Talk:Mercator projection

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)

@Alvesgaspar: I imagine you could answer this easily? Strebe (talk) 21:49, 23 May 2024 (UTC)

• Hi @Strebe: and @Jacobolus:! 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:

  Leitão, Henrique; Alves Caspar, Joaquim (2014). "Globes, Rhumb Tables, and the Pre-History of the Mercator Projection". Imago Mundi. 66 (2): 180–195. JSTOR 24270011.

  [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)

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)

Recent edits

(Copied from my talk page a comment by Hispalois:)

Dear Strebe, I find your revert rather brutal, getting rid of all my edits. You have even deleted a reference to an academic paper that supports what was already written! I am open to discussing your point about keeping Roel Nicolai's point of view, but I am going to reintroduce the other changes. Looking forward to a respectful and constructive conversation.

Hello, Hispalois. I’m sorry to have caused you distress. I did, in fact, miss the citation you added from Leitão and Gaspar. The citation is apt. Here are my concerns with these edits:

• Mercator was both a geographer and a mapmaker. “Geographer” is relevant here; I don’t know why you would remove that. I don’t mind changing “cartographer” to “mapmaker” — I think cartographer is often improperly used as a synonym for “mapmaker”. However, as a map projection inventor and theorist about map usage, Mercator’s work included academic activities beyond the craft of mapmaking.
• The reasons the Mercator came into heavy usage included several traits that you removed, all of which I find relevant to understanding the projection’s purposes and use.
• I found word choices to be worse than the originals. Star charts are not usually spoken of as “built”; “based” is better; “drafted” would be best. Injecting “first” before “presented” is a solecism: In this context, “presented” means first disclosure by anyone, so the word is useless. And so on.
• The matter of Nicolai. One might describe “startling” as peacocking, but otherwise, the content is relevant in the history of the Mercator, since Mercator was influenced by these maps.

In summary, aside from the citation I removed carelessly, I do not agree with the edits. Strebe (talk) 22:41, 21 September 2024 (UTC)

Hello Strebe. Thanks for detailing your concerns.

• I see your point about "geographer". I removed that word because I found "geographer and cartographer" rather long for the introduction but I left it in the text further down. I have no objection to writing "geographer and mapmaker" in both cases.
• Do you mean "the ability to represent north as "up" and south as "down" "? I see how that is valuable for web maps, but for navigation? I am not saying it doesn't but I fail to see how. A reference would help me. Furthermore, I was concerned by the ethnocentrism or recentism of suggesting that north = up is good. The projection would work perfectly with south = up too.
• based --> built --> drafted; perfect for me, I will do the change.
• "first"; In this I would defer to a native English speaker. In my native Spanish, "presentar" does not necessarily imply first disclosure. The main reason I introduced this word, anyways, is that although Mercator introduced the projection, it was not used again until decades later, in what almost amounted to a second introduction.
• On Nicolai's point of view, I find it is given too much prominence, as his theory has not been broadly accepted by the map history community. There was an intense public debate between Nicolai, Joaquim Gaspar and Tony Campbell in Maps in History magazine. Campbell, in his website, states that: "If the prodigious display of cartographic novelties on the earliest surviving chart, the Carte Pisane, might be able to co-exist with that [Nicolai's] argument, the major hydrographic improvements and the significant inventiveness observed in the work of chartmakers during the few decades that followed evidently contradict it."

Hispalois (talk) 07:28, 22 September 2024 (UTC)

The point is not that north has to be at the top, but that it's helpful to have bearing on the globe uniformly match direction angle on the map. –jacobolus (t) 20:26, 22 September 2024 (UTC)

I think I understand better what your objection about Nicolai’s citation is. We view the purpose differently. For me, the point from the Nicolai citation is that it notes the resemblance to the Mercator projection in portolan charts. Tony Campbell’s rebuttal is not about the Mercator resemblance; it’s about Nicolai’s other hypotheses. The Mercator as best-fit has been borne out by multiple independent cartometric analyses. It does not matter to me which of the many available citations we use to note that best-fit correspondence, and I agree there is no need to inject Nicolai’s speculation about how that might have come to be. Strebe (talk) 22:02, 22 September 2024 (UTC)

I think we are in agreement. It would be great to cite one of those earlier cartometric analyses and not deal with the controversy around the origin of portolan charts in this article about Mercator's projection. Hispalois (talk) 20:59, 23 September 2024 (UTC)

How should we pick which mathematical expression to highlight?

