Talk:Transverse Mercator projection

Question
Anybody can contribute an image of a world map in UTM? --Lev 19:44, 7 May 2004 (UTC)

Metric systems and maps within the UTM zones
I am working on systems producing generalised maps (presence/absence and density maps) based on the metric system in UTM-zones. This is fine as long as your data is from within one UTM-zone. But when the data extends over more than one zone several problems occur:


 * how should we have a continous square system
 * how can we keep our precision good enough

The solution on my side is to leave the UTM-based squares and continye by using a system based on longitudes and latitudes. Since it today is quite easy to calculate the square areas of any lon/lat square system density calculations is not a problem.

I would like to have some discussions on this matter on the page describing UTM-sones. What are the shortcomings when it comes to making maps based on UTM-zones...

sample coordinates
please provide some sample coordinates and their latitude/longitude equivalent. Funkyj 06:57, 2005 July 31 (UTC)

nm
Nanometers or nautical miles? —Preceding unsigned comment added by 72.40.45.79 (talk) 09:18, 8 February 2009 (UTC)

Major rewrite
I have prepared a major rewrite of the Transverse Mercator page. The current contents are imbedded within the introductory outline and the first two sections. A new feature is the discussion of the properties of the transverse projection by contrasting them with the normal Mercator. There are two new large sections discussing the derivation of the spherical transverse mercator and a presentation of the formulae for the ellipsoidal transverse mercator. I have called these appendices to head off the casual reader. However I do feel that the mathematics should be up on Wiki even if it is accessible to few readers. If this proposal meets approval (or disapproval) please leave your comments on this page. Please do not edit the proposed page (in my user space) until it is moved up to this page --- if it does! At that stage I would appreciate editing of typos, errors, improvements to the text etc. Checking the maths is more problematic; all I can say is that the formulae come from a pdf article on mercator projections which has been checked by others. The proposed article is here, Peter Mercator (talk) 17:36, 3 April 2010 (UTC)


 * I'm very happy with this rewrite. It is wonderfully detailed. I have not checked all the formulæ, but those I looked at were correct. I would note Ensager and Poder's series development, also. (Engsager, K. E., and K. Poder, 2007 Aug., “A highly accurate world wide algorithm for the transverse Mercator mapping (almost),” in Proc. XXIII Intl. Cartographic Conf. (ICC2007), Moscow, p. 2.1.2.) Strebe (talk) 20:38, 4 April 2010 (UTC)


 * This article increasingly diverges from good Wikipedia practices in the amount and scope of material. It is way too long, and the disparity between the information needed by a casual reader and the minutiæ of series computation is too great. Any number of series developments have been published. They do not belong in this article. Each should be broken off into separate articles, as well as any others people wish to detail. The article needs to get back to the fundamentals of the transverse Mercator. Strebe (talk) 20:44, 13 September 2010 (UTC)

More formulas for meridian distance and Transv Mercator
In 1983 Bowring published (Bulletin Geodesique) formulas for meridian distance that seem to be correct within 0.001 millimeter but are at least as simple as the usual series expansion. I'll add them, unless copyright doesn't allow?

In 1989 he published (Survey Review) formulas for lat-lon to transv-Merc and back; I believe he estimated they were correct with half a millimeter over an 8-degree longitude band, but maybe I'm misreading that. They're simpler too; should I add them to this article? Tim Zukas (talk) 22:56, 11 August 2010 (UTC)


 * Tim, Greetings. A few comments on your mods. Can you add the precise reference for Bowring? I'll look at it next month. Parameter 'n' is already defined above: no need for definition in terms of r, which you haven't defined. I prefer to to write 1/f instead of r, (f=(a-b)/a). The method is neat and I don't think it's easier! It's also probably much slower to compute because of the fractional power of a complex number.  Overall it is much more complicated. Precision is not a problem with the series for they can be extended trivially to higher terms.
 * There are literally dozens of approaches other than Redfearn's and if you want to add more on Bowring go ahead, but possibly include all the stuff together in a new major section under his name. I was going to add another such section on the 'other' Kruger series which are much more accurate than the lambda expansions. My email is accessible through the link 'email this user' on the left panel of my talk page. Peter Mercator (talk) 22:53, 12 August 2010 (UTC)


 * As I recall the Bowring article was page 372-- I'll check. (Page 374, it turns out.)


