User talk:Salix alba/Archive 3

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As a follow up of the discussion above, now that the Starwood arbcom case is over, what do you think of candidating for admin? You've been here for a long time, did a lot of work, and I think you will have very good chances of passing. So what do you think? You can reply here. Cheers, Oleg Alexandrov (talk) 20:03, 30 March 2007 (UTC)

Thank you very much for fixing my erroneous "hangon" tag for Mean information. Bearian 16:13, 18 April 2007 (UTC)

Derivative

Hello Salix alba - you commented on the GA status of Derivative a while ago. In the course of rewriting the article, I believe your comments have been (at least partially) addressed. Your further comments would be much appreciated. Geometry guy 18:43, 31 March 2007 (UTC)

Ah, the King weighs in!

Do you know everything that goes on? You seem to turn up everywhere! Well, it's nice to see you -- even though your math guys nearly gave Pigman a nervous break down. I guess you math people are a story unto you selves. And you will forever remain a mystery. Some one was musing to me the other day about your support of User:Ekajati and I said that I really could not complain because you have been wonderful to me. Sincerely, --Mattisse 20:20, 2 April 2007 (UTC)

P.S. I am referring to Wikipedia:Miscellany for deletion/Wikipedia:Association of Members' Advocates. I realise that was not clear. --Mattisse 20:22, 2 April 2007 (UTC)
Its easy really, I have your talk page on my watchlist - just watching it in case theres any trouble brewing, thankfully things seem quite. Occasionally something of interest pops up.
I'll say nothing on starwood, case closed and its best forgotten.
Pigman and addition was interesting. Yes we could have been a bit more supportive. I think there is something intrinsically dificult with expressing mathematics in english. You can see this as the reason for actually developing mathematical notation. With the mathematical language and notation it becomes posible to express certain concepts easilly, which become quite convuluted in english. --Salix alba (talk) 21:40, 2 April 2007 (UTC)

A question

To man who knows all,

Dinesh and I are working on an article in a sandbox of mine. When the time comes to move it out into the mainspace, how is that done? Can one just rename it and move? Or is it more like blank the page and cut/paste into new article (thereby losing edit history, I presume). Thanks! Sincerely, --Mattisse 00:03, 5 April 2007 (UTC)

P.S. When I was a kid and just into Algebra, I bugged my father a great deal over "what does a function f(x) mean?" ). I couldn't get it through my thick head then, no matter what he said.

In college calculus, everyday a lecturer would walk in, scribble a bunch of stuff on the board and walk out. One day about two-thirds of the way through the semester, a student raised his hand and asked, "Why would you want to know that?" The instructor explained, giving some real world examples, and suddenly everything was clear. Sincerely, --Mattisse 00:15, 5 April 2007 (UTC)

(I know now what a function is -- don't know what the problems was then, except maybe I expected the answer to be more complicated than it was.)

Yes. Moving an article from a sandbox to main space is a prefered way of doing this as it should keep the edit history.
Indeed a good load of examples help at lot. --Salix alba (talk) 09:17, 5 April 2007 (UTC)
Thanks! Sincerely, --Mattisse 15:34, 5 April 2007 (UTC)

Hi, it's Xedi, I saw you nicely modified some things on the Media:x^y2.png image. I'm not at home right now, but would you like me to do another export of it with your additions ? And any other things you would see fit ?

Thanks. --Xedi (not logged)

Not entirely sure what your after. I do have an svg version of the image. I did cheet a bit and used the points on the grid to construct the green curves y=+/- x/2. It would be interesting to see the image of y=x^2, which I think is should be a smooth curve through 1. I don't have access to Mupad so its quite hard to plot other curves on it. --Salix alba (talk) 23:05, 5 April 2007 (UTC)

What!