@Eric Kvaalen just added a parenthetical in § Derivation linking down to § Alternative expressions, and was reverted by @Strebe. I agree with the revert, but it did make me consider again what expression we should use here.

I'm a bit biased perhaps having extensively worked on Gudermannian function, but I think it would be worth explicitly calling this the "anti-gudermannian" or "lambertian", and probably using a different expression. Some readers may want to copy a formula from here to use in their own computer implementation, so I think we should give them the best-behaved expression we can, by default.

Currently we have:

By the integral of the secant function, $$x=R(\lambda -\lambda _{0}),\quad y=R\ln \left[\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right].$$

But I think it would be better if we said something like:

The vertical coordinate ⁠$y$⁠ is proportional to the definite integral of the secant of ⁠$\varphi$⁠, called the anti-gudermannian or lambertian of ⁠$\varphi$⁠, $$x=R(\lambda -\lambda _{0}),\quad y=R\operatorname {gd} ^{-1}\varphi =R\sinh ^{-1}(\tan \varphi ).$$

While mathematically equivalent, ⁠$\textstyle \sinh ^{-1}(\tan \varphi )$⁠ is a much better formula to use in a numerical implementation compared to ⁠$\textstyle {\ln }{\bigl (}\tan {\bigl (}{\tfrac {1}{4}}\pi +{\tfrac {1}{2}}\varphi {\bigr )}{\bigr )}$⁠, especially for small values of ⁠$\varphi$⁠.

We also might not want to belabor the "alternative expressions"; we can mention a couple of the most common alternatives here in case someone is trying to match one to another source they saw, but I'd recommend we explicitly direct people to integral of the secant and Gudermannian function for more detail. –jacobolus (t) 18:30, 21 August 2025 (UTC)

I don’t care (much) which formulation is given first. My objection is to the placement and wording of the intrusive parenthetical; I would have the same problem whatever formulation appeared first. The current form is what shows up most often in the literature I have perused. Strebe (talk) 19:03, 21 August 2025 (UTC)

The current form isn't without some logic: the mercator projection is the complex log of a south-pole-centered stereographic projection. It even makes sense as a computation if starting from the colatitude. The issue with it is that, when starting from latitude, adding ⁠${\tfrac {1}{2}}\pi$⁠ to the angle right off the bat sacrifices all of the relative precision near zero. For many purposes this might not matter, but it seems to me like a poor choice for people to copy/paste into their code. –jacobolus (t) 19:38, 21 August 2025 (UTC)

I agree with Jacobolus that the form given first is not a good choice. I did think of replacing it with one or two of the other, symmetrical, forms, but then I noticed that they were mentioned a bit further down, so I just added that parenthesis. By the way, a few months ago I tried to add a formula for the cosine of half an angle (List of trigonometric identities) for the same sort of reason as Jacobolus invokes, but the powers that be wouldn't allow it. Eric Kvaalen (talk) 07:47, 22 August 2025 (UTC)

Description in terms of complex numbers

I came to this article with the goal of understanding what the mercator projection actually is, in a precise mathematical sense. I found it cumbersome to get this from the article. However, there's an alternate very succinct description, which I found on mathoverflow :

" It is the stereographic projection which sends the North pole to infinity and the South pole to 0, followed by the complex logarithm."

(https://mathoverflow.net/questions/124839/deriving-the-mercator-projection-algorithm)

Not only does this pin down what Mercator projection is, it makes it easy to see that it's conformal. While stereographic projection and complex logarithms are slightly advanced (but certainly well known to someone who has studied undergrad math), I think this would be a superior way of describing things for many readers. I would suggest it be featured (prominently) in the article. 2603:7080:A33D:72EE:98D4:F248:1AFB:A322 (talk) 02:41, 30 August 2025 (UTC)

It’s fine to add that to the section on alternative representations. I’m not sure that the greater share readers would be served by starting with, or emphasizing, the complex form. Strebe (talk) 07:13, 30 August 2025 (UTC)

I think it's worth being explicit about this relatively early, which I think is no less complicated than the coordinate-based version.

