Talk:Mercator projection

Size distortion image
@Erickespinal26 can you explain the reasoning for including this Reddit image? It seems to accomplish the same thing as the animated image already shown, but is much worse because it's hosted externally. I can see that it might make sense to replace the animated image if the Reddit one were uploaded to the Commons, but we hardly need both, do we? Justin Kunimune (talk) 11:54, 20 September 2023 (UTC)

Mercator projection in Song China?
According to this article (I haven't looked it up) Needham's Science and Civilization in China claimed that certain maps (and maybe star charts?) from the Song Dynasty were based on the same principle as the Mercator projection, because they were based on a rectangular grid. An IP editor recently removed (special:diff/1183748240; the sarcastic edit summary "... illustrating 'so-called' discoveries made by the west through mathematical rigors were already well-known in China for centuries" leads me to believe this edit has some ideological motives) an unsourced claim that Needham was incorrect about this. From a cursory skim of the academic literature it seems like the previous claim in the article was right, i.e. Needham was incorrect and these maps don't have anything to do with the Mercator projection, despite using rectangular grids.

Does anyone know more about the research literature on this topic? It would be helpful to write a properly sourced summary of the current scholarly consensus (or lack thereof). The IP editor's version gives in my opinion a misleading impression. –jacobolus (t) 07:28, 6 November 2023 (UTC)


 * The only source I found was this rice.edu page, which seems to be a preprint of a chapter from this book by Richard Smith, which says and, which suggests that there isn't really a projection here.  I don't have access to Smith's actual book to confirm, but based on this I suspect you're right.  I do have access to Needham's book, so I can look more specifically at his claims later this week. Justin Kunimune (talk) 13:10, 6 November 2023 (UTC)
 * Here are a couple other sources: 1, 2. –jacobolus (t) 15:23, 6 November 2023 (UTC)
 * I reverted that edit before I realized this was under discussion. I believe I recall Snyder addressing this, but I don’t have access to my library at present. Sorry for the hasty edit. Strebe (talk) 16:32, 6 November 2023 (UTC)
 * The reason I didn't revert the change outright is that I'm not sure the previous claim (that these maps were based on an equirectangular projection) is correct either. It would be better I think to be very precise in describing Needham's claim(s) (which we should probably describe as "speculation" or similar, since they don't seem to be based on much evidence), and cite some sources about any dispute of that. –jacobolus (t) 18:09, 6 November 2023 (UTC)


