Talk:Great-circle navigation

Questioning that Great Circle is shorter that a direct straight Line on two-dimensional map

Can someone give an explanation why great-circle navigation is the shortest distance between two points? Following a direct "line" seems like it would be shorter, since the curve it would describe would have a smaller radius and thus a shorter arc length between the two points. From Great circle: "A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere. It is the largest circle that can be drawn on a given sphere" (my emphasis.) Since it is clearly true that the great circle is shorter, because planes and ships have been using the great circle method for many years, can someone explain what is flawed about my assumption above, and perhaps add a fuller explanation than what is provided on these pages? 98.244.9.195 (talk) 21:23, 20 December 2008 (UTC)

You don't say what you mean by a direct line. If you are referring to a straight line drawn on a map, the route on the surface of the Earth that corresponds to that line will depend on the type of projection used to draw the map. The Mercator projection that is often used is seriously distorted in most respects although a route drawn along the Equator or a meridian of a Mercator chart will show the shortest route, because those lines follow great circles. The easiest way is to get hold of a terrestrial globe and see how the shortest routes appear. It may help to stretch a piece of string between two points, for example London and New York. treesmill (talk) 17:57, 20 December 2009 (UTC)

Maps are an attempt to represent a three (3) dimensional object with a 2 Dimensional model. Different types of maps have different types of distortions. with some type of maps or shorter distances, there is a very low difference from the maps straight /direct line path and the actual great circle route. A line of latitude- runs east west, and IS NOT a great circle ( the plane this line creates is parallel to the plane of the equator (the only line of latitude (0) that is a great circle. Wfoj3 (talk) 20:27, 1 January 2015 (UTC)

The reason that "great-circle navigation is the shortest distance between two points" is simply that an arc of a great circle is the shortest distance between two points on a sphere — and the Earth is, to a very good approximation, a sphere. This is fairly easy to prove by the use of analytic geometry. (For any two points on a sphere that are not diametrically opposite, there will be one single great circle arc that is the shortest path between the two points. For two points that are antipodal — such as the North and South Poles — there are an infinite number of shortest paths all of the same length.)

It is a deep fact of geometry that it is impossible to represent all distances of the spherical earth or any portion of it on a flat map to the same scale: Any such map would have to be curved like a sphere.

Interestingly, however, it is possible to represent all the great semicircles of an (open) great hemisphere on a flat map so that they are all straight lines.Daqu (talk) 12:59, 22 October 2015 (UTC)

Compare these two versions

The article should give formulas for great-circle initial course and distance, if nothing else. One possible explanation of the formula for initial course is

If a navigator is starting at latitude $\scriptstyle \phi _{1}\,\!$ and plans to travel the great circle to a point at latitude $\scriptstyle \phi _{2}\,\!$, with a longitude difference between the points of L (positive eastward), his initial course $\alpha \,\!$ is given by

$\tan \alpha ={\frac {\sin L}{(\cos \phi _{1})(\tan \phi _{2})-(\sin \phi _{1})(\cos L)}}$

Kaimbridge likes this better:

If a navigator begins at latitude $\scriptstyle \phi _{s}\,\!$ (the "standpoint") and plans to travel the great circle to a point at latitude $\scriptstyle \phi _{s}\,\!$ (the "forepoint"), with a longitude difference between the points of $\scriptstyle \Delta \lambda \,\!$ (positive eastward), his initial course $\alpha \,\!$ is given by

${\begin{aligned}S\!A&=\cos(\phi _{f})\sin(\Delta \lambda );\\S\!B&=\cos(\phi _{s})\sin(\phi _{f})-\sin(\phi _{s})\cos(\phi _{f})\cos(\Delta \lambda );{}_{\color {white}.}\\\tan(\alpha _{s})&={\frac {S\!A}{S\!B}};{}_{\color {white}.}\end{aligned}}\,\!$

The additional complexity is evident, and a quick look shows that the two formulas are equivalent (once we correct the typo)-- no mathematical advantage to either one. As usual we have the choice between telling the reader what he needs to know, and making him a better person by forcing him to hack his way thru the unnecessary deltas and standpoints and forepoints.