If you want a nice source, try Klein (1939) [1908] p. 103 ff. –jacobolus (t) 07:21, 30 August 2025 (UTC)

I added a short sentence to the intro section. I think this will be very illuminating for mathematically sophisticated readers, and won't be very distracting for those who aren't. 2603:7080:A33D:72EE:98D4:F248:1AFB:A322 (talk) 14:08, 30 August 2025 (UTC)

The lead section should not contain things that aren't discussed in the body of the article. The appropriate place for this is somewhere under § Mathematics, and it needs to include sources. –jacobolus (t) 16:05, 30 August 2025 (UTC)

Thanks for the suggestion -- I've added the description in terms of complex numbers to the Mathematics section. And I added the nice Klein source that you found. (I couldn't make the citation look nice, or get the link to the internet archive appear; I encourage anyone who knows how to fix that to do so). 2603:7080:A33D:72EE:98D4:F248:1AFB:A322 (talk) 18:08, 30 August 2025 (UTC)

I rewrote this. @Strebe does that seem too obscure? –jacobolus (t) 20:02, 30 August 2025 (UTC)

It’s fine for me because it’s second nature. But I don’t think it’s a good idea for the general audience of an encyclopedia to put the complex form first. I really don’t like the myriad mathematics articles that essentially amount to having to already know the material in order to understand the presentation. I would guess much less than half the people who make it to, and care about, the formulations are versed in complex numbers, whereas real-valued trig and logarithms are available on any real or virtual calculators that goes beyond a four-banger. Strebe (talk) 00:55, 31 August 2025 (UTC)

I'm not sure calculus (and punting to integral of the secant is necessarily that much more accessible than the complex logarithm though. –jacobolus (t) 03:07, 31 August 2025 (UTC)

I agree with jacobulus. (I do support including the real variables discussion, but the complex picture should appear as well).

" I would guess much less than half the people who make it to, and care about, the formulations are versed in complex numbers"

I don't think that complex numbers are that unfamiliar. Many engineers learn a bit about them. Even if it was only a quarter, I think it's worth including, since it's instantly illuminating for those readers. (I myself know little about cartography, and didn't understand what the mercator projection really was. When i saw the complex description, I understood clearly within a few minutes what it is, why it's conformal, and why it maps rhumb lines to rhumb lines) 2603:7080:A33D:72EE:116E:3256:3D8:6BA4 (talk) 14:12, 31 August 2025 (UTC)

Okay, it looks alright, I've made a few small edits. 2603:7080:A33D:72EE:116E:3256:3D8:6BA4 (talk) 00:48, 31 August 2025 (UTC)

I think your change makes this less accessible. –jacobolus (t) 01:23, 31 August 2025 (UTC)

Okay, that's a reasonable goal, but currently it contains some confusing statements. I don't want to get into a revert war with you, so I'll let you make future edits. If you add more details, they need to be clear and correct, or else it does more harm than good. Someone with a math background should be able to read this and quickly understand what the mercator projection is without having to wade through too much extraneous/misleading stuff (as I was able to do after reading the short mathoverflow post).

1. "the association of each with the corresponding point on the sphere is called the Riemann sphere." This makes it sound like the Riemann sphere is a map from C to a sphere, which it's isn't really.
2. "log is a conformal map of the complex plane onto a two-infinite-ended cylinder" if you want to think of it like this, it should be from the *punctured* complex plane (there's no way to define log 0). That would be an okay way of thinking about, though you can also just take out a ray through the origin (corresponding to a half meridian on the sphere), and then map to an infinite strip. If you just say "compose with log", like I had, the reader can figure this out. Including more detail may be sensible, but it needs to be correct.

2603:7080:A33D:72EE:116E:3256:3D8:6BA4 (talk) 13:59, 31 August 2025 (UTC)

Our article Riemann sphere is, frankly, pretty wrong and bad, but I haven't felt motivated enough to go fix it since it would take a lot of work.

