# Talk:Meridian arc

Uh, about this meridian arc.. How long is it, then? Would be nice to see how far off the mark they were when they defined the meter.

## Major edit

I had originally intended only to clarify the glaring error in the previous version of this page, namely the fact that meridian distance and meridian radius of curvature were confused: an integral is necessary in the definition of meridian distance. Alas, I was carried away with tidying up and adding wee bits here and there. Very little was removed but much was moved around. I hope this edit meets with approval.    Peter Mercator (talk) 18:25, 2 October 2010 (UTC)

The structure and coherency appears improved. Thank you. Some comments:
• I think calculations, if given at all, should be given on the WGS 84 or an IERS ellipsoid. The geographical applicability of Airy seems unnecessarily restricted. I’m sure someone will come in and replace what’s there eventually unless there is some rationale given for Airy specifically.
Yes. I hope some one is willing to do this.  Peter Mercator (talk) 20:37, 4 October 2010 (UTC)
• Wikipedia uses block formatting, so the five space indentations will be removed by a clean-up bot eventually anyway, I’m guessing.
It will probably happen. Unfortunately wiki defaults of layout (and heading fonts) were not defined by printers. Most books I read have paragraph indents. I would go so far as to suggest they should be compulsory on mathematical pages. Otherwise there is no way of telling whether the text following an equation is part of the same paragraph as the equation or starts a new paragraph. Not a principle I am inclined to die for.  Peter Mercator (talk) 20:37, 4 October 2010 (UTC)
Either indentation or double-spacing between paragraphs is normal typography, but not both together. Sadly the math expressions come with several problems, as you’ve doubtless noticed. Strebe (talk) 02:18, 5 October 2010 (UTC)
• I do think the lede needs to retain the astronomical definition.
I honestly didn't understand what that sentence was meant to convey.  Peter Mercator (talk) 20:37, 4 October 2010 (UTC)
• The Comment paragraphs will get integrated into the text or removed eventually, presumably. The comment on notation, for example, appears to refer to the previous revision of the article. I do not think there is any need for that. It may suffice to note that the notation varies in the literature and to state what the article’s conventions are.
Agree. The comment should go.  Peter Mercator (talk) 20:37, 4 October 2010 (UTC)
Again, thanks for the considerable effort. Strebe (talk) 20:58, 2 October 2010 (UTC)

Many thanks to 官翁 for his thorough check. Particularly picking up the error in the e2/n ratio and my omission in the expression for m in terms of elliptic integrals. (A typing error for it's correct in my notes). PLEASE check the statements about the theta substitution. I am fairly sure that it is the complement of the reduced latitude so the square root should be there. Have you checked through the details of the substitution? (I can't bear to write co-reduced latitude in English!)

Greetings. Sorry for my misunderstanding. I replaced with "co-reduced latitude" (or "reduced co-latitude"? This phrase could be found in Karney (2010)). 官翁 (talk) 23:03, 6 October 2010 (UTC)
The phrase "co-reduced latitude" is meaningless and "reduced co-latitude" is ambiguous. The nomenclature is being discussed elsewhere and Karney(2010) may be amended. In the mean time let's stick to "complement of reduced latitude" which is certainly correct.  Peter Mercator (talk) 16:00, 10 October 2010 (UTC)

I have several minor questions about these edits.