 * Needham does claim that several star charts from the Song dynasty used the Mercator projection. Specifically, he says that star charts in the Hsin I Hsiang Fa Yao (1094, see right) used "'Mercator's' projection" (page 277), and he says that the Tunhuang manuscript star chart (10th century) used "a projection analogous to that of Mercator" (page 545).  He doesn't provide much supporting evidence for this.  My gut reaction is that it doesn't look precise enough to identify a projection; there doesn't seem to be any distortion of constellation size or shape near the poles.  Needham also talks about the Yu Ji Tu map, but only remarks on its square grid and its impressive precision compared to contemporary Western T-O maps; he doesn't connect it to the Mercator projection. Justin Kunimune (talk) 20:56, 9 November 2023 (UTC)
 * Thanks a lot for tracking that down. As an aside: you're quoting pages from Needham Vol. 3. Our current article cites Vol 4, p. 359. (I only have copies of vols. 1–3, but not vol. 4+). Do you have access to vol. 4? –jacobolus (t) 21:13, 9 November 2023 (UTC)
 * Ah, you are correct. I do have vol. 4.  The page numbers seem to be different in my edition, but there are a few mentions of the Mercator projection in the section on nautical technology.  In part 3 on page 559–560 he divides navigational technology into three phases and says that China reached phase two – quantitative navigation – before Europe but Europe reached phase three – mathematical navigation, including the invention of the Mercator projection – before China.  On page 568 he raises the possibility that Chinese influence contributed to the invention of the Mercator projection. Then on page 569 he repeats the claim that the Mercator projection was used for tenth-century star charts, adding the detail that "the hour-circles between the hsiu formed the meridians, with the stars marked in quasi-orthomorphic cylindrical projection on each side of the equator according to their north pole distances". Justin Kunimune (talk) 22:32, 13 November 2023 (UTC)
 * Maybe I should write a scholarly article, with cartometric analysis, on this topic, given what a thorn in the side it is. It’s maddening that speculations by an expert in one field about a matter in a different field automatically gets legitimacy. There is every reason to suppose the projection is not Mercator:
 * The domain in question isn’t a sphere; it’s a hemisphere, for which Mercator — if they knew about it — would have been a poor choice;
 * If they knew about it, then they knew that other projections would be much better choices and would have used them;
 * The domain in question isn’t terrestrial or aquatic, and so the redeeming quality of Mercator — straight rhumbs — would be irrelevant;
 * The Chinese cosmology of that period was flat-earth, and so we are to imagine that the cosmographers developed map projections solely for celestial charts, including inventing the principle of straight rhumbs, despite having no use for them in that domain;
 * The source for the claim provides no evidence beyond the implicit “it’s rectangular”, which, of course, is utterly insufficient.
 * I’m probably violating WP:FORUM here, but the only thing keeping the idea from being fringe is the “scholarly reference”. Strebe (talk) 23:52, 9 November 2023 (UTC)
 * I think Needham had this problem somewhat in general. There's been a discussion on the pages about Hindu–Arabic numbers about whether they may have actually originated in China, based on some speculation in Needham's book which was then followed up by further speculation by a couple of other scholars of Chinese descent. But the provided evidence/analysis doesn't go beyond (paraphrased) "both rod numerals and Hindu–Arabic numerals are positional decimal systems, and there was cultural contact between India and China, therefore one probably influenced the other", and none of the obvious historical/mathematical/philological criticisms of this claim were ever really addressed at all. I appreciate that Needham collected a huge amount of valuable material, but I wish he were a bit more careful to qualify pure speculation in the examples I've seen. –jacobolus (t) 00:14, 10 November 2023 (UTC)
 * I think if Needham is indeed the only source to make this claim, then it's something of a fringe theory, and it's fair for us to inject some skepticism into the text. Something like "Joseph Needham, a historian of China, claims that the Chinese developed the Mercator projection in the tenth century, during the Song dynasty, and used it in star charts." Justin Kunimune (talk) 22:56, 13 November 2023 (UTC)
 * I would explicitly use the word "speculate". Along the lines of: "Joseph Needham, a historian of China, speculated that some cylindrical star charts dating from the 10th century, during the Song dynasty, might have been based on the same concept as the Mercator projection. Other scholars remain skeptical of this claim, which was not accompanied by any detailed technical analysis." –jacobolus (t) 23:09, 13 November 2023 (UTC)
 * As far as I know, we don’t have anything to cite to support “Other scholars remain skeptical of this claim”, but I can agree with the rest, including pointing out that no analysis accompanied the speculation. Strebe (talk) 19:27, 15 November 2023 (UTC)
 * Miyajima (1998) is more explicit: "Needham's interpretation of the projection method of the Xin yixiang fayao as Mercator's projection is wrong."
 * –jacobolus (t) 20:10, 15 November 2023 (UTC)
 * Wonderful! Strebe (talk) 08:01, 16 November 2023 (UTC)
 * Maybe, “Joseph Needham, a historian of China, speculated that some star charts dating from the 10th century Song Dynasty and presented on an apparent cylindrical projection were based on the Mercator projection. However, using cartometric analysis, Miyajima finds Needham's interpretation to be wrong. Instead, the charts are on the equirectangular projection.” Strebe (talk) 16:47, 17 November 2023 (UTC)
 * Okay, I put a similar version into the article (and also changed a few other examples of this claim around Wikipedia, though I didn't exhaustively search for it). Feel free to tweak further if you want. –jacobolus (t) 17:35, 19 November 2023 (UTC)
 * Okay, I put a similar version into the article (and also changed a few other examples of this claim around Wikipedia, though I didn't exhaustively search for it). Feel free to tweak further if you want. –jacobolus (t) 17:35, 19 November 2023 (UTC)

Shouldn't the history section start with the context of Portolan charts?
My impression (not knowing much about this) is that the Mercator projection emerged as part of a historical process of developing maps for sailing based on following "rhumbs" (i.e. bearings, found by astronomical observation or a magnetic compass). It seems like it would be worth discussing this at least a little bit for context, and linking to Portolan chart and rhumbline network, etc. One of the reasons that the speculation about Song Dynasty star charts using the Mercator projection are so implausible is that from what I can tell Chinese navigation did not have the same historical or technological context/process [the compass as a needle in a bowl of water was known in Song China, but I'm not sure if it was being widely used by sailors, and I'm not sure if there were any maps with windrose lines all over them] and star charts per se have no particular reason to make rhumb lines straight. Does anyone have any favorite sources about this topic, or enough knowledge to summarize off the top of their head? –jacobolus (t) 17:39, 19 November 2023 (UTC)
 * I imagine you could answer this easily? Strebe (talk) 21:49, 23 May 2024 (UTC)


