Talk:Geographical distance

Accessibility
The article may be more technically correct after the latest revision, but it seems less accessible to the average reader. The nomenclature may be familiar to a professional in the field, but Wikipedia is read by a more general audience. I think the nomenclature can be simplified without sacrificing accuracy.

I think the text "a point of origination, or standpoint, P_s, and a destination, or forepoint, P_f," belongs somewhere other than in the lead. It is essentially a legend. And it may not be immediately obvious that P_s defines both phi_s and lambda_s.

We are not finding bearings here. Do we care which point is the standpoint? Can't we use '1' and '2' for subscripts? They seem more generic. The subscript 's' is quite shorter than the subscript 'f' — I find it to be aesthetically annoying. The '1' and '2' glyphs are the same height.

The 'sigma' symbol appears only here:


 * $$\Delta\phi=\phi_f-\phi_s;\quad\Sigma\phi=\phi_s+\phi_f;\quad\phi_m=\frac{\Sigma\phi}{2};\,\!$$

Introducing an extra symbol for one-time use seems like a needless complication to me. I suggest simplifying that line as follows:


 * $$\Delta\phi=\phi_f-\phi_s;\quad\phi_m=\frac{\phi_s+\phi_f}{2};\,\!$$

Actually, I would rather see:
 * $$\Delta\phi=\phi_2-\phi_1;\quad\phi_m=\frac{\phi_1+\phi_2}{2};\,\!$$

As a user of US English, I am not accustomed to seeing accented characters. The (accented, subscripted, Greek) symbols $$\acute{\phi}_s \mbox{ and } \acute{\phi}_f\,\!$$ are unfamiliar to me and somewhat difficult to read. No accents are used for colatitudes here: http://mathworld.wolfram.com/Colatitude.html or here: http://en.citizendium.org/wiki/Spherical_polar_coordinates

The symbol $$\mathcal{H}$$ seems cryptic. Why H instead of some other letter? And why the unusual shape? If the special symbol has a special meaning in words, it could be helpful to provide it.

Why $$D_x\,\!$$ rather that a simple $$D?\,\!$$ The distance may be measured in any direction. I read D_x as distance along the x-axis, or perhaps an east-west distance on a sphere. I don't see what is gained by adding the subscript. Rather, I see it as a potential source of confusion.

The $$\Delta^2\phi\,\!$$ symbol is unfamiliar to me. I am more accustomed to seeing $$(\Delta \phi)^2.\,\!$$ I am accustomed to seeing the somewhat cryptic notation sin^2(x) as shorthand for (sin(x))^2. I don't recall seeing delta^2(x) before. Sine is a single-argument function, the Delta symbol is not. Delta^2(x) seems like double-shorthand and may be confusing.

Why place the "Polar Coordinate flat-Earth formula" at the top of the "Flat-surface formulae" section? It is the most complicated of the three flattened-spherical-Earth approximations, and probably the least familiar.

Starting the section on "Pythagorean formula with converging meridians" with an inverse hyperbolic cosine may be akin to posting a "do not enter" sign. I almost never use hyperbolic trig functions. I can imagine an average reader stopping there. The equation $$\mathcal{H} = \sqrt{(\Delta\phi)^2+(\cos(\phi_m)\Delta\lambda)^2}\,\!$$ (which I believe to be exact for a sphere) is accessible to many readers, but it is buried beneath complication which might have scared them off before they got there.

My guess is that the "Pythagorean formula with converging meridians" section now includes two versions: I suggest making them separate sections.
 * Converging meridians on an ellipsoid projected to a plane
 * Converging meridians on a sphere projected to a plane

One who wants to use the simplification of a Pythagorean approximation instead of the haversine formula may not be thrilled to find expressions with hyperbolic functions and a natural logarithm. The version for "converging meridians on a sphere projected to a plane" was pretty simple. If it is in its own section, it will again be readily available to average readers.

The FCC formula now appears in the "Flat-surface formulae" section. But it is not a "Flat-surface formula" because it accounts for two radii of curvature. It properly belonged in the section on "Ellipsoidal-surface formulae".

While $$\cos(\phi_m)N\,\!$$ may be an accurate expression for the left hand side of an equation in the FCC algorithm, I usually look for a single variable on the left hand side of an equation that is one in a series for an algorithm. If one is using the sequence of equations here to write a computer program, they need to make up their own symbol for $$\cos(\phi_m)N\,\!$$. I suggest giving them one -- and expanding the equations something like this:


 * $$\begin{align}

K&=\cos(\phi_m)N=111.41513 \cos(\phi_m) - 0.09455 \cos(3\phi_m) + 0.00012 \cos(5\phi_m);\\ D_x&=\sqrt{(M\Delta\phi^\circ)^2+(K\Delta\lambda^\circ)^2}=\sqrt{(M\Delta\phi^\circ)^2+(\cos(\phi_m)N\Delta\lambda^\circ)^2};\end{align}\,\!$$

The symbol $$C\,\!$$ is used in $$C\Delta\lambda\,\!$$ but it is not defined.

I understand the introduction of the $$\phi^\circ\,\!$$ symbol, but it seems messy to me — similar to the issue with accenting. For me, proliferating symbols can make things more complicated, not necessarily clearer. Likewise, unfamiliar combinations of glyphs.

While the article has been improved in some ways by standardization of terminology, I believe that it has also been made less accessible to average readers in some ways.

I am tempted to fix it, but it may take some discussion to get on the same page. One of the goals in creating this article was to make a place for some less-scary formulas. Subscripted, accented, Greek symbols can be scary. Inverse hyperbolic trig functions can be beyond scary.

How do we proceed from here to make the content less intimidating to an average reader? -Ac44ck (talk) 08:47, 7 December 2008 (UTC)


 * Okay, let me give my reasoning.
 * In terms of preferring _s,_f over _1,_2: I think using numbers is vague and can get complicated. For instance, why is the midpoint _m and not _3, or _2 and the destination _3?  What if you want to break it into four points——then the destination is _4.  If you think "standpoint" and "forepoint" are too technical, then how about _b for the "base" and _d for the "destination" (height and shape-wise they are symmetrical)?
 * As for sigma and delta, and even phi and lambda, those are the generally recognized symbols for the given concepts. If you want to keep it tech-lite, you could change
 * $$\phi\,\!$$ = Geographical Latitude (in radians);
 * $$\Delta\phi=\phi_f-\phi_s;\quad\Sigma\phi=\phi_s+\phi_f;\quad\phi_m=\frac{\Sigma\phi}{2};\,\!$$
 * $$\lambda\,\!$$ = Geographical Longitude (in radians);
 * $$\Delta\lambda=\lambda_f-\lambda_s;\,\!$$
 * to
 * $$L_b,L_d\,\!$$ = Geographical base, destination latitudes (in radians);
 * $$M_b,M_d\,\!$$ = Geographical base, destination longitudes/meridians
 * (in radians);
 * $$LD=L_d-L_b;\quad LS=L_b+L_d;\quad L_m=\frac{LS}{2};\,\!$$
 * $$MD=M_d-M_b;\,\!$$
 * $$\acute{\phi}\,\!$$ is intended to mean the complement $$\phi\,\!$$. Again, keeping it tech-lite, you could change
 * $$\begin{align}