• The phrase "may be analagous" to Ell3 integrals is very confusing since the substitution leads directly to an Ell2 integral. Mathematicians never say "may be": a statement is true or false. (I suppose unproven hypotheses are an exception but that's not the case here).
I amended the phrase. Could you check the main article. 官翁 (talk) 23:03, 6 October 2010 (UTC)
Please check change. I think we should stick to new NIST notation. It will be the standard for the next fifty years. The wiki articles will need editing!  Peter Mercator (talk) 16:05, 10 October 2010 (UTC)
• I am not sure whether every possible form of the n-series should be described. My vote would be for the UTM version since it is widely used. Slow convergence is hardly relevant to modern computing. By all means give the earliest known attributions but I believe it would also help to give more recent English references. You are good at finding references on the web so I hope you may be willing.
I intended to note the fact of chronological detail concerned with usage of parameter "n", not to diversify the explanations. If everyone does not care slow convergence, we do not have to use n instead of e in the first place... 官翁 (talk) 23:03, 6 October 2010 (UTC)
• I would wish to restore my comment at the end of n-series which points out that they can be derived from Delambre by substituting e2 in terms of n.
May be it is possible to perform the substitution itself, but very difficult to follow in order to understand. Since the n-series which appears in the DMA document (to order $n^5$ in the original) is completely coincident with that in the Hinks' report (including the comment about err of the coefficient of $\sin8\phi$), I think it is suitable to refer Hinks' report. 官翁 (talk) 23:03, 6 October 2010 (UTC)
• "Helmert . . . derivation has been unclear . . ." What does this mean? Not clear to whom? Helmert, yourself, or all mathematicians. The sentence should cut be cut unless you can convince on this page.
I tried to elaborate my intention with my awkward English in the main article. Could you check it. 官翁 (talk) 23:03, 6 October 2010 (UTC)
• Is the Japanese reference your own paper. Has it been published in English?
Peter Mercator (talk) 21:12, 4 October 2010 (UTC)
To 官翁. If you don't object I will tidy up your English and make some other small changes during the next two or three days. Time to slow down the edits. I still have a problem with the addition of your own (unreferenced) java code: I suspect that this is very much against wiki guidelines in the sense that "Encyclopedic content must be verifiable". I think you should be content to leave the Kawase material with just the simple comment that it reduces to the previous series. Sadly, Japanese references are fairly inappropriate in the English wiki. Peter Mercator (talk) 20:33, 7 October 2010 (UTC)
I see your point. I have just got rid of JavaScript code, and I am waiting quietly for your edit. 官翁 (talk) 22:06, 7 October 2010 (UTC)

Brief comment on your para "Although . . . the process of derivation has been unclear owing to mere extraction ("ausmultipliziert") of $(1-n^2)^2\,\!$ from Bessel's series in order to yield the denominator factors $(1+n)\,\!$. Especially it is difficult to find an outlook towards the methodology of derivation for further higher terms."

"Ausmultipliziert" here implies nothing more that Helmert gets from eq6.4 (in his text) to eq7.1 by multiplying by unity, written as (1+n)/(1+n), and simply "multiplying out" (the literal translation) all the numerator terms. Nothing mysterious here.

It is also very easy to find a general term for the Bessel series which is presented by Helmert in para6. (I haven't accessed Bessel's own paper because it is a "pay to view" page).

I could access it freely. I would like to introduce an alternative meterial which is a reprint version of Bessel's original:
Engelmann, R. ed. (1876): Abhandlungen von Friedrich Wilhelm Bessel, 3, Verlag von Wilhelm Engelmann, Leipzig, 41–47, [1].
I hope you can access it freely. The Bessel's series which appears in the main article keeps the original result as it is. 官翁 (talk) 08:49, 10 October 2010 (UTC)
Thanks. I read this ok but, unless I am mistaken, there is no derivation of his result for m(\phi). I haven't emended the reference yet for it would be good to find a paper with the derivation somewhere. Peter Mercator (talk) 16:15, 10 October 2010 (UTC)
The general terms of Bessel's series have already been derived as I mentioned at the bottom of comment at 08:15, 10 October. The outline of derivation is regarding the integrand of $m(\phi)$ as a generating function of Gegenbauer polynomials $C_n^{3/2}(\cos 2\phi)$ and expanding it to series of trigonometric functions in accordance with well known formulae for instance NIST library[2]. It would not be mysterious to reach the final result by rearranging the suffix of the summation. If you think it is necessary to show the derivation of the Bessel's result and do not object, I think I would introduce the above outline into the main article by setting up the reference of the previous Kawase's report. 官翁 (talk) 08:20, 12 October 2010 (UTC)
I think we should prevent the page getting too heavy so I have reservations about adding another large chunk of mathematics. For Delambre the derivation of the low order series could be outlined in two or three lines. I don't think this can this be done so easily for the Bessel low order series? A reference to an (English) source for the Bessel proof would be nice. (I hesitate to recommend my own notes). But, why not draft your proposed addition (to the "General" section) in your own Sandbox and let me know (here or by email) when it is available. I suspect that the general reader will be satisfied with the stated low order series without any proofs. Only advanced readers will follow the discussion of the general terms.  Peter Mercator (talk) 11:53, 12 October 2010 (UTC)
Thank you so much for your suggestion and instruction at the bottom. The proposed draft is available from here. Although everything is all right in order to obtain first several terms and of course I understand that it is enough for almost all people, it still seemed to me that it was difficult to generalize Helmert's approximation by simple multipling 1-2n^2+n^4 with Bessel's general formula. So I have taken another approach in the above draft. 官翁 (talk) 23:21, 13 October 2010 (UTC)

