesics, what the direct and inverse problems are, that these problems have been solved, and where to go to find the solution.

I'll flesh this idea out over the next week or so. But roughly speaking it would mean

• collapsing the sections Evaluation of the integrals, Solution of the direct problem, and Solution of the inverse problem, into one or two paragraphs;
• removing the equations for m12, M12, and J(sigma), from Differential properties of geodesics;
• performing similar surgery to Area of a geodesic polygon;
• removing the equations for ds^2, dbeta/ds, domega/ds, dalpha/ds from The triaxial coordinate system;
• removing the two equations for ds from Jacobi's solution;
• I would retain the Equations for a geodesic since I think the mapping of the problem onto the auxiliarly sphere is of general interest -- but probably this could be shortened.

cffk (talk) 14:49, 16 February 2017 (UTC)

Can we commit to a definite size limit for the first cut? I propose (a) article byte size (presently 90K) not exceeding 40K (and even that only because of math markup) and (b) prose of maximum 15K. We can then discuss what stays and what gets tossed. Inlinetext (talk) 03:52, 17 February 2017 (UTC)

Inlinetest, my present goal is to improve the article. That will involve shortening it; however I don't think it makes any sense to start with a (somewhat arbitrary) target for the size. I would prefer to take the revision process one step at a time.

fgnievinski, perhaps instead splitting the small flattening case into a separate article "Geodesics on an ellipsoidal Earth", it would make more sense to start an article covering all aspects of 2d adjustments: height reductions, angle reductions, small flattening + small distance approximations for geodesics, small distance approximations for the reduced length (which I see the Leick includes), other approximatings for the geodesic distance, Legrende's and Gauss' results for small triangles, etc. This would be have more "meat" and fill holes in Wikipedia's existing coverage.

More thoughts on proposed cuts:

• Eliminate Figs. 4, 16, and 17.
• Move Fig. 15 to Behavior of geodesics
• In Area of a geodesic polygon, eliminate second Eq. for S12 and Eq. for t(x).

cffk (talk) 23:05, 18 February 2017 (UTC)

R.E.S.P.E.C.T
1. This is not "your" article for you to improve unilaterally. Your overwhelming presence on this page is preventing other domain experts from improving it.
2. You deliberately create long walls of text on the talk page (example) to prevent other editors from having specific dialog.
3. You deliberately set out to create an excessively long article simply to mask your bogus approach to this topic. The summary (lead) is longer than many Wikipedia articles to drive readers away.
It would be best if you self-imposed an initial limit to this article's size especially since I do not want to edit-war while improving this to be a good article (I have already provided several examples for the target size for good maths maths articles for your guidance).Inlinetext (talk) 08:30, 19 February 2017 (UTC)

@Cffk: Thanks for your patience. I think 2D network adjustment is well beyond the scope of the present topic, and it's a topic of only historical importance, so I decline participation, thanks. As for the present article, splitting triaxial ellipsoids into a separate article would seem a no-brainer, would you agree? Other than that, I've made a number of suggestions previously, which now I've implemented in a draft: Talk:Geodesics_on_an_ellipsoid/Draft1. I still think sourcing the present version to your earlier works (arXiv and documentation). Also, I've taken the liberty removing the numerous external links, to avoid undue accusations; if it's to be restored, it should include links to other implementations, too (even inferior ones). May I suggest that you go ahead with your changes in a draft? fgnievinski (talk) 18:53, 19 February 2017 (UTC)

Yes, I'll go ahead with a draft along the lines I indicated (done in a day or two). Then we can compare approaches. cffk (talk) 15:00, 20 February 2017 (UTC)

Could you offer a draft, too? Then we can appreciate your size limit proposal better. Also, Cffk's approach has been published in the Journal of Geodesy, so you cannot call it bogus. fgnievinski (talk) 18:53, 19 February 2017 (UTC)