\acute{\phi}_s&=\frac{\pi}{2}-\phi_s;\quad\acute{\phi}_f=\frac{\pi}{2}-\phi_f;\\ \mathcal{H}&=\sqrt{\acute{\phi}^2_s-2\acute{\phi}_s\acute{\phi}_f\cos(\Delta\lambda)+\acute{\phi}^2_f}\end{align}\,\!$$
 * to
 * $$\begin{align}

L_{c:b}&=\frac{\pi}{2}-L_b;\quad L_{c:d}=\frac{\pi}{2}-L_f;\\ \mathcal{H}&=\sqrt{L^2_{c:b}-2L_{c:b}L_{c:d}\cos(MD)+L^2_{c:d}}\end{align}\,\!$$
 * $$\mathcal{H}\,\!$$ is just meant to denote "hypotenuse" and "Dx" is the common abbreviation for distance (actually "dx", but let's avoid the confusion with the calculus differential notation...likewise with "dL" and "dM" P=).
 * I had always been of the mindset that f(x)^2 equals "(f(x))^2" (i.e., "(x)" is intrinsically part of "f", rather than just a descriptive appendage), though was recently straightened out on that. Likewise, "(f'(x))^2" equals "f'^2(x)", while "((f(x))^2)'" equals "f^2'(x)" (the same with subscript placement, such as herein the "Magnetic Field Vector (GSM Coord)" section).  As such, $$\Delta^2\phi\,\!$$ denotes squaring of the difference, while $$\Delta\phi^2\,\!$$ implies $$\phi^2_f-\phi^2_s\,\!$$ (though if you really find that awkward, $$(\Delta{\phi})^2\,\!$$ is okay).
 * No, "Pythagorean formula with converging meridians" is only dealing with "Converging meridians on a sphere projected to a plane": $$\mathcal{H} = \sqrt{(\Delta\phi)^2+(\cos(\phi_m)\Delta\lambda)^2}\,\!$$ IS an approximation for a sphere, the "true" value is found with $$C\Delta\lambda\,\!$$.  That is why I moved the sections around the way I did:  The FCC version is the elliptical equivalent approximation——to find the actual elliptical equivalent, you need to find the loxodromically differentiated average of T (as discussed on Talk:Earth radius), but that is obviously beyond the scope of this article.  I was debating whether or not to just make it a subsection of "Pythagorean formula with converging meridians", but decided to keep it separate.  In terms of keeping it tech-lite, you could change it from


 * $$\begin{align}
 * $$\begin{align}

M&=111.13209-0.56605\cos(2\phi_m)+0.00120\cos(4\phi_m);\\ \cos(\phi_m)N&=111.41513 \cos(\phi_m) - 0.09455 \cos(3\phi_m) + 0.00012 \cos(5\phi_m);\\ D_x&=\sqrt{(M\Delta\phi^\circ)^2+(\cos(\phi_m)N\Delta\lambda^\circ)^2};\end{align}\,\!$$


 * where
 * $$\Delta\phi^\circ,\Delta\lambda^\circ\,\!$$ equals $$\Delta\phi,\Delta\lambda\,\!$$ in degree form;
 * $$M,N\,\!$$ are the meridional and its perpendicular, or "normal", radii of curvature (expressed as kilometers per degree of arc), in binomial series expansion form, set to the Clarke 1866 reference ellipsoid.


 * to
 * to


 * $$\begin{align}
 * $$\begin{align}

R_{ns}&=111.13209-0.56605\cos(2L_m)+0.00120\cos(4L_m);\\ \cos(L_m)R_{ew}&=111.41513 \cos(L_m) - 0.09455 \cos(3L_m) + 0.00012 \cos(5L_m);\\ D_x&=\sqrt{(R_{ns}LD^\circ)^2+(\cos(L_m)R_{ew}MD^\circ)^2};\end{align}\,\!$$


 * where
 * $$LD^\circ,MD^\circ\,\!$$ equals $$LD,MD\,\!$$ in degree form;
 * $$R_{ns},R_{ew}\,\!$$ are the north-south and east-west radii of arc, or arcradii (expressed as kilometers per degree of arc), in binomial series expansion form, set to the Clarke 1866 reference ellipsoid.




 * I don't know if it was an oversight or not, but you didn't give the (an) equation for $$\Delta\widehat{\sigma}\,\!$$. Again, you could change it to $$\Delta{P}=P_d-P_b=\arctan(...\,\!$$ to keep it "tech-lite". P=)
 * A final suggestion. I'm not comfortable using "spherical" as given, as that means vs. plane or vs. elliptical, two completely different applications!  I would suggest defining the flat/plane as "loxodromic" and arc/spherical as "orthodromic" (you can introduce the terms early on).
 * Overall, I just attempted to straighten out the order and simplify the notation——it's your puppy, so I'm not looking to make a stink! P=)   ~ Kaimbridge ~  (talk) 16:51, 7 December 2008 (UTC)


 * Thanks for your reply. I won't reply to all points now. My overall thoughts at the moment are related to the concept of "premature optimization" in programming, wherein "done" may be sacrificed for "better". It seems that what I am perceiving here is over-generalization. If one is seeking a simple approximation, they may not need a "theory of everything". Giving someone a forest when they want a tree may overwhelm them. It creates winnowing work for them; they must sort through more information than they need to find what they wanted. Not to say that this isn't a place to include a "theory of everything", just that I think it can create a challenge to perceive a "shortcut" if it is a needle in a haystack of generalizations. The sigma symbol is handy for representing the addition of several numbers. I think it is overkill in this article as a symbol for adding two numbers.


 * This is not my puppy. That's what WP:OWN is about:
 * If you don't want your material to be edited mercilessly ... by others, do not submit it.


 * It seems that both KISS and Einstein's "Everything should be made as simple as possible, but no simpler" should be attainable here. When I come across the words "loxodromic" or "orthodromic", I have to look them up. I don't think that the fee for admission to a simplified formula should include a trip to a dictionary to read "pertaining to motion at a constant angle to the meridians", which may not be helpful to a general reader. "Loxodromic" doesn't seem compatible with a straight-line distance amidst "converging meridians". The article on orthodromic left me confused as to how it applies here — even after following the link at the top of that article to great-circle distance.