 * Hi and ! How and why the Mercator projection was developped and proposed, in 1569, is a long story. Even longer is how and when it was fully adopted in navigation and nautical cartography, well into the 18th century. The truth is Gerard Mercator developped the projection, apparently on his own initiative, without any obvious push from mariners. Some useful references are here:
 * Squaring the circle
 * Globes, Rhumb Tables, ...
 * From tables of rhumbs...
 * Assessing Mercator's world map
 * Have fun! -- Alvesgaspar (talk) 18:37, 29 May 2024 (UTC)
 * @Alvesgaspar Thanks. When you say "without any obvious push from mariners" does that mean you think Mercator did not have marine navigation in mind as an application? I thought Nunes's explicit concept of rhumb lines, adopted by Mercator here, was directly intended to be relevant for marine navigation / arose as a consequence of existing marine navigation practices. –jacobolus (t) 18:57, 29 May 2024 (UTC)
 * While we're here, I started reading the first of those sources, and we should probably mention Dee's (c. 1558) Canon gubernauticus on this page, at John Dee, and at Rhumb line. –jacobolus (t) 19:06, 29 May 2024 (UTC)
 * I have to run for now, but there is also some nice material in:
 * [Edit: which I just realized is the second source linked above. Whoops.] –jacobolus (t) 19:29, 29 May 2024 (UTC)
 * Oh, yes! Mercator was aware of the shortcomings of contemporary nautical cartography, and marine navigation was certainly (also) in mind when he engraved his world map. The very title of the map, "...ad Usum Navigantium Emendate Accommodata" is emphatic on that purpose. The problem was that navigtion was still not prepared to accommodate the novel projection: longitude could not be mesured on board, and compass courses were not corrected for magmetic declination. I don't think Dee contributed much to solve the problem. Although his table of rhumbs (Canon Gubernauticus) was very accurate, it was not used by Mercator. -- Alvesgaspar (talk) 19:16, 29 May 2024 (UTC)
 * Oh, yes! Mercator was aware of the shortcomings of contemporary nautical cartography, and marine navigation was certainly (also) in mind when he engraved his world map. The very title of the map, "...ad Usum Navigantium Emendate Accommodata" is emphatic on that purpose. The problem was that navigtion was still not prepared to accommodate the novel projection: longitude could not be mesured on board, and compass courses were not corrected for magmetic declination. I don't think Dee contributed much to solve the problem. Although his table of rhumbs (Canon Gubernauticus) was very accurate, it was not used by Mercator. -- Alvesgaspar (talk) 19:16, 29 May 2024 (UTC)

Irrelevance of Non-Coaxial Mercator
User:Kencf0618 is insisting on including an image from a website that generates non-coaxial Mercator projections. These are not relevant to understanding the standard Mercator projection and confuse the issues with Mercator by appearing completely different. The author’s argument is that it demonstrates the distortions in Mercator, but there are already several pictures that serve that purpose. If there were actually any applications of such variants of Mercator, they would warrant inclusion in a separate section, right after Web Mercator. But so far as I know they’re just a toy and the author has not presented such uses. Since these variants are confusing, uneducational, non-standard, and not useful, they are irrelevant and I do not think they belong in this article. Andy Anderson 17:37, 23 May 2024 (UTC)
 * I disagree that the topic is irrelevant. In concept, it is a good idea and illustrates the inflation problem evocatively. Unfortunately, (1) this is WP:OR and therefore not allowed, including the external link at the bottom; (2) the wording is invented: there is no such thing as “coaxial”, let alone “non-coaxial” in this context; and (3) the implementation is faulty and confusing: the lobes at the extreme level of distortion are not a feature of the Mercator projection. Strebe (talk) 18:34, 23 May 2024 (UTC)
 * I applied the term “co-axial” to indicate that the axis of the cylinder of projection is different than the Earth’s axis. Seemed more descriptive than “Mercator Extreme” :-). Andy Anderson 19:12, 23 May 2024 (UTC)
 * I see now that the “lobes” are intersecting streets stretched nearly infinitely. That leaves only the WP:OR problem. Also, however, the image as given is unacceptably large for an article. Strebe (talk) 21:54, 23 May 2024 (UTC)
 * It takes a while for the lobes to sink in. The Mercator projection is asymptotic such that the closer you get to infinity, the bigger things get. States become as large as continents, then blocks, then individual buildings. This exponential distortion of familiar terrain hits like the blank, white bar of Antarctica does not. And it's all geometry, straightforward. kencf0618 (talk) 22:29, 23 May 2024 (UTC)