 * Perhaps I should make it plainer that "n" is the same in Bowring's formulas as everywhere else in the article. I included the definition in terms of r since a and r are the two defining parameters of some spheroids (including WGS84, I think; far as I know the r I gave for it is exact).


 * (Is f a defining parameter for any spheroid?) Tim Zukas (talk) 00:35, 13 August 2010 (UTC)


 * I know of no datums for which f is a defining parameter rather than 1/f. I do not care for r; I know of no literature that uses it. It is always notated as 1/f. Strebe (talk) 06:15, 13 August 2010 (UTC)

I copied Bowring's formula wrong-- it's fixed now. (I left out the factor (1 - (3/8)n^2) in Z.) Tim Zukas (talk) 16:19, 13 August 2010 (UTC)


 * I do not think Bowring’s formulæ belong in this article. There are many developments for the meridional distance, and the meridional distance is only one component of the full development of the projection. One could imagine alternatives for every step of the development. Meanwhile the section states that it presents Redfearn’s development, not Bowring’s or Krüger’s or Thomas’s or Karney’s. I suggest a much better place for this is a new article describing the meridional distance on the ellipsoid. Strebe (talk) 03:27, 17 August 2010 (UTC)


 * "the section states that it presents Redfearn’s development, not Bowring’s..."


 * True enough-- which is implicit in the dates of their formulas, but I'll make it explicit. Once you have the meridian distance you can proceed with the rest of Redfearn's formulas.


 * Looks like Redfearn's series for meridian distance need the terms in $$n^4$$ to give millimeter accuracy-- I haven't confirmed that yet, but I'm working on it. The series probably have to go thru $$n^5$$ to match Bowring's accuracy. (Edit: the last two sentences are probably wrong; the formulas as given might give millimeter accuracy, and with the $$n^4$$ terms they might even match Bowring's method. Tim Zukas (talk) 20:35, 20 August 2010 (UTC)) Far as I'm concerned Bowring's formula is too useful to ignore here, even if it does rate a separate article as well. —Preceding unsigned comment added by Tim Zukas (talk • contribs) 20:39, 18 August 2010 (UTC)


 * By the way-- weren't Redfearn's series for meridian distance based on $$e^2$$ rather than $$n$$? If so, even less reason to object to substitution of another formula. Tim Zukas (talk) 21:51, 18 August 2010 (UTC)


 * I’m afraid my objections are only increasing. “…seem to be correct within 0.001 millimeter” sounds a lot like original research, and “a useful improvement on the truncated versions of Redfearn's series given above” definitely is— not to mention very likely wrong. Do you not understand that the article is about the transverse Mercator, and not about the meridional distance? What is the point in providing greater accuracy to the meridonal distance when the greater accuracy does not show up in the results of the transverse Mercator calculation because of later truncations? Redfearn didn’t truncate his series earlier because he was lazy or too stupid to go further; he truncated them at the correct places needed to achieve the accuracy he targeted. This is all looking very much out of control and irrelevant to the transverse Mercator. Please step back, take a deep breath, and reconsider. Strebe (talk) 00:53, 19 August 2010 (UTC)

Further comments. Perhaps, in creating the current version, I should have stressed that I had chosen the Redfearn/Thomas formulae because they are the basis of several practical implementations of the TM projection, eg UTM, OSGB, Australia and others; they are also the basis of the many web coordinate converters. I intend to tackle the UTM page and link to this page in a detailed way. As such the use of alternative versions does not make sense, even if they are more accurate, for their use would lead to values in disagreement with published data. Both Redfearn and Thomas calculated to higher orders before dropping terms which were smaller than their design criteria (1mm accuracy). The eighth order terms in the meridian distance are proportional to 0.03sin8\phi mm. The highest neglected terms in OSGB are of less than 1mm in northings and eastings. Note that modern develpments of TM (Podor, Karney) which may replace the present UTM etc are orders of magnitude more accurate and Bowring's formulae would be completely inadequate.