 * The notation $$(\Delta^n x)\,\!$$ seems to have a meaning different from $$(\Delta x)^n\,\!$$ according to this article: First_difference.


 * Others are not so steeped in the concepts of geodesy, I think a "tech-lite" approach is appropriate for major parts of this article, though I would differ on the symbology involved in getting there. Using numbered subscripts can be vague. But I don't see a reason to concern the reader with which end point is numbered '1' or '2' in this article. Why use _m instead of _3 for mean/midpoint? Because it is a derived number from the given end points; it is thus of an entirely different character. I don't see a need for four points along a line in this context, so that doesn't seem to be an issue here. Numbered points can get complicated in a different (and more general) context, but I think they would work fine here. -Ac44ck (talk) 23:23, 7 December 2008 (UTC)
 * Okay, point taken about $$\Delta^nx\,\!$$ (forgot about the special $$d^nx,\Delta^nx\,\!$$ derivative/difference degree meaning!). Just did some minor format cleanup of your revision.
 * A couple points:
 * Why not use the more conceptual $$K_\phi,K_\lambda\mbox{ or }K_{ns},K_{ew}$$?
 * I reordered the polar hypotenuse as $$\sqrt{\theta^2_1\;\mathbf{-}\;2\theta_1\theta_2\cos(\Delta\lambda)\;\boldsymbol{+}\;\theta^2_2}{\color{white}\frac{\big|}{.}}\,\!$$ to illustrate the factoring relationship: $$(x-y)^2=x^2-2xy+y^2\,\!$$! P=)  ~ Kaimbridge ~  (talk) 18:13, 10 December 2008 (UTC)


 * I think there are more discrete sentences than suggested by the current use of semicolons.
 * I prefer to see separate concepts on separate lines. Screen space is cheap. Having the definitions for "delta phi" and "delta lambda" on separate lines makes them easier for me to find.
 * The indentation of D suggests that it is introduced by the text above the values for R.
 * I prefer to avoid two-letter subscripts when possible. I like the notion of using $$K_\phi,K_\lambda$$, but the rendering of the subscript in $$K_\phi$$ may not look close enough to $$\phi\,\!$$ without causing me to pause and ask, "What is that?" In some cases, the top of the glyph isn't closed and the line doesn't extend above the top of the circle. I tend to see the subscript as $$\psi,\,\!$$ which I find to be confusing. This is a screen shot of what I see for two markups:[[Image:Phi and psi.png]].
 * I agree that there is some elegance to the ordering $$\sqrt{\theta^2_1\;\mathbf{-}\;2\theta_1\theta_2\cos(\Delta\lambda)\;\boldsymbol{+}\;\theta^2_2}{\color{white}\frac{\big|}{.}},\,\!$$ but:
 * It doesn't factor neatly into two quantities, like (x+y)(x-y), else we would probably write it as factors rather than with squared terms;
 * I think it obscures the similiarity between this expression and the Pythagorean theorem;
 * It is not the order used in the article on the law of cosines.
 * I don't think we need to specify the units for phi_m in the FCC formula. The user is already working in degrees. If their trig function can handle degrees, they don't need to be burdened with the conversion to radians.
 * I probably should have written the note for "Polar Coordinate flat-Earth formula" to say "colatitude values are in radians". It seems like a waste of effort to convert a latitude value to radians and then subtract it from a messier number than 90. How the user gets to a colatitude in radians doesn't matter, but writing theta = pi/180(90 - phi) may be more confusing than helpful — or not. I am uncomfortable with the present form, but I'm not sure the best way to fix it. -Ac44ck (talk) 21:47, 10 December 2008 (UTC)

Pythagorean formula with parallel meridians
NOTE: THE FOLLOWING FORMULA IS SIMPLY WRONG AND NEEDS FIXING OR REMOVING

(Take for example latitude = 60 degrees, delta-phi = 0 and delta-lambda = 1/60 degrees. Then the formula gives 1 nautical mile but the actual distance is 0.5 nautical miles!)

Moved the above comments from the article. If it is wrong, someone please correct it --Mdebets (talk) 09:33, 10 May 2009 (UTC)
 * I'm not sure you worded the example right, but what I believe you are getting at is that, since lat is at 60° and cos(60°) = .5, then the result should be half. But look at that and the next section:  This section deals with parallel meridians, meaning the width of long difference is the same at the poles as it is at the equator.  The next section takes into account cos(lat).  ~ Kaimbridge ~  (talk) 13:28, 10 May 2009 (UTC)
 * Unfortunately, I can't help you there. I just removed the comment above, which someone else added into the article, and put it here. My mathematical knowledge is rather limited. --Mdebets (talk) 18:13, 10 May 2009 (UTC)
 * Guys, I am going to remove the "parallel meridians" since it is only correct near equator and almost useless. If you don't mind, please delete this talk section after that. The reference to 2 is also wrong. What 2 says is something else, actually relevant for the Eliptical projection, and was designed for Cartesian coordinate system in the vicinity of the point, not for plugging lattitude and longtitude.85.65.152.100 (talk) 19:31, 4 October 2010 (UTC)
 * Good catches. I wonder about this, though:
 * Ultimately, the best accuracy for planar approximation can be achieved if you can obtain correct Cartesian coordinates in the local tangent plane.
 * What are "correct Cartesian coordinates?" Various kinds of projections are possible, including one based on a formula which follows. In which case, why not use the formula directly? - Ac44ck (talk) 04:31, 5 October 2010 (UTC)
 * Explaining. Correct Cartesian coordinates, in any plane, including the local tangent plane to the sphere, imply among other things orthonormality. That is, moving 5 units on one axis makes you travel the same absolute distance than moving 5 units along the other axis. Which is not so for the system (Lat, Lon) in the original version, because everywhere outside equator a single degree in Lat and a single degree in Lon mean different distances, and therefore cannot be plugged into the Pythagoras formula. (If they are, this results in an answer incorrect by factor of cos(Lat) at most). The correct ratio between Lat and Lon units in the local projection is cos(Lat) (Lon to Lat. Lat units are larger than Lon ones) which is the basis for the second, correct formula. By the way, the precision estimation cited is not correct for neither one, because the tangent plane projection to the sphere does not give you real local X,Y. The estimation is probably fully correct for the elliptical approximation. I believe we should reference more data regarding applicability of both approximations, because the truth is that the spherical one is sufficient for most practical purposes, such as location based services, GIS software, etc... 212.179.37.196 (talk) 09:09, 6 October 2010 (UTC)