The integrand involves the product of the two series given in eq6.3. (See my derivation in modern notation at [3]). The series are of the form

\begin{align} & a_0 + a_1z^1+ a_2z^2+ a_3z^3+ a_4z^4 + \cdots\\ & a_0 + a_1z^{-1} + a_2z^{-2} + a_3z^{-3} + a_4z^{-4} +\cdots \end{align}

where the coefficients are known binomial coefficients augmented with a power of n. (a_k is of order n^k) The coefficient of sin(2n\phi) after integrating comes from the coefficient of (z^n + 1/z^n) in the product of the above series. For example the coefficient of sin6\phi is (a0a3+a1a4+a2a5+...) This is an infinite series but if we truncate at some power n^N the series terminates with the product of two a's for which the sum of their indices is N (or N-1). It would be easy to construct the general term . . . and I suspect it would be less mysterious than that of Kawase! Please try it out.

In general Helmert's series has the best convergence (factor of n^2, so better by a factor of 16 than Dalembert). I suspect that Helmert also knew of the general formula but presented only the finite series that were more than adequate for practical purposes. Look how many numerical examples he gives. I suspect he also knew of the complete ausmultipliziert form which is of course that of Hinks and UTM. Note also how Helmert derives the Dalembert e2 series in eq7.2. The other way is just as easy.

More edits tomorrow. Peter Mercator (talk) 15:56, 9 October 2010 (UTC)

You mentioned that "In general Helmert's series has the best convergence." I also completely agree with your point. I am interested in the reason why Hinks and UTM (as well as [4]) did not adopt the Helmert's series. If the reason were your comment; "Slow convergence is hardly relevant to modern computing", I think it does not seem to be scientific attitude . . .
My comment on convergence was misleading. Improvement of convergence is always vital in big computing programs. It's just not a serious problem in calculating these truncated series on a modern machine. In my own notes I was trying only to construct the series as used by osgb (and my source was Rapp in Survey Review).  Peter Mercator (talk) 16:25, 10 October 2010 (UTC)
Although I seeked for the material about meridian distance, I have not still been able to find the document which introduced the Helmert's series with his name except of Rapp (1991): Geometric Geodesy, Part I [5], 36–40. Of course there are several example which introduce only seties itself without direct reference of Helmert such as Комаровский (2005) [6], 69–71 and Hofmann-Wellenhof et al., (2001): Global Positioning System: Theory and Practice [7], I could not find the Helmert's reference in them. So I suspect that many people were not conscious of the exsistence of Helmert's series or concerned about the ambiguity of the procces of its derivation. It was one of the trigger for me that I have decided to introduce the exsistence of Helmert's series on Wikipedia.
By the way, it has been mentioned that a general term for the Bessel series would be derived as
$m(\phi)=a(1-n)^2(1+n)\sum_{j=0}^\infty\left(\prod_{k=1}^j\delta_k\right)^2\left\{\phi+\sum_{l=1}^{2j}\frac{\sin 2l\phi}{l}\prod_{m=1}^l\delta_{j+(-1)^m\lfloor m/2\rfloor}^{(-1)^m}\right\},$
where
$\delta_i=-\frac{n}{2i}-n,$
in the Kawase's report. But this series is still inefficient . . . 官翁 (talk) 08:15, 10 October 2010 (UTC)