I agree that the Bowring formulae would be better in a separate article covering all aspects of meridian distance. I can envisage a page covering the history of meridian measurements, mentioning the Greeks, Newton, Cassini; the arcs in Finland, Peru, France, India etc; the relation of arc meaurements to reference ellipoids; and formulae of all kinds. Could be an interesting page but it would overlap a little with articles such as Figure of the Earth, Reference ellipsoid and Datum; these should be treated as a group. Peter Mercator (talk) 10:10, 19 August 2010 (UTC)


 * Here's the meridian-distance-formula comparison, on the GRS80: first figure is distance in millimeters as given by Redfearn's truncated series, second is the decimal part of the actual distance, third is the decimal part of Bowring's.


 * 15 deg: 1658989589 .3432.34767.34735
 * 22.5 deg: 2489167311 .0967.08445.08406
 * 30 deg: 3320113397 .8651.84501.84464
 * 45 deg: 4984944377 .8373.85798701.85798698
 * 60 deg: 6654072819 .3617.36744.36781
 * 75 deg: 8326937587 .2164.17234.17265
 * 90 deg: 10001965729 .1298.2304570923.2304570914


 * Tim Zukas (talk) 22:20, 25 August 2010 (UTC)


 * The first figure in the above comparison is "distance in millimeters as given by Redfearn's truncated series", which was the series shown in this article at the time. Anybody want to do the comparison for the series that's in the article now? Tim Zukas (talk) 18:45, 7 September 2010 (UTC)


 * I see there is already a page on meridional length: meridian arc. Bowring’s material belongs there, not here, as per the discussion. I have removed this material as extraneous to Redfearn’s method. Strebe (talk) 00:48, 29 August 2010 (UTC)

Bowring's formulas are now together in a separate section. He also gives formulas for scale factor and convergence, but probably nobody has any need for those? Tim Zukas (talk) 15:42, 7 September 2010 (UTC)


 * I have at last had a chance to study the Bowring meridian paper. The formulae he presents are based on the n^4 series given earlier in this article. He first degrades the series by altering the last term so that he can force the series for m into a fairly a simple form: F(1+Gn exp(2i\phi))^N where F,G,N are determined by a comparison of coefficients. He then rewrites the result in terms of the reduced latitude and then rewrites it again as the complex expression quoted by Zukas. The final result is less accurate than the power series in n (to fourth order). The motivation for this rewriting is to shorten the length of the calculation of m. Modern computing power makes this a futile endeavour.


 * The projection equations are, as pointed out below, no more than a rewrite of the fourth order Redfearn series after dropping at least two of Redfearn's terms and applying Schodlbauer's principle to (slightly) improve the range of validity. (I checked in detail only the series for the easting (x) but the same tricks are used in all four series).


 * My contention is that the Bowring papers are no more than Redfearn disguised and as such they offered nothing new in the 1980s and are even less useful today, now that higher order exact solutions avalable. As I mentioned below, I doubt whether Bowring series were ever used but I'm open to contradiction by a single reference. (Consult, for example, the paper by Stuifbergen[ http://www.dfo-mpo.gc.ca/Library/337182.pdf]; he proves that O(n^4, e^8)  series for m is satisfactory. This paper will be discussed in my proposed new sections outlined below).The Redfearn formulae, despite their limitation to 1mm accuracy over a narrow range,  will continue to be of importance whilst they remain in the definitions of OSGB, UTM, Geotrans etc.