 * Aren't "correct Cartesian coordinates" only in "the local tangent plane to the sphere?" But where is the "local" of the tangent? Origin? Destination? Midpoint? Always, regardless of the separation between the points?
 * If Geographical_distance is essentially the same thing, why not use it directly instead of going through the effort of constructing an x-y plane based on the same idea?
 * http://www.movable-type.co.uk/scripts/gis-faq-5.1.html
 * If the locations are not already in Cartesian coordinates, the computational cost of converting from spherical coordinates and then using the flat-Earth model may exceed that of using the more accurate spherical model.
 * This statement seems unverified:
 * Ultimately, the best accuracy for planar approximation can be achieved if you can obtain correct Cartesian coordinates in the local tangent plane.
 * I suggest we delete it.
 * The fact that the accuracy of the parallel meridians formula degrades with increasing latitude would not seem to disqualify it from appearing in this article. It probably needs a big caveat; but it is useful near the equator.
 * Even if it is inaccurate in calculating absolute distances at higher latitudes, it still may be adequate for calculating relative distances in a small region on the globe.
 * This article was created to be a repository for various approximations that are admittedly worse than, but simpler than, those in the great circle distance article. I think it is appropriate to restore the simplest formula based on parallel meridians to this article. - Ac44ck (talk) 02:17, 7 October 2010 (UTC)


 * I took out the parallel meridian section again. The formula is simply wrong.  The "accuracy degrades with increasing latitude" only in the sense that the accuracy of a speedometer which is stuck at zero degrades as the car speeds up.  If someone is working with data from near the equator then she can sent cos(phi_m) = 1 in the formula in the "Spherical Earth projected to a plane" section.  Similarly, a Swede might take cos(phi_m) = 0.5 for rough calculations.  Oh and I switched to the symbol for π instead of using PI and sending the reader off to read about Ludolph.cffk (talk) 15:20, 5 August 2011 (UTC)

Link to FCC
I don't understand User:Ground Zero's objection to the previous link to the FCC. It redirected to the Federal Communications Commission article.

Note:
 * http://www.fcc.gov/mb/audio/includes/63-amfmrule.htm
 * The FCC itself does not keep a public database of its rule sections. That task is performed by the Government Printing Office for most agencies. These rules and regulations are compiled in the Code of Federal Regulations (CFR).

The page linked above links to here:
 * http://edocket.access.gpo.gov/cfr_2009/octqtr/pdf/47cfr73.208.pdf

which contains a wordier version of the calculation. - Ac44ck (talk) 01:05, 9 May 2010 (UTC)


 * I don't know, I redirect fine, too——maybe there is a browser specific bug that screws up redirects?  ~ Kaimbridge ~  (talk) 02:23, 9 May 2010 (UTC)


 * Thanks for stepping in and fixing this. I wasn't catching on from his edit summaries that the redirect may not be behaving as expected. Nor did it occur to me that a broken browser may be the issue. Blessed are the peacemakers. - Ac44ck (talk) 04:45, 9 May 2010 (UTC)

Hey, Wikipeople: Where did that term R/100 come from? It doesn't appear in the FCC version. Also, is the 295 mile limit because of the increasing mathematical error from projecting the radius of curvature to a flat surface, or is it because the FCC deems stations farther than that to be non interfering? — Preceding unsigned comment added by 76.232.10.59 (talk) 19:45, 7 January 2014 (UTC)

About the singularity and discontinuity of latitude/longitude
I think this page gives a nice overview and presentation of useful equations. The equations are intuitive and work as expected for most Earth-positions, but unfortunately there are some positions that give problems due to the singularity and discontinuity of the latitude/longitude representation (as for any other set of Euler angles). The problems arise e.g. when making deltas of latitude and longitude, and when taking mean of latitudes (corresponding to the three first equations). Examples:


 * a) If the two positions are close, but on either side of the 180th meridian, the longitude difference will correspond to a distance the other (long) way around the Earth (e.g. 359.999 deg). This was probably not the intention.


 * b) If one position is at a Pole, the longitude difference (delta) is undefined (since longitude is undefined at the Poles).


 * c) If there are two positions a and b at opposite sides of a Pole, e.g. lat_a=89 deg, long_a=0 deg and for b: lat_b=89 deg, long_b=180 deg, then the calculated mean latitude is still 89 deg, but the true midpoint is actually at lat=90 deg.

I have seen many examples through the years where these deltas and the mean are used without considering these issues. I have seen it in code for radar tracking, autopilots, and even Kalman filter for navigation where difference of two positions is used as a measurement. I would estimate that 70 - 80% who implement these equations are not aware of these limitations. The most famous example is probably the F-22 crossing of the 180th longitude meridian (http://www.v3.co.uk/vnunet/news/2184227/f-22-flight-software-crash/). So, I definitively think the issues with the latitude/longitude representation should be mentioned in this article. Then the readers can decide themselves if they have to worry about it, or perhaps their application is never in these locations anyway, so it’s no problem.

I thought it was a good idea to mention these issues directly after the deltas and mean was introduced, and for those who have to worry about it, I included a link to an article discussing these issues, and containing a solution. What I added was the following:


 * Note: The three above equations (for deltas and mean) may not give the correct answer for positions near or at the  Poles or positions at each side of the ±180° meridian, due to  singularities and  discontinuity at these positions. To achieve calculations that are valid for all Earth positions, consider replacing latitude/longitude with another horizontal position representation.

However, this addition was reverted by Ac44ck with the following comment: “The deltas are fine; the other formulae may not use them well”. OK, for the example a) above, I agree that the delta might be considered to be fine, and that the other equations could add specific code to handle the date-line. So my comment could perhaps be inserted at each of the remaining equations instead.

However, all the remaining equations are based on the deltas and mean, and if these three are as expected, all the equations will give the expected answer. Thus I still think it is best to mention the issues directly after these three equations, rather than repeat the same every time they are used. But I fully agree that I should change the wording; so my suggestion based on the feedback from Ac44ck is to change it to the following:


 * Note: The three above equations (for deltas and mean) may not give the expected answer for positions near or at the  Poles or positions on either side of the ±180° meridian, due to  singularities and  discontinuity of latitude/longitude at these positions. This could affect the accuracy or correctness of the subsequent formulae. If calculations that are valid for all Earth positions are needed, consider replacing latitude/longitude with another horizontal position representation.

