Talk:Geodesics on an ellipsoid/Archive 1

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I've just moved this article from my sandbox. Please let me know what you think.

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Footnotes are for additional explanations. Citations use Harvard parenthetical referencing.

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cffk (talk) 20:57, 19 August 2013 (UTC)

Outstanding -- thanks for taking the time. Fgnievinski (talk) 05:24, 21 September 2013 (UTC)

phi & varphi are same? Jackzhp (talk) 07:15, 27 November 2013 (UTC)

Yes, I consistently use \phi in displayed formulas, φ in in-line equations, and \phi in figures (generated by Matlab) to denote the geographic latitude. Unfortunately, they variously display as the curly and straight variants of the Greek letter. I don't plan to try to "fix" this because what actually gets displayed will vary depending on the fonts available on your browser. cffk (talk) 13:01, 27 November 2013 (UTC)

I will note that other symbols display inconsistently in different contexts: a, f, g. Again the actual differences will depend on your browser's fonts. cffk (talk) 13:08, 27 November 2013 (UTC)

This is an outstanding article. I have had the pleasure of working with cffk (Charles Karney) on a couple of published papers and have benefited from the experience. He has asked for my comments (and suggestions for improvement) which I regard as a compliment. I first make some general comments here and some specific comments in later sections. General comments: (1) Introduction to the topic (study of geodesics) is concise (5 paragraphs) and establishes the historical connection with some of the great names of science and mathematics and the practical applications of mapping and measuring the earth. As well as 'subdividing' the study into ellipsoids of revolution and triaxial ellipsoids. (2) The Notes and References are extensive and well documented. Notes in particular are useful for those souls actually working through various aspects of the study; perhaps developing equations or turning equations into software. (3) Diagrams are excellent, especially those in the sections Behaviour of geodesics and Envelope of geodesics. Whilst similar diagrams can be found scattered in the literature there is nothing (published) that I know of that has them all in the one place with a common style. Very well done. (4) The outline of the development of the equations for a geodesic (on an ellipsoid of revolution) is reasonably complete and with the aid of the references (Rapp 1993 in particular) an interested 'student' could probably fill in the gaps if pressed. (5) The evaluation of the integrals (numerical solution of direct and inverse problems) is succinctly treated and a special mention should be made of the connection with Legendre and the early study of elliptic integrals and numerical techniques for evaluation. E03267 (talk) 04:15, 11 December 2013 (UTC)

Riemann curvature tensor

I want to see Riemann curvature tensor, Ricci tensor for the ellipsoid surface. Jackzhp (talk) 04:31, 27 November 2013 (UTC)

I don't use the Riemann or Ricci tensors myself. However in their application to a 2D surface, they both presumably can be expressed in terms of the Gaussian curvature K, since that's the only intrinsic measure of curvature. Note that I now give the explicit formula for K for an ellipsoid of revolution. cffk (talk) 13:01, 27 November 2013 (UTC)

Suggestions

Here are some suggestions. Sorry it's not detailed. Please ask for clarifications as needed. Fgnievinski (talk) 23:32, 24 December 2013 (UTC)

Secondary sources

there's plenty of primary sources. can we find statements in classical books such as Torge; Clarke; Bomford; Vanicek? or newer syntheses, such as Smith; Hooijberg; Grafarend; Strang and Borre; Weintrit and Neumann; Hofmann-Wellenhof, Legat, ‎and Wieser?

Computations

segregate computations into sub-sections, like this? 1.1 Equations for a geodesic 1.2 Behavior of geodesics 1.3 Differential behavior of geodesics 1.4 Envelope of geodesics 1.5 Computations 1.5.1 Evaluation of the integrals 1.5.2 Solution of the direct problem 1.5.3 Solution of the inverse problem 1.5.4 Area of a geodesic polygon 1.6 Software implementations

Geodesics on a triaxial ellipsoid

split into new article? define tri ellip as X^2/a^2 + Y^2/a'^2 + Z^2/b^2 = 1 more consistent to ellip of rev, X^2/a^2 + Y^2/a^2 + Z^2/b^2 = 1 are Ellipsoidal coordinates same as in the linked article?

Map projections

include figures of maps

add third type: retroazimuthal projections add new application: finding Mecca?

Applications

add figures (steal from other articles)

Lead

too long; didn't read.

repeat some of the figures of applications in the article lead?

More of recent articles

e.g., [1], [2]. Also Panou et al. (2013) could be discussed in the biaxial ellipsoid.