Thanks. (A) My interest is confined to elevating this to a "Good Article" for Wikipedia, and not for creating a review article in a scholarly journal. (B) The approach adopted by Cffk includes solving the path of geodesics using the method of 'auxiliary sphere' and is admittedly "merely a convenient tool for solving for a particular geodesic", for which limitation Karney's approach has been criticised. I could go on, but would direct your primary attention to understanding why this article gets so few visitors and what can be done to make it a good article. Inlinetext (talk) 20:50, 19 February 2017 (UTC)
Inlinetext, if you aren't willing to create a draft yourself, you should stop commenting on this article. Jrheller1 (talk) 21:43, 19 February 2017 (UTC)
I don't understand your point (B) about the auxiliary sphere. Your favored approach, Vincenty (1975), uses exactly the same construction. See Vincenty's definitions of lambda and sigma; his equations (1), (2), (5), (8), (9), (12), (14), (15), (16), (17), (18), (20), (21) are all equations for great circles on the auxilary sphere. cffk (talk) 15:00, 20 February 2017 (UTC)
I've already indicated my cut. Charles ignored it.
(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.Inlinetext (talk) 05:57, 20 February 2017 (UTC)
Inlinetext, No I didn't ignore it. I responded on Feb. 7 "I don't think a strategy that begins by deleting more than half the article makes sense." If you want to advance the conversation, you need to detail how you see the article coming together following the mass deletions. Preparing a draft would be a good way forward. cffk (talk) 15:00, 20 February 2017 (UTC)
The only published criticism of Karney that you have quoted is Tseng (2015a). It should be cited in the present article as an alternative approach, including an additional citation to Tseng's more recent self-contradicting comments in their software documentation [1]. We could cite Tseng (2014), too, no problem. Tseng (2015b) is about great elliptic arcs, which often get confused with geodesics, so we should create a section clarifying the distinction. fgnievinski (talk) 23:01, 19 February 2017 (UTC)
Since there are extensive admissions on this talk page (and elsewhere) documenting serious contraventions of core Wikipedia policy while developing the article, perhaps this article needs to be stubbed and developed afresh ? Inlinetext (talk) 05:57, 20 February 2017 (UTC)
Inlinetext, I disagree with your characterization about how this article was developed. However, that's irrelevant; we should concentrate on the article in its current state and how it can be improved. That doesn't start with deleting all the content! cffk (talk) 15:00, 20 February 2017 (UTC)
Let me restate my position. I am interested to improve this article to achieve good article status. There is a Snowball's chance of that happening if we use the current article as the base. There are 2 possibilities, (i) drastic cuts or (ii) clean slate. Inlinetext (talk) 17:33, 20 February 2017 (UTC)
To further clarify your misunderstandings. I don't favour Vincenty's approach or any other approach. I favour listing Vincenty's equations (only because they are the de-facto standard whether you admit it or not) and directing readers to suitable derivations (i) with or without auxilary spheres or (ii) series expansions or numerical integration, approaches. It is frankly improper and unethical to promote your own approaches and equations via this article. for instance Panou(2013) essentially avoids both the auxilliary sphere approach and uses numerical integration and claims extension to the triaxial where Clairaut's equation does not hold; and his paper is a clear and simple exposition of the problem and its solution, as are papers of other authors. Inlinetext (talk) 17:33, 20 February 2017 (UTC)
We should cover more of these other approaches; the integration in 3D space via Cartesian coordinates is remarkable (Panou and Korakitis, 2016). fgnievinski (talk) 04:32, 21 February 2017 (UTC)
For sure, it's nice to know that geodesics can be computed by simple methods. Indeed, the figures of geodesics on a triaxial ellipsoid in this article were also calculated (by me) by integration of the ODE in cartesian coordinates (posed as the equations of motion of a particle constrained to the surface by the centrifugal force). See also Teodorescu (2009) who uses the same method.
The method is however impractical (too slow!) for an ellipsoid of revolution. According this paper (which only addresses the direct problem), solving for geodesics requires about 0.3 us per step. To achieve 1 um accuracy requires 2000 steps for long lines, i.e., 0.6 ms running time. This compares unfavorably with Vincenty (1975) or Karney (2013) with a running time of about 1 us. The paper makes the point that by solving the problem as an ODE you get way points at the rate of 1 point per 0.3 us. Fair enough; however Vincenty (1975) can be and Karney (2013) has been adapted to produce way points efficiently (at the rate of 1 point per 0.37 us); and, in this case, the number and positions of the way points are at the user's discretion (not dictating by the needs of the ODE solver). cffk (talk) 15:12, 22 February 2017 (UTC)
I've mentioned this in my draft: [2]. fgnievinski (talk) 16:11, 24 February 2017 (UTC)