Hope you find this solution satisfying :) Isaac Euler (talk) 19:49, 27 June 2010 (UTC)


 * I agree that more discussion of the limitations would be appropriate. My objection concerned a statement that the definitions are troublesome. The definitions are what they are.
 * The first case of mentioned above is about which way the difference in longitude is measured. This doesn't matter in the "Polar coordinate flat-Earth formula"
 * Cos(359.999 deg) and Cos(0.001 deg) are the same.
 * The second case mentioned above has one location at a pole. The difference in latitude must be zero in that case. Then the first two formulae are good.
 * The third case above is about points close to the pole with longitudes 180° apart. This is problematic.
 * The article already says:
 * A planar approximation for the surface of the earth may be useful over small distances. The accuracy of distance calculations using this approximation become increasingly inaccurate as:
 * The separation between the points becomes greater;
 * A point becomes closer to a geographic pole.
 * Perhaps this needs clarification or additional emphasis. As a dissertation on singularities, etc., may be much for the intro to the formulae, a separate section may be a better place to elaborate on limitations, problems at the poles, etc. - Ac44ck (talk) 06:39, 28 June 2010 (UTC)


 * Good idea, I have now put it in into a separate section. Now, the readers should be aware of the assumptions/limitations before using the formulae. Isaac Euler (talk) 13:19, 29 June 2010 (UTC)
 * Agreed. However, doesn't the accuracy suffer at great distances if one uses "another horizontal position representation?" The claim that they "are valid for all Earth positions" seems exaggerated. If one needs accuracy over greater distance, the references to great circle distance and Vincenty's formulae seem applicable.
 * A brief interpretation of the fancy words seems in order for the general reader. I'll take a stab at one of them. A one-phrase explanation of the other isn't obvious to me just now.- Ac44ck (talk) 04:29, 30 June 2010 (UTC)


 * About long distance: I agree, it is not sufficient to use any other horizontal position representation than lat/long. Among the alternatives mentioned at the “Horizontal position representation”-page, n-vector is the only one that gives correct answer also for long distances. The distance achieved when using n-vector is the same as the great-circle distance (but without the Pole singularities). To avoid this confusion, I changed the text to link directly to n-vector.


 * If I used some words that you think need more explanations, you could perhaps list them, and I will try to rewrite or add explanation. Isaac Euler (talk) 14:50, 30 June 2010 (UTC)


 * As it stands, the note in the article suggests to the unwary reader that formulas may derail in certain circumstances. Perhaps this is true for those flat-earth approximations (I naturally didn't waste time trying to analyse them) but can anyone find an example where Vincenty's or Bowring's or Lambert's formulas give drastically wrong answers? Vincenty doesn't always converge, but if it converges it's correct-- right?


 * Sure, longitude is undefined at the pole, but the longitude difference between two points is defined well enough, even if one of them is a pole. If one point is 89N 0W and the other is 89N 180W the formulas remain quite unruffled. If the two points are just across the 180th meridian the formulas all work swimmingly-- or don't they? If not, show us an example.


 * In the absence of an example, we should make it clear to readers that the various spheroid formulas work fine, even close to the pole and close to 180 deg longitude. Tim Zukas (talk) 22:34, 11 August 2010 (UTC)


 * You're approaching it wrong.
 * It's not the longitude difference, $$L\,\!$$, or auxiliary $$L\,\!$$, $$\lambda\,\!$$, that is the focus, but rather the auxiliary angular distance, $$\sigma\,\!$$, that is the problem: Along the equator, $$\lambda=\sigma=\tfrac{a}{b}L\,\!$$, so if $$b\,\!$$ is half of $$a\,\!$$, then $$\lambda=\sigma=180^\circ\,\!$$ at $$L=90^\circ\,\!$$.  As you angle away from the east-west equator to the north-south meridian, $$\lambda\,\!$$ and $$\sigma\,\!$$ decrease in comparative value (along a meridian, $$\sigma=\Delta\beta\,\!$$).   ~ Kaimbridge ~  (talk) 13:47, 12 August 2010 (UTC)


 * You say sigma is the problem; now we just need you to show an example. Give the lat-lons of two points for which one of the formulas gives a wrong sigma. Tim Zukas (talk) 23:57, 12 August 2010 (UTC)


 * Actually, I'm addressing the "antipodal effect".
 * Try this: Let $$\phi_1=\phi_2=.001; L=179.2^\circ\,\!$$ and calculate.  Now let $$L=179.999^\circ\,\!$$ and try finding that.  You can't (at least using Vincenty's formula, as is).  But, in both cases, divide $$L\,\!$$ in half, calculate the distance and double it and it will equal the result when using the whole $$L\,\!$$ value (you can use Williams' Great Circle Calculatorif you need).  The "effect" kicks in at about 179.396°.   ~ Kaimbridge ~  (talk) 19:59, 12 August 2010 (UTC)


 * Far as we know Vincenty gives the correct answer for longitude difference 179.2-- agreed? When we shift to lon diff 179.999 his formulas don't converge-- they give no final value of sigma or lambda, and hence no distance. Ed Williams' calculator notes that the formulas didn't converge, but the note doesn't persist, and he doesn't explicitly say "So the figures you see here are just some tentative guesses that didn't turn out to be right, and it seems we can't answer this problem."


 * You seem to be saying we can get the right answer for lon diff 179.999 by doubling the distance from lat 0.001 lon 0 to lat 0.001 lon 89.9995-- which we certainly cannot. I'll work on the actual answer tonight-- you remember Vincenty's direct formulas always converge, so we can get the answer with a bit more work. Tim Zukas (talk) 23:57, 12 August 2010 (UTC)


 * It turns out that if we start from 0.001 N, 0 W on the GRS80 spheroid and proceed 20003710.2179 meters along a geodesic with an initial azimuth of 359.905294 degrees we'll end up at 0.00100000 N, 179.99900000 W. I haven't checked, but I expect Ed Williams' calculator will agree with that. Tim Zukas (talk) 01:28, 13 August 2010 (UTC)


 * Yeah, I worded it wrong (starting with one of the lats should be -.001!). Let me try again, with more detail and better organized:
 * $$\begin{align}

\phi_1&=-.001^\circ,&\phi_2&=0,&L&=89.6^\circ:\;\;&s&=9974.22637507474;\\ \phi_1&=0,&\phi_2&=+.001^\circ,&L&=89.6^\circ:&s&=9974.22637507474;\\ \phi_1&=-.001^\circ,&\phi_2&=+.001^\circ,&L&=179.2^\circ:&s&= 19948.45275014948;\\ &&&&&&&(\alpha=89.999998285^\circ)\end{align}\,\!$$
 * Now increase $$L\,\!$$:
 * $$\begin{align}

\phi_1&=-.001^\circ,&\phi_2&=0,&L&=89.9995^\circ:\;\;&s&=10018.698511639905;\\ \phi_1&=0,&\phi_2&=+.001^\circ,&L&=89.9995^\circ:&s&=10018.698511639905;\\ \phi_1&=-.001^\circ,&\phi_2&=+.001^\circ,&L&=179.999^\circ:&s&=20037.39702327981;\\ &&&&&&&(\alpha=89.9990033389^\circ)\\ &&&&&&s&=20003.93136632067;\\ &&&&&&&(\alpha= 0.0950179774^\circ);\end{align}\,\!$$
 * In the first example, where $$L=179.2^\circ\,\!$$, there is only one possible answer, while the second, where $$L=179.999^\circ\,\!$$, there is a corridor of possiblities between $$\alpha{\approx}0.095^\circ{\to}89.999^\circ\,\!$$ (of course, only the shortest distance, which is the most north-south, is the true geodesic! P=)  ~ Kaimbridge ~  (talk) 15:51, 13 August 2010 (UTC)


 * "...there is a corridor of possibilities..."-- none of which is given as an answer by Vincenty's formula. In your example two geodesics connect the two points, one being shorter than the other and therefore being the right answer to the problem; the point is, Vincenty's inverse formula doesn't give a wrong answer (it doesn't give any answer); when you suggest that it can give wrong answers, you're worrying the reader pointlessly.