Your archived! Must be change in the wind. (I have my suspictions.) Sincerely, Mattisse 00:27, 6 April 2007 (UTC)

Indeed, just beging to go self-employed. Theoretically this means less time for wikipedia. --Salix alba (talk) 01:10, 6 April 2007 (UTC)
But better for you. Once I discovered I could not work for others, the world opened up. I became blessed. May the same happen to ou. Freedom! Sincerely, --Mattisse 01:20, 6 April 2007 (UTC)

Oh

I've already decided to ignore the rest of it. Ridiculous stuff! At least my case has be closed today which is all I have been wanting anyway. Thanks so much for your message and your concern. It means a lot to me. You have been a good friend and I appreciate you. Sincerely, Mattisse 22:56, 8 April 2007 (UTC)

For you, with deep felt appreciation

 The Surreal Barnstar For being a true friend through thick and thin. No Barnstar good enough to give to you for being my kind guiding light of reason in this wilderness. Mattisse 00:58, 9 April 2007 (UTC)
Cool ;-) --Salix alba (talk) 01:00, 9 April 2007 (UTC)

1^infty

I accidentally left out an edit summary when I undid a reversion of yours; I left an explanation here. CMummert · talk 17:02, 9 April 2007 (UTC)

Whoopse, sorry. --Salix alba (talk) 19:18, 9 April 2007 (UTC)

Thanks!!

I was angry that my AMA Advocate refused to let me contact him for two months and then made disparaging remarks about me on talk pages. That is what this latest dust up is about. The person who scolded me for being angry at my Advocate is the one making the posts on my page now. But my new Advocate is very calming and has a quieting effect on me, similar to you, and I will follow his advice and yours. Thank you so much! Sincerely, --Mattisse 19:50, 11 April 2007 (UTC)

I getting closer to doing what I agreed with you I wouldn't do. In fact, I think I will do it. People should know how incredibly sloppy and unprofessional they are. And I don't really care what happens to me. I know I can alway write and edit for Dinesh no matter what. And since I am trashed anyway, why not trash an organisation that does evil by existing. If I were the only person who felt this way it would be one thing, but there are massive amounts of legitimate people who want that thing shut down. I would be crusading for the good! Sincerely, Mattisse 21:00, 15 April 2007 (UTC)
Nooo Don't do it. I don't think I could handle the fallout. --Salix alba (talk) 10:03, 16 April 2007 (UTC)
I have not yet. I have sent emails to my Advocate and a former Advocate of my sockpuppet adversary (who is my friend) and finally posted on My Advocate's page as he does not acknowledge email. If I let this go, the next sock puppet ring (as it surely will spring up again and still has not been thoroughly eradicated) will use this. They will and the whole thing will start up again. One thing I have learned, and should have learned a long time ago, is how to stand up for myself and not just let myself be walked over passively. If I do not stop this now there will never be any hope for me. I have done nothing wrong, except the relatively comparatively mild act of having sock puppets on my computer for a short time and not being fully aware that was happening. I was punished for that and have done nothing wrong since. That was in September, I believe. This is May. There are still sock puppets after me -- fortunately at the moment there are some who look out for me and email me when a sockpuppet is changing my email address and doing other sly things that i do not detect. You do not understand. And you are not aware of all that has happened. I would be sorry to lose you as a friend but I am not going allow this to continue without standing up for myself. Sincerely, Mattisse 14:49, 16 April 2007 (UTC)
You say let my edit reputation be enough. I have firmly learned that no one looks up things like that, no one looks at evidence. I have almost 18,000 mainspace edits out of 21,000 edits -- which seems to me that is evidence that I mainly edit and don't spend my time harassing. No one actually looks at what I do. They opine and accuse. Only the arbitrators look. Editorial reputation means nothing around here. It just leaves you open for more ways to be stalked and attacked. Ekajati/999/Hanuman Das person could easily stalk and harass me through the watchlist. I have almost 4000 articles on my watch list -- not feasible for me to even keep track of his stalking me. A real editor is a piece of cake when a person of his nature has decided to get you. And he has not given up. Sincerely, Mattisse 15:02, 16 April 2007 (UTC)
Pick a fight here and you will get a reputation. Maybe it does need reform, do you have the skills to acheive that? BTW your wacthlist is the one thing which is private on wikipedia, no one apart from the tech crew can read then, not even admins. --Salix alba (talk) 15:54, 16 April 2007 (UTC)
• I do not understand your comment above. You are saying that these should remain on the record? [1]& [2]?They will be used against me when the next round of sockpuppet attacks starts on me. As far as the watchlist, all I am saying is that it is easy to stalk a person and their articles if you have a short watchlist. Since mine tends to have close to 4000 articles on it I cannot protect myself and the articles I am working on with the same level of scrutiny. Are these unreasonable statements, do you think? Sincerely, Mattisse 14:44, 17 April 2007 (UTC)
I'm saying that if your continue to press Silk, there will be plenty more on record than there is now. it will look like your are the sort who will never back down and take things to extreme levels taking up everybodies time, there is a policy about this WP:POINT. People have been blocked for violating this and your coming close to that threshold.
As for watchlist consider cleaning it out, I had over a thousand and now I have about 300 articles which I consider most important. Life has been easier since then. --Salix alba (talk) 17:08, 17 April 2007 (UTC)
It's not that easy. There are no articles that I think are "most important" - or at least almost all of the articles on my list are equally important. I'm not one of those "watchlist" people that is specifically tracking something. And frankly, I don't want to become one. And I do de-list articles when I have no interest in them. --Mattisse 20:54, 18 April 2007 (UTC)
P.S. I have apologised profoundly to Silk. --Mattisse 20:56, 18 April 2007 (UTC)

