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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.
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)
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!)
I have several minor questions about these edits.
Brief comment on your para "Although . . . the process of derivation has been unclear owing to mere extraction ("ausmultipliziert") of from Bessel's series in order to yield the denominator factors . 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).
The integrand involves the product of the two series given in eq6.3. (See my derivation in modern notation at [4]). The series are of the form
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)
(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))
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.
cffk ( talk) 10:27, 7 August 2012 (UTC)
I've made a start on these changes on my user page at User:cffk/Meridian arc. The only sections I'm changing are
which become
I still have some editing to go (I hope to be done by 2014-09-15), but, for the most part you should be able to see where I'm headed. Some highlights:
Comments are welcome, of course. cffk ( talk) 20:26, 7 September 2014 (UTC)
Made a couple of changes... cffk ( talk) 20:48, 8 September 2014 (UTC)
OK, I've finished. I'm sure some errors have crept in and that the explanations could be improved. Feel free to fix. Please hold off on major changes until they've been discussed on this Talk page. cffk ( talk) 08:19, 12 September 2014 (UTC)
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)
There is presently a proposal to merge the historical section with History of Geodesy. I am against this for the present article is nicely self contained. I prefer a little duplication rather than a spaghetti wiki structure. Any comments? Or will someone remove the merger banner. Peter Mercator ( talk) 14:37, 13 January 2015 (UTC)
Hi, I recently added ( [9]) parentheses around the arguments of trigonometric functions in order to remove the ambiguity but got reverted by two editors stating that "mathematicians would not use brackets in this way" and that the brackets were "clutter" and that "nobody who could use these formulas is confused by this notation".
I'm afraid I do not agree. Mathematicians (and teachers) do normally use brackets the way I used them here. And omitting them in cases like this one makes it unnecessarily difficult to interpret the formulas and causes confusion, except for in those who do "know what they meant".
Sure, there are cases, where parentheses are optional and where it is down to personal preferences or an attempt to achieve consistency in style within an article if they are used even though formally not required. However, the given case is a very clear-cut example where they are not optional, and where omitting them causes ambiguity. Without the brackets, these terms simply become incorrect, as formally there is no rule in mathematics which would give implied multiplication a higher operator precedence over explicit multiplication. In fact, the opposite is true and a student omitting the brackets in cases like this one would fail to get the mark. Anyone trying to parse the terms according to established operator priorities would get a wrong result, that's why the parentheses are needed here. Of course, the formulas would not make sense this way here, but it is only by "not making sense in this context" that a reader could deduce that the author must have meant something different. That's hardly a good basis to explain stuff.
In an encyclopedia for anyone it is our goal to express ourselves as clearly and formally correct as possible. The goal is to simply state what we mean and not to let the reader guess what we could have possibly meant. We cannot take it for granted that a reader might have seen such formulas often enough to guess what was meant (but not stated).
I am well aware of the fact that some authors tend to omit the brackets anyway, even in cases where they really must not, but that's just bad jargon and showing an inappropriate attitude towards readers. It's nothing that should be followed in general, and in particular not in a project like this.
-- Matthiaspaul ( talk) 14:50, 23 November 2015 (UTC)
{{
cite book}}
: Invalid |display-editors=4
(
help)
[1]
Please see: Talk:Earth's_circumference#Requested_move_27_April_2021. fgnievinski ( talk) 01:30, 1 May 2021 (UTC)
I am concerned about the proliferation of this term in the geodesy-related articles. “Meridian” in geodesy doesn’t refer the entire great ellipse; it refers to the half ellipse from one pole to the other (as explained in Meridian). At least, this is true in all the literature I am familiar with. In astronomy, “meridian” refers to the full celestial great circle. Historically the situation was a little more muddled, especially given how important astronomical observations were to geophysical measurements, but I don’t see that ambiguity showing up in the past century. I worry about confusing readers. Strebe ( talk) 17:44, 4 May 2021 (UTC)
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||||||||
|
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.