As originally understood (that is, before people reading e.g. Wikipedia got the wrong idea) the Riemann sphere is a literal sphere onto which points of the complex plane are projected by inverse stereographic projection, so that each (extended) complex number is associated bijectively with a point on the sphere. The projection itself is not the sphere, so we shouldn't try to give that impression. Maybe I can rephrase in a clearer way.

The Riemann sphere is not, as our says, the same as the the extended complex plane.

Whether you think of the infinite ends of a cylinder as points (at infinity) or not is down to point of view. If you think of them as points, then the origin and point at infinity of the extended complex plane are mapped onto them. If you think of them only as limits, then the origin and point at infinity of the extended complex plane must be included. I'm not sure throwing in "punctured" is going to be that helpful to readers though. –jacobolus (t) 17:36, 31 August 2025 (UTC)

-The allusion to the Riemann sphere is better now. I actually think the article on the Riemann sphere is okay -- you can think of it as the extended complex numbers (and topologized correctly, these do form a sphere).

-You can add in points at infinity if you like, but then it's no longer a "cylinder" (any cylinder should be homeomorphic to S^1 times an interval). If you want to extend the domain of log to all of C, then you have to add in a point to the target, and then the target is no longer a cylinder. (For mapping purposes, you don't in fact want to have points of infinity, since how do you get a flat map out of such a thing). The current phrasing in the article is wrong, and would become correct if " plane" were replaced by "punctured plane", as i suggested. 2603:7080:A33D:72EE:116E:3256:3D8:6BA4 (talk) 19:30, 31 August 2025 (UTC)

I just don't think "punctured complex plane" is going to be universally understood, and for someone who doesn't know what that means it could be a confusing distraction, whereas for someone who does already, it's probably obviously implied.

In any case, you can, if you like, add an explicit point at positive and negative infinity to your cylinder in just the same way you can add a point at infinity to a plane, or, say, a pair of points at infinity to either end of an extended real number line. Or you can alternately say that the logarithm of 0 or ∞ is undefined. More or less as a matter of personal preference. For the purpose of drawing a map as a picture on a physical piece of paper, you obviously have to cut off some finite portion, so the points at infinity aren't really relevant. –jacobolus (t) 20:07, 31 August 2025 (UTC)

Okay, I'm alright with it now. (Punctured complex plane is very standard, and intuitive, terminology. I guess it doesn't hurt it to write it out, though the current phrasing is not very grammatical.)

You are right that you can add points to various things. But it changes their character -- a cylinder with a point added is no longer a cylinder, just as a plane with a point added is no longer a plane.

Thanks for working with me to add in the Complex logarithm section! (I do wish it could be a bit more prominent, but I appreciate there are other opinions...) 2603:7080:A33D:72EE:116E:3256:3D8:6BA4 (talk) 21:56, 31 August 2025 (UTC)

Yes, but "standard and intuitive" for anyone who went through a whole undergraduate math degree is not necessarily known more widely than that. I don't think there's anything ungrammatical, but I agree this is somewhat awkward (which is why I'd just as soon leave the caveat out). –jacobolus (t) 01:33, 1 September 2025 (UTC)

I think this conversation illustrates the complexities of the complex-valued model that bely the notational brevity. Before I say more, let me reaffirm that I am fine with it, and I am fine with our current explanation of it, and I am fine with where it sits in the mathematical section. I’m just saying this to beat the dead horse of “I’m glad this is not first.” — Not to mention that Mercator and Wright did not arrive at the projection this way at all.

The only thing I am not fine with is that we have no mathematical expression for the projection that isn’t buried deep in exposition. This is in contrast to practically every other page on a map projection, where generating formulæ are given immediately and with a minimum of fuss and generally with the most accessible mathematical functions: those functions requiring the minimum mathematical knowledge in order to use. Someone looking for that in this article is going end up drooling with glazed eyes. I really think we need this or something very like it immediately under the mathematics heading:

$x=\lambda ;y=\ln(\tan({\frac {\varphi }{2}}+{\frac {\pi }{4}})$

We don’t have a convention for including or excluding radius R; in general, I think it’s better without. Strebe (talk) 00:22, 2 September 2025 (UTC)