Coaxial is in the Oxford English Dictionary: "Mathematics. Sharing a common axis". The image can readily be expandable with a click, and the external link should certainly remain. That said, and continents notwithstanding, all geography is local... Perhaps we should make it interactive? That would neatly get around the WP:OR issue... And Mercator: Extreme at least as instructive as Nukemap! kencf0618 (talk) 22:20, 23 May 2024 (UTC)


 * The term for “non-coaxial” in this context is “oblique”. As for the link at the bottom of the page, it falls squarely in the WP:NOBLOGS prohibition. The WP:OR assessment is more complicated. I don’t know how making it interactive helps. Strebe (talk) 01:33, 24 May 2024 (UTC)

One source for this type of projection is:



–jacobolus (t) 06:09, 24 May 2024 (UTC)

Nukemap is CC-BY-4.0; Mercator: Extreme CC-BY-3.0. Furthermore it's open source on Github. Secondarily, the exemplary example of exponential distortion is linked on the site itself: the Arc de Triumphe. Finally, why don't the salient objections to Mercator: Extreme apply in full measure to Nukemap? kencf0618 (talk) 12:08, 24 May 2024 (UTC)


 * I don't understand what point you are trying to make. –jacobolus (t) 13:57, 24 May 2024 (UTC)


 * Nukemap is not relevant to justifying the inclusion of “Mercator: Extreme”: Nukemap has an article because it is based on WP:RELIABLE sources and found to be WP:NOTABLE, not because of subjective statements about how useful it is. If you find other content or articles that do not meet minimal standards, those do not amount to an argument to permit even more. kencf0618, you have ignored the WP:NOBLOGS argument, which the link to your site seems to me to be in clear violation of. Unless other editors contradict that, I will delete it. With respect to the illustration, don’t get me wrong: I like it for the reasons I gave. However, its qualifications are in a gray zone, having no formal validation for correctness. It does help that the code is available for inspection, which is what puts it into the “maybe” category for me. My opinion, as just an editor, is that the image, if it to be included, needs to be of similar size to any other on the page, needs to crop out the application UI elements, and needs to move the informative text into a much more compressed space. Andy Anderson, would those improvements, along with this discussion, allay your concerns? Strebe (talk) 17:33, 24 May 2024 (UTC)
 * Agreed that the image needs to be smaller, if that were the only issue I would have fixed it myself. As I said previously, because it is not standard Mercator and is not even recognizable as Mercator without some thought, it needs to be in its own section, "Oblique Mercator” as you suggest, with its own informative description, and really should include some justification of “why bother”. I know there are a lot of map projections out there that are not commonly used, but they were typically developed for particular purposes, e.g. transverse Mercator which is useful in the region of particular meridians. Andy Anderson 17:55, 24 May 2024 (UTC)
 * Oblique Mercator with the code it is. Arc de Triomphe, anyone? kencf0618 (talk) 20:33, 24 May 2024 (UTC)

Terminology in section Properties
The section Properties begins with a very nice diagram illustrating the different possible orientations of the cylinder axis. It uses the term "standard" to refer to the commonly understood representation of the Mercator projection with the Earth and cylinder axes coincident. Jacobolus wants to use instead the synonym "normal" that he asserts is the proper technical term for this orientation. I can find both terms in use, e.g.. However, the paramount consideration here is user understanding, and using a different term from the one in the diagram will certainly be confusing. I have no other objection to the use of "normal", but if it is to be used then the diagram must be changed at the same time. That is Jacobolus' responsibility. -- Andy Anderson 16:42, 26 May 2024 (UTC)