 * For all the above reasons I propose that the Bowring material be deleted. I do not even think it warrants a page of its own. Peter Mercator (talk) 21:37, 15 September 2010 (UTC)


 * I cannot find a single academic citation of Bowring nor any implementation actually in use. I do not see how the material could meet Wikipedia’s requirement of “notability”. Strebe (talk) 23:01, 15 September 2010 (UTC)

An easy improvement: more spacing
Compare the readability of the formulas in the sections on Bowring's "Newer Formulas for the Ellipsoid" as of 23:24 on 10 Sept 2010, and then as of 01:05 on 11 Sept, after the formulas have been scrunched together. Clearly, with the additional spacing between them the formulas were easier to comprehend-- probably they should have still more spacing. So what is the objection to added white space? Does it cost somebody something, or does it wear something out faster?

(No doubt the formulas in the other sections of the article would likewise benefit from a bit more white space, but I'll leave that to their authors.) Tim Zukas (talk) 22:55, 11 September 2010 (UTC)
 * Perhaps it is a matter of browser rendering. On Firefox/Macintosh, these new edits of yours leave too much whitespace, much more than any article or convention I have seen. It’s jarring and does not provide adequate material in a single view. Strebe (talk) 16:45, 13 September 2010 (UTC)


 * You're right, I did the editing in Int Expl and Firefox does expand it vertically 10-15%, which is too bad-- so we have a choice between Firefox giving a bit more white space than needed, or IE showing the formulas as a discouraging jumble. You would like "adequate material in a single view" (meaning "all the formulas on one screen with no scrolling"?); I'd say that would be nice if the sequence of formulas were short enough to permit it, but if it isn't short enough then readability takes priority over eliminating scrolling. (If the readers have made their way thru the rest of the article they must be used to scrolling.) How's it look to everyone else? Tim Zukas (talk) 17:42, 13 September 2010 (UTC)


 * "...too much whitespace, much more than any article or convention I have seen." I might be using less white space than the typical printed journal. Compare the layout of the formulas in Vincenty's article http://www.ngs.noaa.gov/PUBS_LIB/inverse.pdf . They're not trying to fit everything onto one page, nor should they. Tim Zukas (talk) 18:29, 13 September 2010 (UTC)


 * It is not a matter of fitting “everything” on one page; it is a matter of seeing a satisfying density of information. Vincenty does not support your position, insofar as my browser renders the transverse Mercator article and Acrobat renders the PDF article. The Vincenty paper is normal. Firefox, Chrome, and Safari render the Redfearn maths well but give way too much whitespace to the Bowring material. I tried IE 8 on Vista, finding the Redfearn material a little cramped, but also still finding the Bowring material too sparse vertically.


 * I have to think this matter has been debated ad nauseam on Wikipedia talk pages, given how much mathematics appears in articles. We should defer to Wikipedia normal practices. I do not see important articles devoting so much whitespace to formulæ. Strebe (talk) 20:39, 13 September 2010 (UTC)


 * We agree the Vincenty article is normal, and it has a full linewidth of white between most of its equations. You may well be right that "Wikipedia normal practice" is to put formulas too close together, so that the reader's eye has to hack its way thru the thicket to make out the sequence of formulas that the reader hopes to follow-- lessee, this formula needs nu, where's that-- need epsilon for that, where's the formula for that-- but consistency isn't everything. Tim Zukas (talk) 22:04, 13 September 2010 (UTC)