A similar situation with the distance formula: the straightforward version is

The central angle $\sigma$ between the two points is given by

$\cos \;\sigma =(\cos \phi _{1})(\cos \phi _{2})(\cos L)+(\sin \phi _{1})(\sin \phi _{2})$

And Kaimbridge's version is

The central angle between the two points, $\Delta \sigma$, is given by

$\tan(\Delta \sigma )=\tan(\sigma _{f}-\sigma _{s})={\frac {\sqrt {S\!A^{2}+S\!B^{2}}}{\sin(\phi _{s})\sin(\phi _{f})+\cos(\phi _{s})\cos(\phi _{f})\cos(\Delta \lambda )}};{}_{\color {white}.}\,\!$

In this case the more complex formula is indeed more accurate if the navigator's origin and destination are a few hundred meters apart, or within a few hundred meters of antipodal. That rates a mention in a footnote-- if you're trying to navigate your ship on a 1-kilometer great circle trip, or on a 20000-km great circle trip, you should investigate the more complex formula. If your trip is between 2 km and 19998 km, don't bother-- the simple version will do fine. Tim Zukas (talk) 03:02, 27 February 2011 (UTC)

Yeah, sorry about the typos.P=|

Originally I was just going to put Az as the second equation, but I realized breaking it down to SA and SB, allows them to be used for the central angle (AD), too.

If it is pure simplicity you are seeking, you could reverse the order of presentation and use AD in calculating Az:

$\Delta \sigma =\sigma _{f}-\sigma _{s}=\arccos(\sin(\phi _{s})\sin(\phi _{f})+\cos(\phi _{s})\cos(\phi _{f})\cos(\Delta \lambda ));{}_{\color {white}.}\,\!$

$\sin(\alpha _{s})={\frac {\cos(\phi _{f})|\sin(\Delta \lambda )|}{\sin(\Delta \sigma )}};{}_{\color {white}.}\,\!$

(albeit, only 0->90°, rather than 0->180° or 360°)

And, if you wanted to get just a little bit more complicated, rather than Az, you could define the great circle, or "arc path" (AP or A) and then be able to find the Az at any point along it:

$\sin(\mathrm {A} )={\frac {\cos(\phi _{s})\cos(\phi _{f})|\sin(\Delta \lambda )|}{\sin(\Delta \sigma )}};{}_{\color {white}.}\,\!$

$\sin(\alpha _{p})={\frac {\sin(\mathrm {A} )}{\cos(\phi _{p})}}={\frac {\cos(\phi _{s})\cos(\phi _{f})|\sin(\Delta \lambda )|}{\cos(\phi _{p})\sin(\Delta \sigma )}};{}_{\color {white}.}\,\!$

Or just, I suppose, ignore mention of AP (though it is the equational definition of a given great circle),

$\sin(\alpha _{p})={\frac {\cos(\phi _{s})\cos(\phi _{f})|\sin(\Delta \lambda )|}{\cos(\phi _{p})\sin(\Delta \sigma )}};{}_{\color {white}.}\,\!$

As for "standpoint/forepoint", how is "_s,_f" more complicated than "_1,_2" (there isn't any involved explanation, just the brief defining mention)? ~Kaimbridge~ (talk) 16:20, 27 February 2011 (UTC)

No need to seek simplicity-- it's right there in front of us. All we have to do is not deliberately avoid it by throwing in lots of unnecessary greek letters and terminology. The terms "standpoint" and "forepoint" are no use to a navigator, and moving from point "1" to point "2" is simpler than moving from point "s" to point "f". No need for delta sigma or sigma-sub-s or sigma-sub-f or delta-lambda or any other lambda.

The simple formula for distance belongs in the article, along with a mention that it may not work well for 1-km trips or for 19999-km trips. The well-conditioned formula can be included (preferably in a footnote) for those who are planning 1-km trips, but the article should make it clear that that most readers can skip that section. The more complicated course formula (with "SA" and "SB") has no advantage for those readers. Maybe not for other readers, either.

No doubt you're aware of the disadvantage of using the law of sines to calculate initial course-- if the sine of the course comes out 0.9999 we don't know whether course is 89 degrees or 91 degrees. So the article should include some other formula, and once it has the other formula the law-of-sines calculation merits a footnote at most. (In an article on navigation, that is. If you're writing the spherical trig article, that's different.)

The article could mention that along the great circle the sine of the course angle times the cosine of the latitude is a constant, but how does that help the navigator? Is there anything in your last few equations beyond that? Tim Zukas (talk) 18:46, 28 February 2011 (UTC)

The Greek lettering is just the physical denotation of the concepts: When I see $\scriptstyle \Delta \lambda \,\!$, I usually just think "long diff", not "delta lambda, which means longitude difference". Early on I preferred to denote as Lat, Long, AD, Az, etc., but was advised it is best (cosistency-wise) to stick with the Greek characters: $\scriptstyle \phi ,\lambda ,\Delta \sigma ,\alpha \,\!$, etc. (which is why I think $\scriptstyle \Delta {V}=V_{f}-V_{s}\,\!$ is better than just $\scriptstyle V=V_{f}-V_{s}\,\!$).