Major tidy now completed. Page going up now. At the end of the day it seems to me that Bessel's step was the most important. Helmert only made a small change from the Bessel formula. Grateful for check of typos and other infelicities -- as well as comments here.  Peter Mercator (talk) 16:31, 10 October 2010 (UTC)
I amended the fractional coefficient of $n^5$ in $D_6$ of Bessel series. In addition, I added $n^5$ terms to UTM series since both the original Hinks' report and DMA-TR TM 8358.2 display to $n^5$ order, which is equivalent or inferior to the Helmert's series. On the contrary to your impression, it was much easier for me to follow the process of derivation of Bessel's series than that of Helmert's. I would appreciate it if you could elaborate the outline of Helmert's small change of simply "multiplying out" all the numerator terms. 官翁 (talk) 23:01, 11 October 2010 (UTC)
Am I missing something? After deriving the Bessel series Helmert just reorganizes the result:
\begin{align} (1-n)^2(1+n)D_0 &=\frac{1}{1+n} (1-n)^2(1+n)^2D_0 =\frac{(1-n^2)^2}{1+n}D_0\\ &= \frac{1-2n^2+n^4}{1+n} \left(1+\frac{9}{4}n^2+\frac{225}{64}n^4+\cdots\right) \end{align}
gives $H_0/(1+n)$ on multiplying out. (Similarly other terms). This is a trivial step. The Bessel series is the important one for it already has the n^2 convergence. The factor befor the bracketed series need be evaluated once only so, although it is simpler for Helmert, it is no big deal. The big time saving is the fast convergence.  Peter Mercator (talk) 14:50, 12 October 2010 (UTC)

## Summary of errors in series

(This is in preparation for a discussion of possible changes in the article.)

All errors are converted into true distance (meters)

Delambre, Bessel, Helmert, UTM refer to the sections of the Meridian Arc article. "Helmert via parametric latitude" refers to the expressions Helmert gives for the meridian arc in terms of the parametric latitude. (Bessel gave the same direct series in his paper on geodesics. Helmert also gives the reverted series, beta(mu), in the section on geodesics.)

"mean rad" give error in estimate of mean radius (leading term in expansion) = 1/4 meridian / (pi/2).

"distance" gives max error in meridian distance.

"rectif lat" gives max error in rectifying latitude, mu (converted to distance) using formula obtained by dividing out leading term and reexpanding. (Adams gives the formula in terms of phi.)

"inverse" gives the max error in the reverted series for the rectifying latitude, i.e., solving for phi (or beta) in terms of mu. (Adams gives the formula for phi(mu).)

Two ellipsoids are considered

A a = 6378137 f = 1/298.257223563 (WGS84)
B a = 6400000 f = 1/150 (NGA's SRMax)

order 4           5           6         (e^2 or n)

Delambre
A   5.71E-5     3.79E-7     2.52E-9     mean rad
B   0.00176     2.33E-5     3.08E-7
A   8.85E-5     5.87E-7     3.91E-9     distance
B   0.00273     3.60E-5     4.76E-7

Bessel
A   6.83E-10    6.83E-10    2.43E-15    mean rad
B   4.27E-8     4.27E-8     6.04E-13
A   3.04E-7     1.05E-9     1.11E-12    distance
B   9.55E-6     6.57E-8     1.39E-10

Helmert
A   5.57E-13    5.57E-13    6.14E-19    mean rad
B   3.48E-11    3.48E-11    1.52E-16
A   6.34E-8     9.44E-11    1.48E-13    distance
B   1.99E-6     5.90E-9     1.84E-11
A   8.88E-8     1.22E-10    2.07E-13    rectif lat
B   2.79E-6     7.64E-9     2.58E-11
A   5.86E-7     1.60E-9     5.58E-12    inverse
B   1.83E-5     1.00E-7     6.93E-10