Hopefully the reorganization segregating the computational aspects will help in introducing competing alternatives, e.g., spherical closed-form plus numerical integration of the excess, as per Sjorberg. Fgnievinski (talk) 02:58, 25 December 2013 (UTC)

Response

Thanks for the suggestions. I will try to incorporate them. Some comments:

I checked the secondary sources you suggested and the treatments of this problem are cursory at best. The modern English-language "authority" remains Rapp's course notes. However, I will insert references to other sources where appropriate.

Notation for triaxial ellipsoids: Using a, a', and b will break the symmetry in all the formulas for the triaxial ellipsoid. Also this notation is geared towards the limit of an oblate ellipsoid; for a prolate ellipsoid, (a,b,c) would be replaced by (b,a,a'). Probably the best thing is to alert the reader to the change in notation near the beginning of the triaxial treatment.

Recent articles: I will expand the treatment of recent articles. However, it is striking how few of the modern treatments of the problem include code that allow the reader to judge the proposed methods. In the case of the papers you cite:

Sjoberg and Shirazian (2012): I have requested the Matlab code from Shirazian. I believe that the basic iterative technique for the inverse method is just Vincenty (regular + his antipodal method); so it may fail to converge in some cases.

Rollins (2011): The inverse problem is cast as a root finding problem in Mathematica, but no details are given. (And the reader who does not have Mathematica is left to devise his own method for finding the root.) The paper incorrectly presumes that all previous authors solved the problems in terms of integrals over latitude, where the integrands diverge at the vertices, necessitating stitching together the solution for long lines. The reality is that this technical problem was avoided by most authors starting with Legendre in 1806. Rollins' integrals are different because he uses a different independent variable, theta, related to sigma by tan(theta) = tan(sigma) / sqrt(1 - k^2*e^2). But it's not clear what is gained by this choice.

Panou et al. (2013): Panou kindly did provide me with his Matlab code. This fails to obtain a result in some cases (very short lines and nearly antipodal cases) and the error can be as large as 1 km for nearly meridional lines. I've no doubt that this defects could be fixed. However ...

A big problem with Panou's approach is the timing. For randomly chosen pairs of points, the mean running time for Panou's code is 11 s. This compares with 2.6 ms for a single line (and 10 us/line averaged over many lines) for the native Matlab implementation for the GeographicLib algorithms. This large gap in performance makes all the difference to being able to solve complex problems in terms of geodesics. E.g., computing median boundaries (which involve many distance calculations), which might be done in a minute using the Bessel/Helmert series, will instead take a year using numerical integration. I suspect that a similar timing penalty will apply to Sjoberg and Shirazian's method. I know that it does apply to Schmidt's (2000) method as used by WTrans.

So for the "average reader" who just wants to do some geodesic calculations, the viable alternatives are surely just Vincenty and GeographicLib (or proj.4).

cffk (talk) 16:00, 30 December 2013 (UTC)

I added a mention of the retroazimuthal projection together with Hink's application of determining your position given the range and bearing to the Rugby clock. cffk (talk) 14:45, 31 December 2013 (UTC)

I added Bagratuni (1962), Krakiwsky & Thomson (1974), and Jekeli (2012) as additional secondary sources. I see that I already link to ellipsoidal coordinates (via confocal ellipsoidal coordinates); however, I added another link in the caption to the figure. cffk (talk) 20:03, 3 January 2014 (UTC)

NGS and the inverse problem

The National Geodetic Survey now provides a semi-endorsement of Karney (2013) at Inverse/Forward Computation Utilities. It also similarly endorses Sjoberg and Shirazian (2012) (but that paper does not provide a link for the code) and Pittman (1986). Pittman does provide code, however his method fails to work when one of the points is near a vertex. See Problems with Pittman geodesic.

The NGS online tool (which uses Vincenty with his antipodal fix) goes into an infinite loop with the following inputs:

   User defined ellipsoid: a = 6378137, 1/f = 298.257223563
   lat1 = N31.394417440639
   lon1 = E000.0
   lat2 = S31.275540610835467
   lon2 = E179.61560163120291

Finally, Lee (2011) compares 18 methods for solving the inverse problem. Unfortunately, the paper is in Korean.