CMSP

Salix alba,

your comments on the AfD page were fair, but even more your expansion of the article convinces me that it should be kept. Thank you. Jd2718 23:50, 11 April 2007 (UTC)

Indeed, improvements to the article are often the best result of an AfD. Care to change your vote? --Salix alba (talk) 08:47, 12 April 2007 (UTC)

Hi Rich. What do you think about the knot theory article? Do you think it'll pass an A-class nom? Or is there some serious issue with it? --C S (Talk) 12:10, 15 April 2007 (UTC)

Category:Complex systems

Thank you for your contribution to Category:Complex systems in the past. There is currently a Call for Deletion for this category. If you would like to contribute to the discussion, you would be very welcome. Please do this soon if possible since the discussion period is very short. Thank you for your interest if you can contribute. Regards, Jonathan Bowen 14:18, 16 April 2007 (UTC)

Edit conflict

Hi, I was in the process of moving the discussion with 141.211... at derivative to User talk:Geometry guy when we had an edit conflict. Since your reply mostly concerned with the personal aspects (I was going to make exactly the same point about anonymity - great minds!), I have moved it over. I hope that is okay. Geometry guy 16:33, 18 April 2007 (UTC)

Sure no problem. --Salix alba (talk) 16:40, 18 April 2007 (UTC)

Tidying up other peoples user pages

Okay, I was really confused by this. Did this guy REALLY edit out my redundant email address? We're all for tidying up Wikipedia, but this seems really invasive. For example, I'd remove over half your witty identification buttons, because i think they are retarded, but that seems like it'd be crossing some personal boundary. Please leave my user page alone. MotherFunctor 23:38, 18 April 2007 (UTC)

You have mail

You have email --Mattisse 12:17, 19 April 2007 (UTC)

Don't care

I'm leaveomg Wikipedia any way. Just learned that Fred Baruder thinks I am a sock pupppet so no help now. He sides with the sockpuppets(as you do also.) I don't care any more. You just get punished for trying to be a good editor and rewarded for being a sockpuppet. Lost all respect for good edits and no longer care what I post as it is clearly hopeless. Not interested in being foolishly good any more as that just makes me a sap. I want to burn bridges so I can't come back. I have notived Dnesh that i camnot help him anymore. Then I won't do anything serious as I Wiki withdraw for a few days amd get my mental health back. In yor closed world all is realatively calm, but I am without protection and way too aged to handle this. Don't care any more. Wish they woud be upfront and ban me if they think i am a sok puppekt, but they prefer the agonising torutre method. This is a bad place. --Mattisse 08:00, 20 April 2007 (UTC)