Likewise, in terms of standpoint/forepoint, once the reader understands the standpoint is the first point and the forepoint the second, s/he is likely to think $\scriptstyle \phi _{s}\,\!$ as "lat 1". Regarding azimuth, yes I'm aware of the limitations of using sin instead of tan or cos (that's why I added "albeit, only 0->90°, rather than 0->180° or 360°"). And since this article is about "great circles", I'm just suggesting that it might not hurt to formatically define it.

Hey, until recently, this article was strictly qualitative and didn't have any formulation in it, the reader could just go click the Great circle distance or geographical distance links for more technical info! P=)  ~Kaimbridge~ (talk) 19:46, 1 March 2011 (UTC)

"The Greek lettering is just the physical denotation of the concepts"

And un-Greek lettering isn't?

"I preferred to denote as Lat, Long, AD, Az, etc., but was advised it is best (cosistency-wise) to stick with the Greek characters"

Yes, to be consistent with other Wikipedia articles we should definitely include the usual quota of pointless complexity. If we want to be consistent with an article that likes simplicity (e.g. Vincenty's http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf ) we can call the longitude difference L instead of delta-lambda, and the arc length on the sphere sigma instead of delta-sigma, and latitudes phi-sub-1 and phi-sub-2.

"in terms of standpoint/forepoint, once the reader understands the standpoint is the first point and the forepoint the second..."

Why require him to understand it? The terminology is utterly superfluous for this article (at least); Vincenty managed without it and so can we. Tim Zukas (talk) 20:43, 1 March 2011 (UTC)

"And since this article is about "great circles", I'm just suggesting that it might not hurt to formatically define it." It wouldn't hurt to "define" the great circle, but your formulas so far haven't done that. We can mention cos-lat-times-sin-course being a constant, but that doesn't help the navigator find points along the great circle-- the lat for a given lon, etc. Tim Zukas (talk) 20:51, 1 March 2011 (UTC)

Much simplified version

I've redone all the spherical trig formulas to make the notation consistent, to simplify the presentation, and to use accurate formulas for all quantities. cffk (talk) 14:27, 19 July 2013 (UTC)

Tim: Your "simpler version" needs fixing. If I apply it blindly to the example I give, I get $\phi _{V}=146.74^{\circ }$. That can't be right(!), so I stick an absolute value in and get $\phi _{V}=33.26^{\circ }$. But now I get $\lambda _{V}=-8.07^{\circ }$ which is the longitude difference between the starting point and $\phi =-\phi _{V}$. The tangent versions not only are more accurate, but also get the quadrants right! It seems that a decent compromise would be to list the simpler equations for $\sigma _{12}$ (your acos formula), $\alpha _{0}$ ($\sin \alpha _{0}=\sin \alpha _{1}\cos \phi _{1}$), and $\phi _{p}$ ($\sin \phi _{p}=\cos \alpha _{0}\sin \sigma _{p}$), where the right quadrant is given. Your last equation should be expressed as an equation for latitude, $\tan \phi _{p}=\cot \alpha _{0}\sin \lambda _{0p}$ (once again this gets the right quadrant). These could all be given as alternatives within the earlier sections obviating the need for an additional section. (By the way, characterizing the great circle via $\alpha _{0}$ instead of the vertex latitude has the great advantage of distinguishing east and west going paths.) I'll have a crack at making these changes so you can take a look. cffk (talk) 09:19, 22 July 2013 (UTC)

OK, I made the changes. I realize that even with these changes more equations are involved than your version. However, I think that's inevitable if you are looking for a way of working out the problem which doesn't sometimes land you in the wrong hemisphere. Can the "simpler version" be removed now? cffk (talk) 09:59, 22 July 2013 (UTC)

Tim: I've incorporated much of the material in the "simpler version" section into the preceding two sections:

• The first paragraph is moot. I now give the simple formulas (with the atan versions in footnotes).
• The equations for $\alpha _{1}$ and $\sigma _{12}$ are now the same.
• I include the comment about the accuracy of the great circle distance.
• As I explained above, reliable formulas require referring the measurement from the node instead of the vertex. I now give a simple formula for $\alpha _{0}$.
• The equation for $\lambda _{v}$ doesn't uniquely determine it (and sometimes returns the wrong result), so I give a reliable equation for the longitude difference to the node, $\lambda _{01}$.
• More useful than an equation giving longitude in terms of latitude (which typically has multiple solutions), is the equation for the latitude in terms of the longitude which I give above.
• I include the caveat about a spherical earth and point out how these solutions are used as an aid in solution the geodesic problem for a spheroid.