 * Or do you still hope to find an example where Vincenty gives a wrong answer? Tim Zukas (talk) 19:12, 13 August 2010 (UTC)


 * I did forget that when you program the formulas that use reduced latitude, you need the tangent of the latitude-- so the program needs to include a separate provision that if the latitude is 90 degrees then the reduced latitude is 90 degrees also. Tim Zukas (talk) 16:48, 12 August 2010 (UTC)

Another approximation
Another approach is suggested here:
 * http://stackoverflow.com/questions/1664799/calculating-distance-between-two-points-using-pythagorean-theorem

For point 1: x1 = r*cos(lat1)*cos(long1) y1 = r*cos(lat1)*sin(long1) z1 = r*sin(lat1) Likewise for point 2. Then d= sqrt(dx^2 + ...) But d is the length of the chord of the great circle. Central angle = 2 * arcsin(d/2/r) Great circle distance = r * central angle in radians. For small values of x, in radians: sin(x) ~ x or arcsin(x) ~ x. For small values of (d/2/r): Great circle distance ~ r * 2 * (d/2/r) = d. - Ac44ck (talk) 04:33, 7 October 2010 (UTC)
 * For antipodal locations, great circle distance ≈ $$ \pi \,\! $$ R - (d / 2R) * sqrt(4R^2 - d^2). - Ac44ck (talk) 11:30, 9 October 2010 (UTC)

Sometimes helpful to convert to n-vector
The section “Singularities and discontinuity of longitude/latitude” contained the following sentence: “This minor problem is unavoidable as long as you're starting with latitude and longitude “

This is correct for some formulas, such as the exact formulas for great circle distance. However, there are several cases where a conversion to n-vector and then back to lat/long is useful, e.g. when calculating $$\phi_m\!$$ which is needed some places in the main article: In general, assume that we have two or more positions given as pairs of lat/long, ($$\phi_1,\lambda_1\!$$), ($$\phi_2,\lambda_2\!$$), ... and we need to find the mean position ($$\phi_m,\lambda_m\!$$). Taking the mean of the latitudes and the mean of the longitudes, would give approximately correct answer if the positions are within a small area far away from the Poles and away from the 180° meridian. But near these locations, the mean position will typically get a wrong value when calculated this way. If the positions instead are first converted to n-vectors, we can simply take the mean of those vectors, and find a new n-vector for the exact mean position. This will work anywhere on the Earth. Finally, this new n-vector can be converted back to ($$\phi_m,\lambda_m\!$$).

Thus, I have rewritten the sentence about “unavoidable”. Isaac Euler (talk) 09:15, 16 August 2011 (UTC)

This is fine. However, I think the n-vector terminology is a little over the top. Gade's paper is just describing what people have been doing for centuries; namely avoiding making silly mistakes. A systematic treatment of how to average angles and other directional quantities is given in K. V. Mardia and P. E. Jupp, Directional Statistics, (J. Wiley & Sons, 1999). cffk (talk) 22:03, 20 July 2012 (UTC)

Edits to ellipsoidal-surface formulae section
I reworked this section substantially to improve the balance and add several missing references to the original work. I also include a plug to my paper which has recently appeared in the Journal of Geodesy. The whole business of geodesics on the earth still gets short shrift in Wikipedia since the subject is spread across multiple articles while getting no decent coverage in any of them (and the Geodesic article has been hijacked by mathematicians so that it is of no help to geodesists). As I find the time, I might try to remedy this. cffk (talk) 22:13, 20 July 2012 (UTC)

Tunnel distance and chord length
I suggest including text to note that the endpoints of a straight tunnel lie on a great circle. The relationship may not be clear to a casual reader. Not taking exception to the validity of the formula for estimating the error in the tunnel distance, but two questions: Further description of the error could help the user decide whether the distance between their points of interest is "short" for the purposes of this error estimate. - Ac44ck (talk) 23:30, 15 June 2013 (UTC)
 * 1) Where may the derivation of this formula be found?
 * 2) Is this a minimum error?
 * It's a two-minute derivation, see this link. The minimum relative error is 0.  I will include $$D \ll R$$ as the definition of short. cffk (talk) 01:06, 16 June 2013 (UTC)

July 2013 suggestions for merging articles
This section is for discussing Fgnievinski's tags suggesting moving articles. cffk (talk) 10:34, 30 July 2013 (UTC)

I think it's OK to merge Great circle into this article. The tunnel distance section is rather silly. Rather than junk up a great circle article with it (a tunnel isn't a great circle!), I would prefer to see this section moved into a footnote (or removed entirely). I think it's a good idea to move the "Ellipsoidal-surface formulae" section to a new page on ellipsoidal geodesics particularly the "Exact method" subsection. I'm rather more dubious of the "Approximate methods" section. I've been mulling over starting a new article on ellipsoidal geodesic for a year now (see my comments two sections up from July 2012) and you can see what I'm thinking in User:Cffk/sandbox/Geodesics on an ellipsoid. I was planning to take a month or two to do a decent job covering the subject (not just the geodetic applications) with lots of inline citations and lots of pictures. My idea is to explain the problem and its solution without necessarily giving the reader a ton of formulas which implement the solution (that's what the original papers are for). Assuming the other editors are patient enough, I would ask them to wait until I'm much further along with the new article before worrying about how to merge existing material into it. cffk (talk) 22:17, 29 July 2013 (UTC)