May I had my opinion on the white space: it is horrendous. Lots of material could be chopped out of here. Laying out Latex is an art which is easy to get right when fixed in pdf form (see my latex in ref 13) but is very difficult on web pages. I have many comments to make on the Bowring section. (a) It should point out that it is derived from Redfearn and offers no improvement of accuracy. (b) Shorter formulae are not an essential requirement today with the availabilty of modern computers. (c) The method isolates the spherical terms and rewrites them in terms of the exact TM on sphere terms. This is an important point and should be made clearer. It would be better if theta4 and theta5 (spherical terms) were placed directly into the formulae following. (d) Notation. It isn't necessary to explain a,b,k_0,phi,m again in this section. (e) 'r' is completely superfluous and numerical values are not helpful in a 'general' discussion. Bowring could be applied to any ellipsoid. (f) epsilon should be given its usual name. (g) Reduced lat should be named where defined. (h) Stick to x,y as projection coords. E,N make sense only after discussing grid conventions. It also helps comparison with other series. (i) Use lambda rather than omega to assist comparisons. (j) High accuracy m is irrelevant when truncation errors are almost 1mm. Bowring also references less accurate formulae for m. (Moreover evaluation of m from an elliptic integral can be carried out to arbitrarily high accuracy). (k) There is no such thing as a trillionth of a metre in SI units. (l) 'real coefficient of the imaginary part' is nonsense. Normal maths simply writes this as Im(z) so that one has theta=psi-Im(Z). (m) Finally. Has anyone ever used these formulae? I suspect not. Peter Mercator (talk) 23:27, 13 September 2010 (UTC)


 * "(a) It should point out that it is derived from Redfearn and offers no improvement of accuracy."


 * None is claimed.


 * "(b) Shorter formulae are not an essential requirement today with the availabilty of modern computers."


 * Agreed they're not "essential".


 * "(c) The method isolates the spherical terms and rewrites them in terms of the exact TM on sphere terms. This is an important point and should be made clearer."


 * If the reader is unaware of the important point, how will he be prevented from using the formulas?


 * Bowring is untested in the literature. I doubt there is anything wrong with it, but it certainly seems a lot of space, time, and discussion is devoted to a formulation that, after all, delivers little over the classical, well-analyzed, deployed implementations. Bowring does not meet Wikipedia’s requirements for notability, yet a huge chunk of the article is devoted to enticing the reader to use the formulation. Peter Mercator seems hostile toward its inclusion, and, after poking around a lot and seeing the massive expansion in article size (also a violation of Wikipedia guidelines), I too am opposed. Series formulations should be moved out into separate articles, if they warrant inclusion at all.


 * Addressing your question more specifically, it is particularly wrong to direct readers to a formulation whose only purpose is to arrive at numbers, devoid of pedagogical value, when the purpose of the encyclopædia is pedagogical and when the method being purveyed has no notability. It simply seems tendentious. If readers cannot even compare the Bowring formula to Redfearn because the notation differs and important correspondences are not noted, then there is no pedagogic value at all. It belongs on some other website. People can search the Web wonderfully easily these days. Strebe (talk) 01:59, 17 September 2010 (UTC)


 * You seem to be saying Redfearn's method is pedagogical, and notable, and Bowring's is not. It will be impossible to explain why Redfearn's formulas (the formulas themselves, as given here) are more pedagogical; if it's pedagogy you want, we could always include Bowring's derivation (or Redfearn's?) but few others will be interested. Comparing "notability" sounds tough too; we'd have to say Vincenty's 1975 formulas for geodetic distance are less "notable" than Rainsford's 1955 formulas, wouldn't we? But Rainsford would likely be the first to agree that Vincenty's formulas are the ones that belong on Wikipedia.


 * Is Redfearn still around, by any chance? Would be nice if we could ask him which formulas he now prefers. Any reason to think he'd prefer his? Tim Zukas (talk) 00:58, 29 September 2010 (UTC)


 * No, I am saying being able to compare approaches is pedagogical. Second, it’s not hard to compare notability here: I can’t find a single citation of Bowring. I can find plenty of Redfearn. Vincenty versus Rainsford is irrelevant here, but part of the job of editors is to come to a consensus about notability, so if there were some reason one needed to be chosen over the other, one would be chosen. And lastly, Redfearn’s and Rainsford’s opinions aren’t encyclopædic, and therefore would be irrelevant even if we had them, so I don’t understand where you are going with that. Strebe (talk) 02:38, 29 September 2010 (UTC)


 * In "What Wikipedia Is Not", it says "Wikipedia is an encyclopedic reference, not a textbook. The purpose of Wikipedia is to present facts, not to teach subject matter." So as far as I can see your objection to lack of pedagogy is backwards-- pedagogy is just what Wikipedia doesn't want. Just the facts, ma'am. Wikipedia wants the reader who hopes to learn how to use Transverse Mercator coordinates to be able to find the necessary formulas in the article; Wikipedia wouldn't prefer a more pedagogical set of formulas, even if there were any reason to think Redfearn's formulas were more pedagogical than anyone else's.