I've made a first cut implementing my proposal at Talk:Geodesics_on_an_ellipsoid/Draft2.

As you will see I've drastically shortened the material covering the solutions of the direct and inverse problems. I'm trying to take the point of view of a professor (e.g., Jekeli, 2012) who has to cover the topic in a single 1-1/2 hr lecture. In this case, it make no sense to go into the details of these solutions. Much better to derive the fundamental equations and to gives the students a good grasp of the properties of geodesics plus pointers to where to find the solutions should they need them.

fgnievinski, I intentionally didn't read your draft carefully before starting mine. I thought it would be better to start with two more-or-less individual efforts. I'll read your draft over the next week and then perhaps we can both propose how the two efforts can be combined.

cffk (talk) 20:11, 20 February 2017 (UTC)

I remind you that you are ignoring my proper concerns, and continuing on the same path of play-acting as a professor for this article. Kindly don't waste your time "deriving" these fundamental equations in this article, since Wikipedia policies do not permit such displays of personal scholarship. Inlinetext (talk) 20:33, 20 February 2017 (UTC)
I've made a draft implementing your proposal at Talk:Geodesics_on_an_ellipsoid/Draft3. I'll leave you to take care of your points (4) Trim the lead section by 50% (5) Remove many of the older references and add in some more recent ones. And of course you'll have to "insert the minimal bridging text, formulae and images to cement this article together" so that your proposal can be evaluated. cffk (talk) 22:18, 20 February 2017 (UTC)
Inlinetext should show us a concrete draft or else their comments should be ignored. fgnievinski (talk) 04:03, 21 February 2017 (UTC)
@Cffk: Why not split Geodesics on a triaxial ellipsoid from Geodesics on a spheroid (Geodesics on an ellipsoid of revolution)? fgnievinski (talk) 04:06, 21 February 2017 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── This is a possibility... The arguments for splitting would appear to be

• The triaxial section makes the article too long.
• It's irrelevant to the concerns of those interested in the biaxial problem.

To this I would say

• Let's wait and see how long the article ends up being without a split and to what extent the triaxial material contributes to the length.
• I think a case can be made that an understanding of the more general triaxial case contributes to an understanding of the special biaxial case. Specifically:
• the generic geodesics on a oblate (resp. prolate) ellipsoid correspond to the circumpolar (resp. transpolar) geodesics on a triaxial ellipsoid;
• when a triaxial ellipsoid degenerates to a biaxial ellipsoid, one of the stable closed geodesics merges with the (exponentially) unstable closed geodesic to give the family of meridional closed geodesics;
• the stability of the meridional geodesics is therefore a compromise between stability and exponential instability, namely linear instability (a geodesic crossing the equator at azimuth +epsilon, linearly drifts westward on an oblate ellipsoid);
• the degenerate c -> 0 limit corresponds to a billiard ball bouncing in a circular (biaxial) or elliptical (triaxial) table is a simple and illuminating model for geodesics that probably appeals even to those without much mathematical background.