 * Is the first sentence a question? If so, Great circle, which is about a theory of describing shape, may be distracting in an article on calculating distances.
 * Why is the tunnel distance section silly? As I added it, my objectivity may be impaired. As I agree it isn't a great-circle distance, I don't think this section belongs in the the Great-circle distance article.
 * I think the other methods described in the 'Approximate methods' section are not on-topic in the Vincenty's formulae article, which is about a particular approximate method.
 * I think both types of articles are useful: 1) collections of formulas and 2) explanations of theories. Some are interested in both. Often, one's current need may be for one or the other. Including many formulae in an article on theory may clutter it. Or various related formulae may be strewn on several pages. Consolidation was the reason I created this Geographical distance article as well as the Darcy_friction_factor_formulae article. I see both articles as relevant to those wanting to implement a calculation in a computer program. The complexity/accuracy trade-offs may change as the program develops. The articles can be resources at all stages of development.
 * This Geographical distance article was created to de-clutter Great-circle_distance article, as discussed here: Talk:Great-circle_distance; and to give a home to formulae that may not fit elsewhere.
 * Unlike the Great-circle distance article, the proposed 'Geodesics on an ellipsoid' article is not specifically about calculating distance. I favor retaining relevant formulae here. - Ac44ck (talk) 06:52, 30 July 2013 (UTC)


 * I reworded my comment about Great circle so it's clear it's an answer. It would be more accurate to say that I don't think this section belongs in the Great circle article.  I agree with Ac44ck that it shouldn't be imported verbatim into Geographical distances; perhaps it can just go away.  I don't feel that strongly about the tunnel distance section; my describing it as "silly" (sorry!) reflects that talking about tunnel distance is somewhat out of place in an article on geographical distances which, by definition, aren't Euclidean distances. cffk (talk) 10:34, 30 July 2013 (UTC)

The article on geodesics on an ellipsoid has now been created. cffk (talk) 21:33, 19 August 2013 (UTC)

Bowring
Tim, I think Bowring is grabbing far too much acreage in this article. Why repeat the statement of the problem (defining phi1 lambda1, etc)? 2/3 of the Bowring's formulas are devoted to solving for the great circle using Delambre. This is already covered in the great-circle navigation page. Isn't it better to refer the reader there rather than repeating "Delambre in disguise". That way the reader can understand the nature of Bowring's approximation, how it relates to other approximations, etc. Surely this is better that a canned cookbook recipe which the reader is supposed to mindlessly adopt? If anyone really needs to see the form Bowring put these formulas in, they are in Rapp (1981).

I took out your non-neutral POV remarks about geodesics: "stuck", "complicated"; let the reader judge. Also "lat-lon" is your own private abbreviation and doesn't belong in Wikipedia. I understand the meaning well enough. But what about someone for whom English is a second language?

Here's an example of a 120km line which Bowring underestimates the length off by 1.8mm: Point 1 = (45.3,0), Point2 = (44.175157309504712,0), WGS84. Presumably Bowring was using the advertisers definition of "up to".

cffk (talk) 04:29, 19 August 2013 (UTC)

One reason for preferring an explanation of the method over a mere repetition of the formulas is that it suggests ways in which the method can be improved (to give 1mm accuracy over 180km or even 380km). See http://geographiclib.sourceforge.net/1.32/geodesic.html#geodshort

Finally, who do you think is likely to benefit from the listing Bowring method? Presumably it's someone I can't envision any circumstances where I would recommend the use of this algorithm nowadays (however, to repeat, the method is of some interest). cffk (talk) 12:03, 19 August 2013 (UTC)
 * 1) who needs better accuracy than 1% (which they would be able to get from a great circle calculation),
 * 2) who understands the difference between a sphere and an ellipsoid,
 * 3) who indeed wants the geodesic distance rather than the travel distance (from Google maps),
 * 4) who needs more than a handful of results (which they could get from an online geodesic calculator),
 * 5) who can't install software on her computer (otherwise she could download the NGS calculator or a library for geodesic calculations),
 * 6) who nevertheless has some programming skills,
 * 7) who is sure that she'll never need results for points more than 150km apart,
 * 8) who is happy to transcribe an algorithm she doesn't really understand,
 * 9) and who is satisfied with minimal testing (because she doesn't have access to a large number of exact results)!


 * "Point 1 = (45.3,0), Point2 = (44.175157309504712,0)"


 * Can't argue with that.


 * "a canned cookbook recipe which the reader is supposed to mindlessly adopt"


 * Or to describe it another way, a set of formulas that are complete, easy to follow, and ready to use. But it's true, I hadn't noticed that Rapp's Chapter 6 was online-- that's why I added the formulas, after the earlier link that showed them disappeared. Tim Zukas (talk) 23:02, 19 August 2013 (UTC)


 * "it suggests ways in which the method can be improved (to give 1mm accuracy over 180km or even 380km)"


 * So far it doesn't-- you're going to add the suggestions?


 * "I can't envision any circumstances where I would recommend the use of this algorithm nowadays"


 * Because simplicity doesn't matter, you mean-- you probably agree no formula better combines simplicity and accuracy for points within sight of each other? But no matter, the user can always download something comprehensive to do his calculations and he won't have to deal with anything complicated. So what formulas does the article need to include? Probably none-- no flat-surface, no sphere, no ellipsoid?


 * (By the way, one thing the article does need more of-- it should always give some indication of each formula's accuracy. Why bother giving flat-surface or other formulas without telling the reader how far off they are?) Tim Zukas (talk) 00:10, 20 August 2013 (UTC)

You ask a good question. Here's what I think this page should contain in the way of formulas: For the rest I would prefer that the methods be explained and contrasted with links to the original papers or other sources for the details.
 * the formula using projected coordinates (currently this is treated in terms of the lamest of projections). However if the coordinates start off in terms of eastings and northings, let's say for the UK ordnance survey maps, then using Pythagoras on these (and possibly dividing by the scale at the midpoint) is a good starting point.  (Error is roughly the amount the scale changes by over the line.)
 * the formula for the spherical case (or a pointer to the corresponding article). (Error is roughly 2*flattening.)
 * the formula for the differential distance (see Figs. 2 and 3 in geodesics on an ellipsoid) in terms of the principal radii of curvature. This would be an introduction to ellipsoidal geodesics and to ...
 * the averaged version of the this which I would give symbolically (not with the FCC's numerical values -- that should be a link).

So why not include Bowring's formulas? I try to imagine various categories of people consulting Wikipedia to find out how to determine distance. Here are some:
 * the high-school student doing a project on the migration of humming birds. Bowring is inappropriate in this case (a false sense of accuracy given the subject).  Better would be computing the distance with map coordinates or the spherical formula depending on the coordinate system being used.
 * a recent graduate newly hired by some company in some vaguely GIS-related field. Bowring is inappropriate in this case too.  The company is probably already using a package which includes geodesic distance calculations and that should be used to keep everything consistent and to minimize the amount of code that needs to be maintained.  Even if this is not the case, deploying a formula with limited applicability is a mistake because 3 years from now the recent grad will have moved on and no-one will remember about the restriction to 150km.  (Spherical or ellipsoidal geodesic would be better choices depending on the application domain.)
 * an enthusiast/dabbler (or fellow Wikipedia editor). Much better than giving the formulas would be explaining where they came from (which is what I attempted).  (And add a link to Rapp.)
 * a GIS professional who needs to be reminded what Bowring's method is. Ditto.