 * ("No, I am saying being able to compare approaches is pedagogical." You mean comparing Redfearn's approach with Bowring's? You're saying the article should contain both? The article would be even better with more approaches?)


 * I looked up "Notability"-- it refers to the topic, not the content of the article. Wikipedia wants to know, Is an article on Transverse Mercator justified? Is the topic Notable? Once it's been decided that yes, it is, then Notability is no longer an issue, and we don't need to waste time trying to decide whose formulas are more notable than whose. Tim Zukas (talk) 16:52, 29 September 2010 (UTC)


 * I hate to break it to you, but facts are pedagogical. “Having pedagogical value” ≠ “reads as a textbook”. I did not advocate “…creating or editing articles that read as textbooks, with leading questions and systematic problem solutions as examples,” which is Wikipedia’s explanation of “not a textbook”. Try searching on "consistent notation" within the site:en.wikipedia.org to learn how often this is discussed and resolved in favor of consistent notation. Consistent notation is part of presenting facts effectively. With your interpretation, it would be our duty to go sabotage anything that might help anyone learn anything as long we leave just the facts alone.


 * With apologies for using alternative terminology, how about you look up “prominence”?“…Fairly representing all majority and significant-minority viewpoints that have been published by reliable sources, in rough proportion to the prominence of each view.” Bowring has no prominence.


 * Lastly, no; my advocacy is plain: Neither series should be presented on this page. I know Peter Mercator is preparing to move Redfearn elsewhere. If he does not remove Bowring at that time, then I will. You are welcome to move it to some other article, of course, though that one would likely suffer from a “notability” problem. Strebe (talk) 18:38, 29 September 2010 (UTC)


 * "facts are pedagogical"...okay, sounds like you have fortunately discarded your earlier objection to "direct[ing] readers to a formulation whose only purpose is to arrive at numbers, devoid of pedagogical value, when the purpose of the encyclopædia is pedagogical". We're agreed that whatever pedagogy is, each set of formulas has enough of it, for Wikipedia anyway-- so no need to worry about that any longer.


 * "Neither series should be presented on this page." So you propose a "Transverse Mercator" article with no formulas, and a separate article "Formulas for the Transverse Mercator" (which will not repeat the explanation of the nature of the projection?). The former has little use except as an introduction to the latter, so hard to see how they're better separate. You figure Wikipedia prefers that one long article be split into two articles adding up to a bit more length? Even tho the table of contents of the long article immediately directs readers to whatever they need?


 * It's true, of course, the article needs pruning; if we're worried about length, the best plan would be to move the formulas for the ellipsoid to a separate article, but include the necessary explanation of the projection-- then call that article "Transverse Mercator" and dump this one. Tim Zukas (talk) 20:24, 29 September 2010 (UTC)


 * The formulas you talk about are truncated series. They are not the definition or description of the transverse Mercator; they are merely different ways of approximating it out of a theoretical infinite number. Dozens of such series and variants of them are extant in different national and regional projection systems. No, I am not advocating a separate article to hold all of the different formulations of the transverse Mercator, and I cannot imagine how you came to that conclusion. I’m advocating a separate article for each notable (prominent?) usage. There are several reasons for that: The article is too long; the “implementations” generally do not even go by the name “transverse Mercator”; and details of the implementations do not add to the “facts” about the transverse Mercator itself, which is supposed to be the topic of the article. At some point the closed-form representation needs to get added to the main article; they are the formulæ that belong in it.