Now readers have a simple and reliable set of equations with consistent notation and explanatory figures in the preceding two sections. For these reasons, I've removed the "simpler version" section. cffk (talk) 01:03, 24 July 2013 (UTC)

Example

Speaking as someone wishing to use this page as a reference (without any great prior knowledge), I would like to mention that I found the examples, when combined with the explanations, quite confusing (for instance, there is mention of substituting values into the equations and using equations to calculate, but without mention of which equation). If someone were able to label each equation which is to be used, and then reference those labels in the example, that would be really useful. I have been reading this and experimenting for about an hour and am still not certain I have used the equations (which are really nice and succinct and diverse) correctly.--141.0.62.90 (talk) 13:05, 2 October 2013 (UTC)

In the third equation of the way-points explanation, there is a reference made to σ1, which is not previously defined (is it the same as σ01?). There is not σ1 on either of the diagrams...--141.0.62.90 (talk) 13:16, 2 October 2013 (UTC)

Added A Section

I added a section, entitled Related Topics and Extensions on Oct. 31, 2014--NavigationGuy (talk) 19:32, 1 November 2014 (UTC). The motivation is that a "navigator", either a person or computer, needs to have a greater repertoire than one equation. Comments/critiques are welcome. — Preceding unsigned comment added by NavigationGuy (talk • contribs) 01:03, 1 November 2014 (UTC)

At present, your new section is an uncomfortable fit with the rest of the article. It reads mostly as a wish list for additions to the article and as such perhaps it belongs in this talk page rather than in the article itself. A few comments:

• The article already deals with your first two bullets (vertex and latitude as a function of longitude).
• Longitude as a function of latitude is elementary and perhaps it would be useful to add this. (Of course, there may be 0, 1, or 2 solutions.)
• The rest of your list are problems of spherical trigonometry of varying degrees of complexity. Aren't the interesting ones already covered in other articles?
• The last paragraph is a little odd. "Lost to history" surely just means that someone hasn't done their homework. The "exceptional" publication is only exceptional for a meaningless reason.
• Section 7.4 of Geyer seems to deal only with the cartesian problem where signals propagate in straight lines instead of along the surface of the earth.
• Surely, in serious navigation applications the ellipsoidal model of the earth should be used.

cffk (talk) 20:57, 1 November 2014 (UTC)

NavigationGuy, I've removed your section because there's been no progress integrating it with the rest of the article in the last 2 weeks. I encourage you to develop your ideas in a sandbox in your user area and to post a link to it on these talk pages inviting comments from other editors. cffk (talk) 20:14, 16 November 2014 (UTC)

Old School means to determine Great Circle navigation Routes

If you look in section D of the Sight Reduction Tables for Marine navigation (Pub #229) it describes how using the tables there to determine the great Circle navigation route. to see my reference go to: http://msi.nga.mil/MSISiteContent/StaticFiles/NAV_PUBS/SRTM/Pub229/Vol_1/229V1P1.pdf That's how any navigator 50 or so or older was taught Wfoj3 (talk) 20:27, 1 January 2015 (UTC)

History of great circle navigation

Surely there is an interesting history regarding how and when great-circle navigation routes were adopted and used by sailors.

This article certainly would benefit from having such a section.Daqu (talk) 12:42, 22 October 2015 (UTC)

Its not that interesting bro 118.211.108.92 (talk) 06:10, 25 March 2017 (UTC)

Proposing merger

I've just proposed a merger with Orthodromic navigation. Unless I'm missing something, both articles give complementary information for basically the same concept. Fun fact: Great circle navigation (without dash) redirects there. Cato censor (talk) 20:35, 26 August 2016 (UTC)

I don't see anything useful in Orthodromic navigation that isn't already in this article. I recommend deleting Orthodromic navigation and replacing it with a redirect to this article. cffk (talk) 02:52, 4 September 2016 (UTC)

Keep or Merge There is no good reason to outright delete. No compliance with WP:Before. 7&6=thirteen (☎) 13:09, 25 March 2017 (UTC)

 Done Klbrain (talk) 09:50, 5 May 2018 (UTC)

Broken example?