Most of these points are not made in the current article (maybe they should be). However I would argue that having the two cases presented together nevertheless encourages readers to explore further.

The bottom line: let's wait and I'm inclined to keep the two cases together. However, I'm not dogmatic about this.

cffk (talk) 15:12, 22 February 2017 (UTC)

### Comparing versions

We know have three versions in play

I should start by emphazing that although there's room for improvement in every article, the existing article is not in need of urgent revision. It addresses a rather specialized topic (unlike for example geographical distance) and will most likely be consulted by readers with some mathematical skills. It's not obviously wrong. The presentation is more-or-less coherent. Inlinetext says it's too long; however, its present size is well within the guidelines. And so on...

The major change in fgnievinski's version is to remove the treatment of trixial ellipsoids. In addition, he's rejigged the sectioning somewhat. I've already described why I would prefer to keep the triaxial case included. I would also say "what's the harm in keeping it?" It doesn't add that much to the article and the reader who is only interested in the biaxial case and easily skip over the triaxial section.

The big change in my version is the collapse of the sections describing the nuts and bolts of solving the direct and inverse problems into a few paragraphs of prose. Someone who's interested in the details can easily find them in the references. And, I would contend, there are more important aspects of geodesics than these somewhat "grubby" details. I refer you to this discussion with Tim Zukas on how much detail to include about Bowring's method in Geographical distance#Ellipsoidal-surface formulae. In the end, I let Tim win this fight (under the banner I refered to above: "what's the harm in keeping it").

In the specific case of calculating geodesics accurately we have Vincenty's method. This is simple enough to list in detail in Vincenty's formulae. I have misgivings about this article (it just replicates the formulas in Vincenty's paper). Also, I wouldn't advise that the casual reader to code up these formulas (and probably make half-a-dozen typos along the right). (Here, I would draw a contrast with the formulas for great-circle navigation. I think it is a good idea that these are given in detail. These are simple, useful in their own right, and underpin the solution of the geodesic problems.) Perhaps a justification for Rapp (back in the 1990s) to cover Vincenty's method in detail in his lectures was that there was clearly a need for improvement and laying out the method in detail would provide his students with a starting point.

The second "accurate" method is the one I give in Karney (2013). This is that much more complicated (the series go to 6th order, there's a need to calculate the reduced length, etc.), that there's no realistic way this can be covered in Wikipedia. Arguably the current article makes a mistake in trying to go halfway. Furthermore, given that my method delivers a more-or-less "complete" solution, there's little need for most practioners to delve into the details of the method; instead they can pick up the method from their favorite GIS package. Here, I think that Jekeli (2012) has it exactly right: cover the properties of geodesics; point out that software exists for calculating them exactly; finally, deemphasize approximate methods.

On approximate methods, I refer back to my discussion on Bowring's method. The interesting aspect of his paper is not the set of formulas (which I can't envision anyone wanting to use nowadays), but the underlying method (fit a sphere as snugly as possible to the ellipsoid). A separate Wikipedia article describing approximate methods from this perspective would be useful. However, I don't have the background to initiate such a project.

cffk (talk) 19:00, 1 March 2017 (UTC)

Unless I hear objections, I'm inclined to adopt my Draft2 as an interim measure. It shortens the article the article and lessens the exposure given to my paper (both things that Inlinetext has wanted, I believe). It defers the question of whether to split off the triaxial case (see above for why I think it should not be split off). I haven't heard any feedback on the general thrust of this draft -- namely not to include the specifics of the solutions to the direct and inverse problems and instead leave the interested reader to consult either Vincenty's formulae or Karney (2013); I think that this makes the article more useful for the majority of users. cffk (talk) 13:52, 8 March 2017 (UTC)

I've replaced the article with Draft2. cffk (talk) 18:41, 15 March 2017 (UTC)