Wikipedia, at its best, can allow readers to see the links between different topics. This promotes understanding, can spark interest in related fields, etc. By merely listing Bowring's formulas "cookbook style" you have had a possibly interesting topic boring and wrung all the "poetry" out of the subject. Indeed this does a disservice to Bowring because the paper is more interesting for the methods than for the actual formulas. You've created stovepipes for Lambert's and Bowring's methods separating them from each other and from related subjects. Here's a partial list of the connections you miss as a result:
 * To the great-circle formulas. Didn't your reader just implement these last week!!  Surely they should use these as a subroutine (and learn about that wonderful piece of technology).
 * Since R' is the radius of Gaussian curvature to Gauss's theorema egregium.
 * A generalized Mercator projection (with shifts and scales).
 * To conformal mapping of the ellipsoid to a sphere (it's interesting that you can do this without having the equators coincide, isn't it?)
 * To Taylor series as a way of computing differences.

By the way, the improvements that might be apparent from such a presentation are All of these occurred to me once I understood Bowring's method (by reading his paper). None of them are apparent from the recitation of the formulas (and Bowring, in a mindset focused on programmable calculators, manages to obfuscate Delambre's formulas).
 * using the midpoint of the latitude range as the origin for the Taylor expansion (this double the range of applicability)
 * reformulating the problem in terms of the reduced latitude (since the reduced latitude scales more closely with the meridian distance). This doubles the range of applicability again!
 * applying the method to other surfaces, e.g., to a triaxial ellipsoid.

In all these cases, the object isn't really to come up with a better formula for short distances what will actually be used (the general geodesic solution takes care of this), but to understand better the nature of distances and surfaces, etc. A good analogy might be the article on the sine function. At some point you give the Taylor series for sin(x). But you would be crazy to suggest to the reader that this is a good way to evaluate the sine function (instead of using the built-in function). Instead Taylor series are important in all sort of analysis (radius of convergence, relating sine and sinh, etc.).

cffk (talk) 13:59, 20 August 2013 (UTC)


 * Tim: I went ahead and combined + shortened the Bowring material again adding a pointer to Rapp for details. cffk (talk) 21:33, 28 August 2013 (UTC)

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 * The external link here, a paper by Frank Ivey presented to the North-East SAS Users group, seems not to be a very useful one. I proposed removing it. cffk (talk) 12:49, 9 January 2017 (UTC)


 * I've removed this external link. cffk (talk) 01:36, 16 January 2017 (UTC)

Short-distance/flat-surface approximation
I think that the short-distance/flat-surface approximation should not be derived from that $$|\Delta \phi|$$ and $$|\Delta \lambda|$$ are small but that $$|\Delta \phi|$$ and $$|\cos\phi_\textrm{m} \Delta \lambda|$$ are small. Note that $$|\Delta \lambda|$$ is not justified to be small in the case of high latitude. 240D:1A:7FC:9200:34D3:B0DD:63D3:8646 (talk) 09:28, 10 June 2024 (UTC)

Does anyone know the foremost study in literature about the short-distance/flat-surface approximation? It should be calculated as the tunnel distance by substituting $$\lambda_1 = \frac{\Delta \lambda}{2}, \ \lambda_2 = -\frac{\Delta \lambda}{2}$$ as follows: $$\begin{align} & \Delta X = \left( N(\phi_\textrm{m} + \frac{\Delta \phi}{2})\cos(\phi_\textrm{m} + \frac{\Delta \phi}{2}) - N(\phi_\textrm{m} - \frac{\Delta \phi}{2})\cos(\phi_\textrm{m} - \frac{\Delta \phi}{2}) \right) \cos\frac{\Delta \lambda}{2} \approx -M\left(\phi_\textrm{m}\right) \sin\phi_\textrm{m} \Delta \phi \cos\frac{\Delta \lambda}{2} ,\\ & \Delta Y = \left( N(\phi_\textrm{m} + \frac{\Delta \phi}{2})\cos(\phi_\textrm{m} + \frac{\Delta \phi}{2}) + N(\phi_\textrm{m} - \frac{\Delta \phi}{2})\cos(\phi_\textrm{m} - \frac{\Delta \phi}{2}) \right) \sin\frac{\Delta \lambda}{2} \approx 2 N\left(\phi_\textrm{m}\right) \cos \phi _\textrm{m}  \sin\frac{\Delta \lambda}{2} ,\\ & \Delta Z = (1-e^2) \left( N(\phi _\textrm{m} + \frac{\Delta \phi}{2})\sin(\phi _\textrm{m} + \frac{\Delta \phi}{2}) - N(\phi _\textrm{m} - \frac{\Delta \phi}{2})\sin(\phi _\textrm{m} - \frac{\Delta \phi}{2}) \right) \approx M\left(\phi_\textrm{m}\right) \cos\phi _\textrm{m} \Delta \phi ,\\ & {D_\textrm{t}}^2 \triangleq (\Delta X)^2 + (\Delta Y)^2 + (\Delta Z)^2 \\ & \approx \left(2 N\left(\phi_\textrm{m}\right) \cos\phi_\textrm{m} \sin \frac{\Delta \lambda}{2}\right)^2 + \left(M\left(\phi_\textrm{m}\right) \Delta \phi\right)^2 (\cos^2 \frac{\Delta \lambda}{2} + \cos^2 \phi_\textrm{m} \sin^2 \frac{\Delta \lambda}{2} ) \\ & \approx \left(2 N\left(\phi_\textrm{m}\right) \cos\phi_\textrm{m} \sin \frac{\Delta \lambda}{2}\right)^2 + \left(M\left(\phi_\textrm{m}\right) \cos \frac{\Delta \lambda}{2} \Delta \phi\right)^2 .\\ \end{align}$$ 240D:1A:7FC:9200:2DA6:90E2:798F:920E (talk) 12:49, 14 June 2024 (UTC)

Further well-known simplification is achieved when $$|\Delta \lambda| \ll 1$$ in the case of low or middle latitude: $$\begin{align} {D_\textrm{t}}^2 \approx \left(N\left(\phi_\textrm{m}\right) \cos\phi_\textrm{m} \Delta \lambda\right)^2 + \left(M\left(\phi_\textrm{m}\right) \Delta \phi\right)^2. \end{align}$$ 240D:1A:7FC:9200:DD76:7FBD:EE4C:7775 (talk) 12:21, 17 June 2024 (UTC)