 * …sounds like you have fortunately discarded your earlier objection to "direct[ing] readers to a formulation whose only purpose is to arrive at numbers, devoid of pedagogical value, when the purpose of the encyclopædia is pedagogical" That again does not follow, since you do not bring the context of the discussion into your comments. I did not object to the inclusion of Redfearn even though its only purpose was to arrive at numbers. I objected to the inclusion of a second set of formulas whose only purpose was to arrive at about the same numbers when the reader could not even readily compare the two formulations because their notations differed. Strebe (talk) 22:17, 29 September 2010 (UTC)


 * "It would be better if theta4 and theta5 (spherical terms) were placed directly into the formulae following."


 * Bowring kept them separate; I'm guessing most people who compare the alternative layouts will agree he was right.


 * "(d) Notation. It isn't necessary to explain a,b,k_0,phi,m again in this section."


 * No, we can always make the reader hunt them up. Taking up five extra lines to give the reader all the info he needs is indeed wasteful.


 * "(e) 'r' is completely superfluous..."


 * See the new section, below.


 * "(f) epsilon should be given its usual name."


 * Bowring used epsilon; what's the point of switching?


 * "(g) Reduced lat should be named where defined."


 * That definitely falls under the heading of "Interesting info for those who are interested in such things, but if you just want the essentials, you don't need this." You agree the formulas will work fine without the interesting info?


 * "(h) Stick to x,y as projection coords."


 * British National Grid and US State Plane Coords use E and N. It's true, I should have stuck to Bowring's E' and N' notation, since the formulas give distances from the central meridian and equator, rather than actual Easting and Northing.


 * "(i) Use lambda rather than omega to assist comparisons."


 * No big objection to that-- but I'm guessing there won't be many comparisons made.


 * "(j) High accuracy m is irrelevant when truncation errors are almost 1mm."


 * True enough. The point is Bowring's meridian distance formulas are simpler; the increased accuracy is no big deal.


 * "Bowring also references less accurate formulae for m."


 * Hm. You think he decided his 1983 formulas are wrong after all?


 * "(Moreover evaluation of m from an elliptic integral can be carried out to arbitrarily high accuracy)."


 * Feel free to supply the additional terms needed.


 * The elliptic integral can be computed by iteration using common transcendental functions. See AGM series in Abramowitz & Stegun or Bulirsch’s refinements for computers. Also, papers by Carlson (“Carlson symmetric form”) for highly general methods, both real and complex. Strebe (talk) 01:59, 17 September 2010 (UTC)


 * "(k) There is no such thing as a trillionth of a metre in SI units."


 * No one said there was. You think "a picometer" would be better? Why?


 * Because “trillion” means different things in American English and British English. Strebe (talk) 01:59, 17 September 2010 (UTC)


 * "(l) 'real coefficient of the imaginary part' is nonsense."


 * Yeah, "real coefficient of $$i$$" would be more correct.


 * It is simply “imaginary part”. Strebe (talk) 01:59, 17 September 2010 (UTC)


 * "(m) Finally. Has anyone ever used these formulae? I suspect not."


 * If you mean, Do they work?, the answer's yes. Their greater simplicity will be evident to anyone who programs the alternative sets of formulas; if millimeter accuracy isn't good enough for the user then he will have to look elsewhere, but most people will find no advantage to the traditional formulas. Tim Zukas (talk) 20:48, 16 September 2010 (UTC)


 * Other than that the traditional formulæ are well-tested, actually used in national and international systems, and computational results can be compared on large test sets. It is not good enough for the newcomer to be “not worse”. I agree Bowring’s formulation is easier on the eyes, for the most part. It’s just not an important advance, particularly for this era. Strebe (talk) 01:59, 17 September 2010 (UTC)

Suggested revisions
I am presently rewriting the section on the ellipsoid version. Initially I thought that the Redfearn series provided the major implemention of TM (Transverse Mercator). I now realise that it is just as important to include the 'other' series developed by Kruger. It is much more accurate and moreover it is now in use by a number of national agencies (France, Finland, Sweden . .). Higher accuracy series have also been developed which may well be implemented by national agencies and commercial (oil) companies. These solutions are accurate to within a few nanometres within  4000km from the central meridian and will allow the use of very wide zones. In addition I feel it is important to include a discussion of the closed form (exact) TM projections. These have now been evaluated to horrendous accuracy and they provide an invaluable tool for the error analysis of the truncated series.