 * Agreed. Strebe (talk) 18:46, 26 May 2024 (UTC)
 * @AndyAnderson If you want feel free to remove or fix the diagram. Repeated addition of "standard" all over the text is awkward and unhelpful. It should say something like "in normal aspect" (or if you demand "in standard aspect" is probably an okay substitute), once, at the top of the section. –jacobolus (t) 06:30, 27 May 2024 (UTC)
 * When the paragraph only refers to the "standard" Mercator but not to the other variants its use is not only helpful but necessary. Andy Anderson 15:57, 27 May 2024 (UTC)
 * @AndyAnderson You need to immediately desist from edit warring, or you can be temporarily blocked from this page. See Edit Warring. If you like we can revert to the version from a few days ago while this gets hashed out. –jacobolus (t) 18:03, 27 May 2024 (UTC)
 * @Jacobolus -- You are the one who keeps changing it back, without responding to my clear rationale as described in my changes or here in the Talk. So who is the one who is “warring”? -- Andy Anderson 18:12, 27 May 2024 (UTC)
 * The standard protocol when there is a dispute based on new changes is to revert the changes causing the dispute and have a discussion, see WP:BRD. What is not okay is repeatedly reverting to enforce your preferred new version that other editors disagree with. My offer here is that if you prefer we can go back to a stable version of the text of the section from a week ago or whenever. –jacobolus (t) 18:58, 27 May 2024 (UTC)
 * I changed the diagram to say "normal". Does this address all parties' concerns?  Justin Kunimune (talk) 23:43, 27 May 2024 (UTC)
 * Thanks Justin. –jacobolus (t) 02:14, 28 May 2024 (UTC)
 * @Jacobolus If you pay attention to the Talk, you will see that it was only you, me, and streib commenting on this, and streib agreed with me. Andy Anderson 00:27, 28 May 2024 (UTC)
 * That's fine. Wikipedia is about building consensus, and edit warring behavior is disruptive of that process. Every editor is encouraged to make "bold" edits which they believe are improvements, but when there is any kind of dispute where two or more editors disagree about the best outcome, the common approach, unless there is some overriding concern (copyright violation, defamation, ...), is to revert new controversial changes and then discuss them on the talk page until consensus can be reached. This substantially reduces squabbling and finger pointing while leaving time for discussion. Consensus is about dialog and compromise, and is not a vote. (Aside: I read the recent discussion carefully, and don't need help "paying attention".) –jacobolus (t) 02:13, 28 May 2024 (UTC)

Standard lines
The use of “standard lines” complicates the text in a way that I don’t think is useful to readers. The literature typically does not use the term, nor even “standard parallels”, with respect to the Mercator. The terms aren’t exactly meaningless in this context, but they are more complicated than the simplistic implication in the text here that they are the paths with “no distortion”. The whole conversation around distortion is misleading. Distortion in a conformal map is not analogous to distortion in nonconformal maps and should not be discussed or treated in the same way. What is important in a conformal map is the rate of change of scale factors, not the “absolute” scale factor, which is arbitrary. In equatorial aspect, the Mercator has zero rate of change of scale factor at the equator, regardless of what nominal scale you assign the map. You can arbitrarily set the nominal map scale to something else such equator’s scale factor is not unity, but when you do that, you assign two opposite latitudes a scale factor of unity while their rate of change of scale (which is their actual contribution to “distortion”: the variation of scale across regions) remains whatever value greater than nothing that it was. Let us please stick with the description of scale increasing away from the equator, not the away from “standard lines” — which isn’t even true if the standard lines are not the equator, since scale increases toward the equator but then decreases again when it crosses. Strebe (talk) 19:05, 26 May 2024 (UTC)