P.S. I have 18,000 mainspace adits and for that I get torture and no help or support. I get the picture I am not a complete fool. I fell like going back and removing every one of those edits. The foolish work I put into it. I was really stupid. --Mattisse 08:04, 20 April 2007 (UTC)
P.S.S. Thanks for outing me as a female. One email has pointed out to me that the extremely personal and vitriolic nature of the personal attacks against me seem sexist to him. Of course, he comes from a country where women are not treated so crudely. Rosencomet and Jefferson Anderson are attacking me as a sock puppet again. What was that about the road to hell being paved with -- what was that it was paved with? I am so angry with you because I was so fond of you and trusted you (almost, although I could tell you did not want to see the real story). Sincerely, Mattisse 17:39, 22 April 2007 (UTC)
Ouch. --Salix alba (talk) 18:25, 22 April 2007 (UTC)

You used AWB, "Moving comments to /Comments per WT:WPM using AWB", to mess up the comments at Talk:Axiom of choice. You moved into a subpage a pointer to a section on the talk page rather than the relevant section of talk itself. Now the pointer points to nothing and the section is still on the talk page. JRSpriggs 08:08, 20 April 2007 (UTC)

I overstated the problem. I see now that one can still see the pointer on the main talk page and click it to get to the section. JRSpriggs 08:12, 20 April 2007 (UTC)
Sorry to have bothered you. JRSpriggs 08:07, 24 April 2007 (UTC)

--Mattisse 08:29, 20 April 2007 (UTC)

Encouraging sexual abuse

Several people have mentioned they never would have known my sex if you had not deliberately and gratuitously outed me. In my request for the name change I gave as the reason for the change that I wanted a gender neutral name to prevent the sexual abuse common here toward females. I have been told that the viciousness directed toward me is because I am female. Why did you do that? I do not understand why you wanted to undermine me and my attempts to help myself by preventing this abuse. What could your reasons possibly be? Why? I do not understand such cruelty. --Mattisse 19:14, 22 April 2007 (UTC)

Because people at the time were refering to you as male which I thought seemed a little rude. --Salix alba (talk) 19:29, 22 April 2007 (UTC)
I wanted to avoid rudeness. Why do you think I changed my name to a non female one? It is much ruder to be treated as a female.--Mattisse 20:06, 22 April 2007 (UTC)
My mistake, I didn't realise. Sorry. BTW my name is a Dioecious species. --Salix alba (talk) 20:19, 22 April 2007 (UTC)
Oh, Salix. I apologise to you. The AMA comments made against me have alreadby been used in Rosencomet's and Jefferson Anderson's latest accusation of my sockpuppetry. Rosencomet asked for a private check user from Fred Bauder which was apparently granted (I thought there were procedures for such things but apparently not) and supposedly that shows I am a sockpuppet of two individuals. My newest AMA Advocate (though relatively inexperienced) quickly caught on to what was happening. He is handling it and is very kind to me and is an adult, praise to be, keeping me informed which the others never did. Dinesh's FA article is being attack by Indian extremest, the politics of which I do not understand.
I'm sorry I took all this out on you. Please accept me as I am -- flawed, human, overly emotional at times, even mean as I was to you. But my career was related to the criminal justice system for years, and I do not intentionally break rules, ever. I am overly anxious about any kind of breaking of rules because in a heartbeat I would have lost my profession -- which by the way -- if you want to check my license has not a single complaint lodged against it. It is a pure as the driven snow. You are a good guy, but nonetheless a math guy and we all know that math guys have their limitations. I am sorry. Be my friend. I will fix you Star back. I apologise for the nth time to you. I know you are a good guy -- just a pinch naive (I can't help this last remark, sorry!) Hopefully, still your friend if you let me, --Mattisse 21:03, 22 April 2007 (UTC)
No worries apology accepted. Better you let your angst out on me that elsewhere. --Salix alba (talk) 21:14, 22 April 2007 (UTC)
Thank you. You truly are kind. --Mattisse 21:29, 22 April 2007 (UTC)

