Talk:Earth section paths

Copyright vs Public Domain
This article includes paragraphs taken directly from Gilbertson's paper. The journal's web site says that the paper is copyright by the Institude of Navigation. However, since the author works for the US Government, the paper may be in the public domain. If this is the case, then please include a link to the public domain paper. Presumably there's also a public domain implementation of the methods described in this article. It would be good to include a link for that too. cffk (talk) 15:08, 29 April 2018 (UTC)


 * I will rewrite paragraphs which duplicate the Navigation paper.Chuckage (talk) 13:01, 19 February 2020 (UTC)chuckage

Math notation improvements
I believe that I've fixed these.
 * Include the full equation in a math environment, e.g., $$A$$=-$$B$$ should be written as $$A=-B$$ so that the spacing (and the minus sign) comes out right.

I don't think that there are any.
 * Don't use a star to indicate multiplication.

I don't think that there are any.
 * Use $$r'$$ instead of $$r^'$$.

Done.
 * Use $$e^6$$ instead of $$e6$$, etc.

Done.
 * The $$\Delta$$ notation for the definite integral is ugly. Why not define it in terms of the indefinite integral, e.g., $$s_{12} = s(\theta_2) - s(\theta_1)$$, where $$s(\theta) = \ldots$$? Alternatively, use $$s_{12} = \left.\mbox{the expression for the indefinite integral}\right|_{\theta=\theta_1}^{\theta_2}$$. cffk (talk) 14:39, 2 May 2018 (UTC)

I don't think that there are any.
 * Use $$\left|\Delta\theta\right|$$ instead of $$abs(\Delta\theta)$$.

cffk (talk) 16:27, 29 April 2018 (UTC)


 * I will do my best to not use ugly notation.Chuckage (talk) 13:02, 19 February 2020 (UTC)chuckage

Further glitches:

Fixed.
 * the functions sin and cos rendering in italics in the section "locate P_2".

Fixed. Are there others?
 * randomly not putting parts of equations into math environments, e.g., in the section "section plane".


 * Here are a few:

fixed fixed fixed fixed fixed
 * Section 1.1: Missing mathbf in $$\hat N$$.
 * Section 1.1: Missing math environment: "of (x,y,z)".
 * Section 1.3: Missing math environment: "If p=0".
 * Section 1.3: Missing math environment: "If p>0".
 * Section 2.2: Portion of "s = s(theta)" outside math environment.
 * @chuckage, It would be good if you could proof your own maths! Also don't forget to sign you edits to the talk page. cffk (talk) 15:28, 13 May 2021 (UTC)

Right. Do you know of a tool that would help?
 * No, I don't know of a tool. I find that reading the text carefully helps! cffk (talk) 16:12, 13 May 2021 (UTC)

They should all have been bold, and yes a dot was mssing. Fixed.
 * I'm not sure what the less-than sign in "$$C_2 [1 - (\mathbf\hat{N_1}\hat{N_2})^2] = d_2 - d_1(<\mathbf\hat{N_1}\cdot\mathbf\hat{N_2}$$)" means. (This is also an example where a part of equation, the last paren, is outside the equation.)  Why are some "N"s in boldface and not others?  Is there a dot missing?

cffk (talk) 21:51, 12 May 2021 (UTC)


 * The hats will look better centered in the N's, e.g., $$\mathbf\hat N_1$$ instead of $$\mathbf\hat{N_1}$$. cffk (talk) 15:07, 13 May 2021 (UTC)

Ok. — Preceding unsigned comment added by Chuckage (talk • contribs) 16:07, 17 May 2021 (UTC)

Distance along an ellipse
You give an expansion in $$e^2$$ for arc length along an ellipse as a function of central angle $$\theta$$ and you suggest inverting the process via Newton's method. Better would be to direct the reader to the article on the meridian arc for the solution of these problems. This article offers various series for the arc length which converge more rapidly; I recommend Bessel's (1825) who uses the third flattening $$n$$ as the small parameter and the parametric latitude $$\beta$$ as the independent coordinate. This series can be reverted (Helmert, 1880) so that, given the arc length, the parametric latitude can be found without iteration. cffk (talk) 23:27, 29 April 2018 (UTC)


 * Agreed.Chuckage (talk) 13:03, 19 February 2020 (UTC)chuckage

Let me clarify this. It is not immediately evident how the meridian arc formula applies. It is a different problem. With the proper explanation it may be seen that it can be used, but a redirection would fail to accomplish that. You keep saying reverting a series when I think you mean inverting (because it is in fact a functional inverse). — Preceding unsigned comment added by Chuckage (talk • contribs) 22:36, 7 June 2021 (UTC) Done — Preceding unsigned comment added by Chuckage (talk • contribs) 17:15, 12 May 2021 (UTC)


 * I recommend removing all the details about computing meridian distances. This is a solved described in meridian arc; this is how it should be described in this article. cffk (talk) 09:14, 15 May 2021 (UTC)

I disagree because the context must be made clear. Additionally, there are complications with the inverse series.