• Lee, Y. C. (2011). "측지 역 문제 해석기법의 정확도 분석". Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography (in Korean). 29 (4): 329–341. doi:10.7848/ksgpc.2011.29.4.329. {{cite journal}}: Invalid |ref=harv (help); Unknown parameter |trans_title= ignored (|trans-title= suggested) (help)

cffk (talk) 16:44, 30 December 2013 (UTC)

More details on the NGS geodesic tool, Version 3.0. These results apply to the solution of the inverse problem either for the GRS80 or WGS84 ellipsoids:

• Example where the error in the distance is 0.08 mm:

    lat1 lon1  lat2 lon2
     3.2   0    2.9 169.8

• Examples where the solution of the inverse problem fails to converge:

    lat1 lon1  lat2 lon2
     0.7   0   -0.3 179.7
     2.2   0   -1.8 179.6
     2.3   0   -2.0 179.6
     2.3   0   -1.8 179.8
     2.7   0   -2.5 179.7
     7.5   0   -7.1 179.8
    10.7   0  -10.2 179.8
    11.3   0  -10.8 179.7
    14.3   0  -14.0 179.9
    16.2   0  -15.9 179.6
    16.3   0  -16.1 179.5
    17.8   0  -17.5 179.9
    18.4   0  -18.1 179.8
    21.8   0  -21.6 179.6
    25.1   0  -24.7 179.7
    27.2   0  -27.1 179.5
    29.7   0  -29.5 179.7
    31.9   0  -31.8 179.6
    42.7   0  -42.6 179.6
    43.7   0  -43.4 179.9
    45.8   0  -45.7 179.9
    54.0   0  -53.8 179.9

cffk (talk) 12:28, 6 October 2014 (UTC)

Mathar's solution of the inverse problem for constant altitude

Here are some failure cases of Mather's solution. (The java code is available from http://www.mpia-hd.mpg.de/~mathar/progs/Geod.tar.gz.) When the following endpoints are given as input, NaNs are returned. These were run with -h 0 (i.e., the altitude is zero) and -s 40000. This documents the statements in the footnote to the Mathar citation.

    lat1 lon1  lat2 lon2
      20    0    10 179.99
      30    0    80 179.9
       0    0     0  20
       0    0  0.01 120
    -0.1    0   0.1 120
     -10    0    10 170
      30    0 -29.6 172
      30    0    80   0
      90    0    20  10
      90    0    20   0
      90    0   -90   0

cffk (talk) 04:47, 14 January 2014 (UTC)

149.217.40.222, I've partially reverted your edit. The most interesting aspect of Mathar's work is that a more complex problem (on a surface of constant height above the ellipsoid) can be handled by solving the ODEs. So I've reintroduced this (without the grousing about the lack of convergence). On the other hand, although Mathar published the code for his method (and he's in a distinct minority here), the method is hardly one that can be recommended for routine use. Besides the problems with convergence noted above, high accuracy is very costly. For example, to get the error in the distance between Paris (49N 2E) and Saigon (11N 107E) down to 1m, I need to take 1000000 steps which takes 14s on my machine. (Pittman's method is in the same category. He nicely publishes his code, but there are severe problems with his inverse method; see above.) cffk (talk) 15:34, 14 January 2014 (UTC)

Borre

I think these notes could be cited: [3] I'm not sure what's the best place. They seem to have been published as the last chapters of this book: [4]. Fgnievinski (talk) 01:32, 18 June 2014 (UTC)

I'm familiar with the notes. I think a colleague has the book; I will check tomorrow. cffk (talk) 03:17, 18 June 2014 (UTC)

No, my colleague didn't have the book and the public library can't get it through an inter-library loan. I don't feel that the notes add much to the presentation by Rapp (for example). But possibly they could be cited at the beginning to the section "Equations for a geodesic". cffk (talk) 09:00, 21 June 2014 (UTC)

OK, thanks for checking. Fgnievinski (talk) 02:52, 23 June 2014 (UTC)

I managed to get hold of Borre and Strang (2012) and have included a couple of references to it. The citation includes a link to the pdf to give which is close to chapters 11 & 12. cffk (talk) 02:33, 1 October 2014 (UTC)

Lead

The first paragraph of this article is as follows:

"The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, i.e., the analogue of a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry (Euler 1755)."

This could hardly be less appropriate: Nowhere is the "solution of triangulation networks" explained, and certainly it is of little help to click on the link to the article "Triangulation" that the phrase "triangulation networks" links to. So the whole paragraph is rendered meaningless.

It is a common but huge error among some Wikipedia editors to think that just by linking to an article, one has no need to write clearly. Nothing could be more wrong.Daqu (talk) 20:53, 27 January 2016 (UTC)

I think we only need Triangulation#Planimetric solution. A hint is given later, where it reads "The adjustment of triangulation networks entailed reducing all the measurements to a reference ellipsoid and solving the resulting two-dimensional problem as an exercise in spheroidal trigonometry (Bomford 1952, Chap. 3)." fgnievinski (talk) 21:53, 27 January 2016 (UTC)

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