Helmert via parametric latitude
A   1.22E-9     1.25E-12    1.35E-15    distance
B   3.84E-8     7.85E-11    1.68E-13
A   4.11E-9     1.65E-12    3.94E-15    rectif lat
B   1.29E-7     1.03E-10    4.91E-13
A   9.93E-8     2.38E-10    7.64E-13    inverse
B   3.11E-6     1.48E-8     9.49E-11

UTM
A   1.07E-7     1.81E-10    3.04E-13    mean rad
B   3.37E-6     1.13E-8     3.79E-11
A   2.47E-7     4.17E-10    7.01E-13    distance
B   7.78E-6     2.60E-8     8.74E-11


cffk (talk) 09:44, 7 August 2012 (UTC) (Correct 1/4 meridian to mean radius cffk (talk) 19:08, 1 April 2013 (UTC))

## Some proposals for changes

First of all, congratulations to the editors of this page. It's much more professional than most of the articles in this field—properly sourced, an appropriate level of coverage, etc.

However(!), there's always room for improvement. Here are my \$0.02.

• Explain why the series in n have half the number of terms (invariance under interchange of a and b and φ and its complement). Similarly point out that a/(1 + n) = (a + b)/2.
• Refer to Clenshaw summation for how to sum the trigonometric series.
• The series in n do converge faster, but not for the reason given (namely, the fact that n is a quarter of e2)! In the absence of a real explanation (something to do with the symmetry?), it's fine to just include the statement that the convergence is faster (see previous section). (Also while halving the number of terms is nice, it does not lead to computation efficiently since the coefficients of the trig terms in the series would be evaluated just once per ellipsoid.)
• I'm uncomfortable with "This expansion is important, despite the poorer convergence of series in n, because it is used in the definition of UTM." because it implies (mistakenly, I believe) that this series (with its finite number of terms!) constitutes the definition of UTM. Instead, the UTM document defines UTM in terms of its fundamental properties (conformal, constant scale on a central meridian) and the series is just offered as a possible implementation.

• Series for rectifying latitude μ in terms of φ, obtained by dividing the leading term in Helmert's series.
• The reverted series for φ in terms of μ given, e.g., by Adams in his 1921 monograph.
• Equivalent series in terms of the parametric latitude β. From "Summary of errors" above, it's clear that these converge faster than the corresponding series in φ. In addition these series occur naturally in the computation of geodesics (where β is the latitude on the auxiliary sphere). An historical note: Helmert observed that the series in β converge faster than the corresponding series in φ in section 5.11 of Helmert (1880), "Meridian Arcs by Means of Reduced Latitude." cffk (talk) 15:41, 25 July 2014 (UTC)

### Candidates for removal

• Bessel and UTM series (leaving the references). These have much poorer convergence and I can't envision the circumstance in which they would be preferred over the Helmert series. (I would leave in Delambre's series for its historical interest.)

cffk (talk) 10:27, 7 August 2012 (UTC)

## Meridian distance on the ellipsoid

Be very careful copying length parameterization (curvature,compenent accelerations) formulas from books! Or even explaining them.

You suggested that the distance equation is an easy subst. and while it may be "within tolerance" or for some reason had canceled and be ok: in general it isn't at all.

swok calc p 771 def 15.16 "arc len" x=f(s), y=g(s) so that if s == C (a length) P(x,y) is answer.

However s is NOT L (arc Length of r from 0 to P) as one would assume. s must be an equation that when substituted in r causes r to have arc length c when s(c) is the parameter of r. That's more complicated than s simply s == L == sqrt( dx^2 + dy^2) by far you have to work to find that equation it's not automatic.

(it's more than just knowing the the distance formula, which is a specially adjusted (reimann) integral. and it is NOT the formula which has unit length for s substituded either nor that you substitute unit length or time or angle in)

in other words: if they start saying parameteric parameter. be very wary authors do not always state if "x" is "dx/dt" or "dx" or "dx/ds". the equations are totally different and difficult to convert between in the general sense. — Preceding unsigned comment added by 72.219.202.186 (talk) 05:18, 3 April 2014 (UTC)

The above appears to be mainly nonsense. I have added a link to the derivation of curvature from first principles.  Peter Mercator (talk) 20:34, 3 April 2014 (UTC)