 * Similarly, the details given in Section 5 are superfluous because the intersection of two planes is a known problem covered more clearly in Plane_(geometry). In general this article could be improved by describing clearly the ideas involved and omitting the detailed formulas. cffk (talk) 11:24, 15 May 2021 (UTC)

This paragraph looks new. Anyway, it's only 4 lines, so I disagree again to keep consistent notation.

Earth section paths as an approximation to geodesics
The reader of great ellipse is sent to this page with the promise that a "more accurate normal section may be computed". It would be good to have this improved accuracy documented here. For great ellipses and for points within 10000 km, we have


 * the maximum error in the distance compared to the geodesic is 13.5 m;
 * the maximum deviation from the geodesic is 8.3 km.

It would be good to have the equivalent numbers for normal sections. The cited paper only gives a few examples and then says that the method is highly accurate. Better would be to have some reasonably rigorous bounds as I've given for great ellipses. cffk (talk) 21:14, 30 April 2018 (UTC)

the maximums for the great ellipse are a bit bigger than what you suggest.

If you're willing to use the geodesic to locate the midpoint and use the surface normal there, then this midpoint normal section would have a maximum distance error of 0.5 m, and maximum deviation of about 0.7 km. Chuckage (talk)
 * I would say about 13.5 m, and 8.4 km.
 * For the normal section about: 9.0 m, and 6.7 km.
 * For the MNS about: 0.8 m, and 2.1 km.

I'm not sure how tight something like a Taylor series remainder would be. But that would only be truncation error for Normal sections, and not the comparative error. Did you have in mind differencing remainders, or something else?


 * No, Taylor series shouldn't come into it. Compute a geodesic path exactly (e.g., full double precision) and similarly for the normal sections.  Compute difference in distances and how far apart the two paths are.  Repeat for many such paths.  Quote the worst errors. cffk (talk) 15:13, 13 May 2021 (UTC)

Issues, Feb 2020
Here is some background on the notability and one source issues I've added to the article.

Earth section paths was created by User:Chuckage who is the author of a paper of the same name (published in Navigation in 2012). The Wikipedia article summarizes the paper and presents some extensions to it. However a scientific paper does not deserve a dedicated Wikipedia entry unless it is especially notable. The Navigation paper probably does not pass this test. According to Google, it's only been cited once in the past 8 years. The paper offers earth section paths as providing "good" approximations to geodesics. However "good" is not quantified (there are no rigorous bounds on the errors); the lack of citations indicates that few have adopted this approach; and I can't find any GIS software libraries that include the computation of earth section paths. In any case, algorithms already exist for computing geodesics quickly and accurately, so there is little interest in adopting approximate solutions.

cffk (talk) 16:18, 16 February 2020 (UTC)


 * I find no mention of "one source" in the policies and guidelines. I think that the unifying nature of the page is sufficient basis for its existence.Chuckage (talk) 13:06, 19 February 2020 (UTC)chuckage

I am working on a rewrite that doesn't appear to be self promotional. The issue for me is the abundance of inferior ways to do such things. For example, there is a blizzard of information on spherical trig and how it's useful in navigation. Fundamentally, my purpose was to provide something better. Yet all of that other information is out there misleading the reader into thinking that it should be used! In other words, my purpose was in fact not self promotion, but rather was to do a public service. Chuckage (talk) 20:43, 2 June 2021 (UTC)

See WP:ONESOURCE. Note that this is really just one of the issues about the sources used in the article. The others are


 * that you are the author of this source; this means that it invites more scrutiny, see WP:A;
 * this is a primary source and secondary sources are prefered;
 * the focus of the article is narrow (following that of your paper).

The subject is absolutely verifiable. Just do the math! There are many much more narrow pages out there. — Preceding unsigned comment added by Chuckage (talk • contribs) 18:47, 7 June 2021 (UTC)

On the last two issues, you have not included any decent discussion of Normal Sections which are important in surveying. This subject is extensively covered in several secondary sources, e.g., Helmert (1880), Jordan (Vol 3, part 2), Rapp (Part 1).

cffk (talk) 01:04, 21 February 2020 (UTC)