## Objection to creation of multiple drafts

I am recording my protest to creation of multiple drafts. These are quite clearly stalling strategems to delay and obstruct any version of the article except Cffk's. Such multiple drafts do not encourage consensus but repel it. In any case my objections to and suggestions for Cffk's version are stated on this page. The comments and wiki-stalking behavior of editor 'Jrheller1' are also protested. Inlinetext (talk) 06:30, 23 February 2017 (UTC)

I've proposed the drafts as a way for each of us to demonstrate our vision of how the improved article could look like. I've offered my initial version and Cffk's offered his. Now, where is your version? You can't expect us to read your mind and write the draft that you'd like to have. Please stop complaining and do your homework. fgnievinski (talk) 20:59, 23 February 2017 (UTC)
User:Fgnievinski I have already protested that yours was not a 3rd opinion as you represented it to be, and expand on it to say that you and 'Jrheller1' are tag-teaming with 'Cffk'. I have expressly stated the problem areas in the present article and very well specified the concerned policies which are being openly flouted as a consequence. It is also news to me that every talk page participant is compelled to submit a draft article as part of the discussion process. Cffk would be well advised to read WP:EXPERT and its advice Editing an article in Wikipedia is not like writing an original research article for an academic journal, nor it is like writing a literature review article where you synthesize a story from original research papers; instead, it should be a solid review of the subject as a whole, summarizing what published reviews say. Wikipedia is not a place to publish original research, nor your own synthesis of the research literature, even if it is brilliant. The genre here is "encyclopedia"—each article is meant to provide "a summary of accepted knowledge regarding its subject". (see WP:NOT) Inlinetext (talk) 07:59, 24 February 2017 (UTC)
I fail to see why Jrheller1's and my opinion do not count here as you say. Just because we do not agree with you? I've disagreed with Cffk openly on a number of issues above. fgnievinski (talk) 16:06, 24 February 2017 (UTC)
This timeline shows you never had any substantial interest in developing this article in the past and are only now here to defend Cffk's work against my edits. Inlinetext (talk) 16:21, 24 February 2017 (UTC)
I object, if you look at the talk page, you'll notice that my interest goes back to 2013: [3]. fgnievinski (talk) 18:06, 24 February 2017 (UTC)

Inlinetext, I get it: you want to make drastic changes to this article. However the only concrete step you've taken is to delete half the article leaving it (for the period Jan 17 - Jan 29) in a broken state and postponing the actual work of improving the article. In the circumstances, I don't think it's unreasonable to ask you to complete a draft which illustrates your vision of the finished article. cffk (talk) 14:02, 24 February 2017 (UTC)

That is incorrect. I was in the process of editing the article to render it policy compliant, when 'Jrheller1' began wikistalking and harassing me all over Wikipedia simply to restore your version. I shall await the outcome of the SPI before I resume editing this article. 'Jrheller1's views don't count here because he has recorded that he will not discuss anything with me on this talk page. Inlinetext (talk) 16:14, 24 February 2017 (UTC)

## Appreciation for article

I'm a professional in the field, and I find this article to be a gold mine of useful information. The diagrams alone are wonderful specimens of elegance and precision. I've had recourse to Karney's work outside wikipedia on several occasions, and one of the things that I love about wikipedia is that persons of his stature take this much care to write such good articles in their expertise.

That said, I am concerned that some of the proposals on this talk page might gut much of the good information that I get from the article. While I do agree with moving the triaxial ellipsoid section elsewhere, and with the removal of the historic portraits, the earlier proposal to remove many of the figures (hopefully dropped by now?) would be an especially great loss. Yes, there is a lot of specialized detail in this article. But at the same time, this field is complex, and without presenting these details, the essence of the topic would undoubtedly be lost. For the sections I have needed to read, the detail was exactly what I needed.