With λ1 = −71.6° and λ2 = 121.8° how can λ12 = −166.6° if above λ12 is defined as λ2-λ1?! 121.8°--71.6° is 193.4°! — Preceding unsigned comment added by 194.95.79.11 (talk) 12:52, 11 April 2017 (UTC)

The difference needs to be reduced to the range [−180°,180°]. I've added a note to that effect. cffk (talk) 13:31, 11 April 2017 (UTC)

Thank you! — Preceding unsigned comment added by 194.95.79.11 (talk) 14:48, 12 April 2017 (UTC)

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Not informative for non mathematicians

The article as written fails to explain the topic to non mathematicians. The intuitive lay explanation that the great circle route is obtained by projecting a triangle with the source and destination and center of the earth as points would be helpful in establishing a mental framework before diving into the math. 8.46.75.75 (talk) 17:04, 23 July 2022 (UTC)

Finding way-points references to figures

Thank you for this article. Reading the section "Finding way-points". I'm having trouble resolving the text with the diagrams. Are the notations consistent? As an example, the text refers to an angular distance σ02. I'm not finding that in the referenced figure number one. Regretably, I can only ask the question. I'm not qualified to make corrections or improvements or know for certain that they are warranted. Does the section need a figure of its own? Dclo (talk) 15:30, 9 September 2022 (UTC)

The text in this section includes the definition σ₀₂ = σ₀₁ + σ₁₂. Does this answer your question? In light of this do you recommend any changes to the text? cffk (talk) 15:39, 9 September 2022 (UTC)

Thank you for the rapid reply. Yes, I figured out what it means. It was only that I was lazy, looking for it in the figure. The derivative of the figure at

Navigation problem

solves it for me, but that's me. I can't presume to add it to this article, nor argue that it's better. It's only my way around getting σ02 in the figure. Dclo (talk) 16:28, 9 September 2022 (UTC)

Usage of azimuth

In some sentences, i.e. the 1st sentence of the section "Finding way-points", azimuth is used instead of longitude. I know they are basically interchangeable, but I think we should standardise the term for the whole article.

The example mentioned above: let the longitude of this point be λ0 — see Fig 1. The azimuth at this point, α0, is given by The unknown player (talk) 20:43, 15 February 2023 (UTC)

Ok I was mistaken on which definition of azimuth we were using. However, instead of linking to the main azimuth (Azimuth) page, I would suggest it would suggest it redirects to the geodesic definition of azimuth (Azimuth#In_geodesy). The unknown player (talk) 20:54, 15 February 2023 (UTC)

Great Circle Navigation History

Suggested edit .... example

Great Circle Navigation History

Plain of Jars is a Great Circle Navigation Archeology Site, it is in alignment with Newgrange Ireland, and Machu Picchu Peru.

Jars are probably observatories to study the stars, like a pinhole telescope, aimed at specific sections of the night sky, making Plain of Jars an Archeoastronomy site.

See ... Great Circle Navigation Archaeology

Example 19

Plain of Jars Laos

Newgrange Ireland, Machu Picchu Peru, Plain of Jars Laos

https://goo.gl/maps/gbYtSoxMsDovreqW7 … map

https://drive.google.com/file/d/1QPCrnLs0NZxL3AeKnVyFOHweay7oJ7ev

https://drive.google.com/file/d/192BKxJOJpXdGcU7q3zVQSTpk4UMn1XfB

https://drive.google.com/file/d/1QWeLQJ2aLqjm8dpPj1WGy_HRRvaDotdW

https://drive.google.com/file/d/1uQCzfaxSz6EspjydvH8XT9tBJRYzwZ2l

https://drive.google.com/file/d/1lU3lTtPlO-9SAPkBA3FkSIVI7CbkTQyA

2601:444:300:B070:1D51:A600:CB4F:96B6 (talk) 17:10, 11 January 2024 (UTC)

This looks like "original research". If you want to add something to a Wikipedia article, you need to find "reliable" sources supporting it, such as published books, peer-reviewed journal papers, or major newspapers. This also seems like a fringe theory outside of mainstream scholarship. –jacobolus (t) 20:38, 11 January 2024 (UTC)

Link to Great Circle Navigation Archaeology published in the Minnesota Archaeological Society newsletter, Fall 2019. See ... Marshall Mountain Triple Point map ... https://www.mnarchsociety.org/newsletters/2019/MAS%202019%20Fall.pdf ... page 5 — Preceding unsigned comment added by 76.156.161.247 (talk) 15:23, 14 January 2024 (UTC)

This is (a) not a "reliable source" by wikipedia standards, and (b) does not make any claims, just links to a facebook page. The content of the facebook page is not coherently organized/formatted and I can't make sense of it. –jacobolus (t) 19:43, 14 January 2024 (UTC)

References