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)
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!)
I have several minor questions about these edits.
Brief comment on your para "Although . . . the process of derivation has been unclear owing to mere extraction ("ausmultipliziert") of from Bessel's series in order to yield the denominator factors . 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).
The integrand involves the product of the two series given in eq6.3. (See my derivation in modern notation at [4]). The series are of the form
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)
(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))
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.
cffk ( talk) 10:27, 7 August 2012 (UTC)
I've made a start on these changes on my user page at User:cffk/Meridian arc. The only sections I'm changing are
which become
I still have some editing to go (I hope to be done by 2014-09-15), but, for the most part you should be able to see where I'm headed. Some highlights:
Comments are welcome, of course. cffk ( talk) 20:26, 7 September 2014 (UTC)
Made a couple of changes... cffk ( talk) 20:48, 8 September 2014 (UTC)
OK, I've finished. I'm sure some errors have crept in and that the explanations could be improved. Feel free to fix. Please hold off on major changes until they've been discussed on this Talk page. cffk ( talk) 08:19, 12 September 2014 (UTC)
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)
There is presently a proposal to merge the historical section with History of Geodesy. I am against this for the present article is nicely self contained. I prefer a little duplication rather than a spaghetti wiki structure. Any comments? Or will someone remove the merger banner. Peter Mercator ( talk) 14:37, 13 January 2015 (UTC)
Hi, I recently added ( [9]) parentheses around the arguments of trigonometric functions in order to remove the ambiguity but got reverted by two editors stating that "mathematicians would not use brackets in this way" and that the brackets were "clutter" and that "nobody who could use these formulas is confused by this notation".
I'm afraid I do not agree. Mathematicians (and teachers) do normally use brackets the way I used them here. And omitting them in cases like this one makes it unnecessarily difficult to interpret the formulas and causes confusion, except for in those who do "know what they meant".
Sure, there are cases, where parentheses are optional and where it is down to personal preferences or an attempt to achieve consistency in style within an article if they are used even though formally not required. However, the given case is a very clear-cut example where they are not optional, and where omitting them causes ambiguity. Without the brackets, these terms simply become incorrect, as formally there is no rule in mathematics which would give implied multiplication a higher operator precedence over explicit multiplication. In fact, the opposite is true and a student omitting the brackets in cases like this one would fail to get the mark. Anyone trying to parse the terms according to established operator priorities would get a wrong result, that's why the parentheses are needed here. Of course, the formulas would not make sense this way here, but it is only by "not making sense in this context" that a reader could deduce that the author must have meant something different. That's hardly a good basis to explain stuff.
In an encyclopedia for anyone it is our goal to express ourselves as clearly and formally correct as possible. The goal is to simply state what we mean and not to let the reader guess what we could have possibly meant. We cannot take it for granted that a reader might have seen such formulas often enough to guess what was meant (but not stated).
I am well aware of the fact that some authors tend to omit the brackets anyway, even in cases where they really must not, but that's just bad jargon and showing an inappropriate attitude towards readers. It's nothing that should be followed in general, and in particular not in a project like this.
-- Matthiaspaul ( talk) 14:50, 23 November 2015 (UTC)
{{
cite book}}
: Invalid |display-editors=4
(
help)
[1]
Please see: Talk:Earth's_circumference#Requested_move_27_April_2021. fgnievinski ( talk) 01:30, 1 May 2021 (UTC)
I am concerned about the proliferation of this term in the geodesy-related articles. “Meridian” in geodesy doesn’t refer the entire great ellipse; it refers to the half ellipse from one pole to the other (as explained in Meridian). At least, this is true in all the literature I am familiar with. In astronomy, “meridian” refers to the full celestial great circle. Historically the situation was a little more muddled, especially given how important astronomical observations were to geophysical measurements, but I don’t see that ambiguity showing up in the past century. I worry about confusing readers. Strebe ( talk) 17:44, 4 May 2021 (UTC)