Additionally, I believe the original author has shown remarkable patience in persisting with an attempt to work with other editors -- I would probably have abandoned such an effort from frustration much earlier in the conversation. I'm glad he has not. Mlouns (talk) 19:37, 8 March 2017 (UTC)

I'm a professional, but not in this field. I found the article enormously helpful in that it made vast amounts of time and research unnecessary by providing a central source of the base of knowledge. I have a working module that solves the forward problem and am working to make it robust in general use when initial lat/long may be odd, such as a geoid beginning in the Southern hemisphere and ending in the Northern hemisphere. This work is directly useful in preparing unrelated research for publication. Geoid lengths greater than half a diameter aren't addressed in the article, but this is easily handled in software by breaking up long geoids.

I've been around long enough to see what is going on here, and I do admire the patience of the principal author in the face of unsubstantiated assertions of plagiarism, demands for changes without consensus, unilateral deletions of material, etc. The pictures of the historic contributors is a minor loss but a loss nonetheless, but without the existing figures that illustrate the trigometry I would not have been able to decipher the article to the extent that I could get my arms around the science of geoid mapping and the ambiguities that arise in the trigonometry. I'm looking into a way to solve the inverse problem with vector arithmetic to avoid the singularities at latitudes near the poles and poor numerical conditioning near these singularities. I very much appreciate the perspective of a scholar, the clarity of a teacher, and the practicality of a practitioner in this article.

The nature of the auxiliary sphere could be treated in a little more detail; it seems that azimuth ${\displaystyle \alpha }$ seems to track between the spheroid and the auxiliary sphere, meaning that it is a bit more abstract than simply a sphere containing the spheroid of revolution and tangent at the equator. I find that Napier's analogy can be helpful in finding ${\displaystyle \sigma }$ but is tricky in finding ${\displaystyle \omega }$. I'm looking at ways to use the half-angle formulas to avoid problems with ambiguities. And I find that the use of vector arithmetic for the more difficult problem of the triaxial ellipsoid implies to me that vector arithmetic may simplify some areas of working with ellipsoids of rotation as well. -motorfingers- 20:38, 8 March 2017 (UTC) — Preceding unsigned comment added by Motorfingers (talkcontribs)

@Mlouns: I appreciate your feedback. Much of the discussion here has between specialists in this field and it's very important that we made sure that the needs of the "average" user are met.
Motorfingers, the auxiliary sphere is not merely a sphere which is tangent to the ellipsoid at the equator. Their relationship is given in Fig. 6 in the article. The two complex relationships are between s and σ and between λ and ω which are given by Eqs. (4) and (5) (and to complicate matters further, these relationships have α0 as a parameter). Application of the equations given at the beginning of the section "solution of the direct problem" should allow you to completely solve any triangle problem for triangle NEP. In particular, ω is given by tan ω = sin α0 tan σ (with ω and σ being in the same quadrants for east-going geodesics).
cffk (talk) 16:53, 10 March 2017 (UTC)

## Introduction to geodesics on an ellipsoid

I still find this article intimidating. It seems that other complicated topics have received a companion introductory article, e.g.:

So I'd like to propose the creation of an Introduction to geodesics on an ellipsoid.

For me, the key concepts would be:

- Definition
global: least length
local: zero geodetic curvature

- Finding the endpoint (direct problem)
- integration in Cartesian coordinates
- integration in curvilinear coordinates
- spherical trigonometry
- on the osculating sphere
- on the auxiliary sphere
- series expansion

- Calculating the length and azimuth between two points (inverse problem)
- root-finding (shooting method)

- Drawing the geodesic
- ODE solving

- Reducing azimuth and distance to the ellipsoid
- Differential properties

- Properties
- geodesics on a spheroid are not necessarily plane curves
- geodesics on a spheroid are not necessarily closed curves
- meridians are geodesics
- parallels are not geodesics (except for the equator)
- "Clairaut's relation is just a consequence of conservation of angular momentum for a particle on a surface of revolution."


I'll just leave this comment here for future prospects. fgnievinski (talk) 04:57, 8 April 2017 (UTC)