 * @Strebe Did you maybe ping the wrong person? –jacobolus (t) 06:29, 27 May 2024 (UTC)
 * I’ve seen the use of “standard lines” all over the literature, e.g.:
 * "The line of tangency is called a standard line in map projection. For cylindrical and conic projection the simple case has one standard line, whereas the secant case has two standard lines.”
 * "If there is no distortion at every point of a line/curve, we say that it is a line with zero distortion or a standard line."
 * Standard lines are important as reference lines around which projections can be built with lower average distortion. For my work I sometimes design projection variants where I adjust standard lines for this purpose.
 * Having said that, I agree that introducing the idea here complicates things, so I’ve set up the section on "Properties" so that they are only referenced once in the introductory paragraph to explain the graphic, and in the next paragraph refer to the equatorial tangent for what follows.
 * Proposal: All of this disagreement extends from wanting to explain the accompanying graphic. So let’s find another image that only shows the normal, tangent projection, and move the more involved graphic down to the section "Uses" where it will be important to explain transverse and oblique Mercator. Then we can assume normal, tangent Mercator as the default reference until the article reaches that point.
 * -- Andy Anderson 16:46, 27 May 2024 (UTC)
 * @Andy Anderson, my comment about standard lines reads, does not use the term, nor even “standard parallels, with respect to the Mercator. Therefore, references about standard lines in general do not help. I don’t want to be repetitious, but saying that the Mercator has “no distortion” at the “standard lines” is not semantically clear, nor necessarily a useful statement. The whole pedagogy of developable surfaces with secant or tangent lines is fraught, as M. Lapine likes to point out — and his arguments are unassailable. A lot of us steeped in map projections are pleased that someone is finally trying to tear down this cognitive device that has outlasted its utility. In nonconformal maps, standard lines have utility and meaning, but their association with developable surfaces shouldn’t be emphasized, for the reasons Professor Lapine gives, as well as because the notion doesn’t generalize even to the oblique or transverse ellipsoid, let alone more complicated surfaces. Strebe (talk) 16:44, 28 May 2024 (UTC)
 * After adding the disclaimer: "This is sometimes visualized as a projection onto a cylinder which is secant to (cuts) the sphere, though this picture is misleading insofar as the standard parallels are not spaced the same distance apart on the map as the shortest distance between them through the interior of the sphere." I did a google scholar search, and found the following two 2024 papers making the same point:     ––jacobolus (t) 18:07, 28 May 2024 (UTC)
 * @Strebe I tried rewriting the relevant description to be a bit more precise. Does that help? –jacobolus (t) 22:04, 27 May 2024 (UTC)
 * @Jacobolus, yes, that is an improvement. Thanks. Strebe (talk) 16:47, 28 May 2024 (UTC)

Transverse and oblique Mercator
This section is not accurate. The mentioned uses of the transverse Mercator are all against the ellipsoidal form, not the cylindrical form that is the topic of this article. The only mention of ellipsoidal form is about the equatorial aspect. The same goes for the oblique form. There are no common uses for spherical, non-equatorial Mercator. Strebe (talk) 19:11, 26 May 2024 (UTC)


 * The relevant section to add vis-a-vis "extreme mercator" is something like "Artistic uses", and we can link Pérez-Duarte & Swart 2013 as a source, and maybe some other sources of people doing stuff like projecting panoramic images using the Mercator projection, etc. –jacobolus (t) 06:35, 27 May 2024 (UTC)


 * This reversion ignored the reasons I made for the change. In general, I am dissatisfied with User:AndyAnderson's practice of hasty reversion and apparent WP:OWN behavior, particularly as exercised against User:Jacobolus. I am at the point of considering calling in admin help. As I pointed out, we already have articles that address the ellipsoidal versions, which are the only versions used in practice. Meanwhile, the section talks about what are ellipsoidal uses for the Mercator projection without even stating that. The reasons AndyAnderson gives for the reversion make no sense to me. “Along the meridian” means “north-south”; if for some reason AndyAnderson believes the typical reader is ignorant of that, then I think we could work out a suitable text without leaving my criticisms unaddressed. If the text needs a wikilink for national grid systems or such, that, too, could easily be arranged. These sorts of niggles are not normally the subject of reversions, particularly when the people involved are all experienced editors and are knowledgeable about the subject. Strebe (talk) 18:11, 28 May 2024 (UTC)
 * I don't think we need admin help. While AndyAnderson's account was created a long time ago, they aren't a tremendously heavy Wikipedia editor, so probably (I didn't check) haven't been involved in too many recent disputes. I think we can work this out with talk page discussion. –jacobolus (t) 18:15, 28 May 2024 (UTC)

Shape of great circles in the projection?
What is the shape of great circles in the projection? Are they scaled sinusoids (I have not really investigated this myself)? We could maybe add a mention of this in the article. —Kri (talk) 09:56, 15 July 2024 (UTC)
 * This is not remarked on in the literature, as far as I have seen, probably because the shape is not a simple curve. Strebe (talk) 18:47, 15 July 2024 (UTC)