These considerations show that this article is getting too long and very maths heavy. The best solution seems to be to keep the discussion of the ellipsoidal series solutions as general as possible in this article and create three or four new pages with the details of the various series. (For the two Kruger variants, Lee-Thompson and Bowring -- and possibly others. Peter Mercator (talk) 22:38, 13 September 2010 (UTC)


 * Since no one has commented adversely on the above proposals I have removed much of the mathematical material to a new page for TM-Redfearn. I am working on two new pages, TM-flattening-series, TM-exact. I have also moved the Bowring material (and shall copy the Bowring discussion) to its own page without any further comments on its value.


 * My justifications are two-fold. Firstly, with the two new sections mentioned above, and new additions to the Redfearn material and possible further sections for other variants, the TM page will grow into a rather large and messy article. Secondly, my sympathy with the general reader prompts the construction of a less mathematical page here. From this article he can gain a general survey and then go to more "advanced" pages for technical details.


 * It is possible that the mathematics of the TM-sphere section should move. I favour keeping it because of the value of the figures to even non-mathematical readers. (eg the relation of the graticules and the introduction of convergence).


 * It would be good to have some feedback from other "watchers" on the actions I have taken (and possibly on the Bowring discussion). It would also be good to have some assistance with copy edits, link additions and checks of references etc. (on this page and the TM-Redfearn TM-Bowring pages.


 * Peter Mercator (talk) 22:05, 2 October 2010 (UTC)


 * I find the restructuring to be an improvement. For my own part, I prefer the exact formulation to appear in the main article on the transverse Mercator. It is not usual to separate the mathematical description of the topic from the topic. It is, after all, not so very long in this case. For the same reason I prefer keeping the spherical definition on the main page. I agree with the other rationales stated. Thanks for all the effort. Strebe (talk) 21:29, 3 October 2010 (UTC)

Ellipsoid formulas: imagine you're the reader
Let's say we have come to this article looking for formulas to convert between lat-lon and transverse mercator. We know lat-lon is plotted on a spheroid, and we've learned the WGS84 defining parameters: equatorial radius 6378137 meters, reciprocal of the flattening 298.257223563 (hereinafter referred to as $$r$$). Getting started with the article we find


 * In addition to the parameters $$a$$ and $$e$$ the series employ:

\begin{align} b&=a(1-e^2)^{1/2},\qquad\qquad n=\frac{a-b}{a+b}. \end{align} $$

Looking thru the rest of the formulas we see we're going to need $$e^2$$ and $$n$$... but the author figures we shouldn't need to be told that $$e^2$$ equals $$(2r - 1)/r^2$$, so we have to go someplace else to learn that $$e^2$$ is 0.00669437999012 for the WGS84. Okay, now we need to calculate $$b$$; the author figures we shouldn't need to be told that $$b$$ equals $$(r - 1)/r$$ times $$a$$, so we plug $$e^2$$ into the formula he has given us and find that $$b$$ is 6356752.31425 meters, allowing us to find that


 * $$n \quad = \quad \frac {6378137 - 6356752.31425}{6378137 + 6356752.31425}$$

Think we might be a bit piqued when we someday learn that $$n$$ is just $$1/(2r - 1)$$?

Peter Mercator says the use of $$r$$ is superfluous-- and it's true, it's only useful if you're aiming to give the reader the info he needs as simply and directly as possible. Tim Zukas (talk) 19:53, 16 September 2010 (UTC)

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