I don't know if my comments are welcome here, so I'll keep it short, at least for me. :-) I can't believe Mattisse is accusing you of "outing her as female", much less using the term "encouraging sexual abuse". When was this supposed to have happened? She's been out for at least five or six months that I know of, both as female and a grandmother. I'm glad she apologized.

Just for the record, if Mattisse is listening, I got nothing concerning any of the contentious stuff between us from Salix Alba. Also, neither Fred Bauer nor I ever said she was a sockpuppet; he said a checkuser on her, BackMaun and Alien666 showed that the three share an IP address from time to time. I have never said more than that, merely posed the question as to whether those three (and possibly RasputinJSvengali) were the same, OR if one or more were a returning Timmy12, or something. How you, Salix Alba got in this discussion, as far as I can see, was just by suggesting a few angry and unhelpful words might be better left unsaid and/or retracted, and I can only applaud your good intentions.

Now if she and anyone she either controls, or works with, or who's ear she has, or whatever the relationship is, will just stop instigating conflicts with me and Jefferson Anderson and William M. Connolley (for the life of me, I can't understand why they're being treated the way they are), we can all go back to editing and have a good time. Believe me, it's been a relief to be able to contribute for the past few weeks without butting heads with Pigman, Kathryn, and Weniwediwiki, and I'll bet they feel the same way, and probably don't have any real problems with the editing I've done during that period.

In any case, I've always thought you were fair and well-meaning, even when we disagreed.

Yep, I know... that was short? :-) Rosencomet 22:27, 23 April 2007 (UTC)

Yes you are free to comment here. In my view there is nothing more to do, we have been through arbitration and it is now the time to drop the matter and concentrate on with writing an encylopedia. It is only when one party posts something that the whole argument flares up again. if people can just let this whole matter drop it will be much less stress for all. --Salix alba (talk) 22:46, 23 April 2007 (UTC)

Affine space

I appreciate what you have done to fix up the pecise definition, but it's basically as unsatisfactory to me as before. Partly, what's S? I of course suspect S is V minus some structure, but logically it could be just one point of V for example. That can't be left out if it is to be useful or precise.Rich 17:49, 27 April 2007 (UTC)

Mathematicians page

Hi there - I'd be interested if you have any comments on User:Geometry guy/Mathematicians. Geometry guy 19:22, 29 April 2007 (UTC)

Further to our exchange on WT:WPM, I think it might be quite easy to migrate the data on /Dates, /SortName and /Contribs to /DB by combining a template like "MakeDB" with the AWB facility to append text to talk pages and expand templates. I've ask for clearance to use AWB now, but if you can see how to do it in the meanwhile, please go ahead and create /DB pages! Geometry guy 20:36, 30 April 2007 (UTC)

I've now implemented your template idea at Template:Mathematician data. The template doesn't seem to work for pre-1000 mathematicians which use a hidden "span" to sort their dates. Otherwise it seems to be okay. Thanks for this idea! Geometry guy 17:41, 5 May 2007 (UTC)

Thanks!

I joined the list of WikiProject Mathematics participants. Thanks also for the praise of my Theorem edits. I thought it was good clarification and information I was adding. I must say I am a little surprised that there've been basically no edits since then... I felt sure that since it was the collaboration of the week, it'd be changing quickly. Why do you suppose more people haven't edited the article? Perhaps most who look at it already think it's good enough?

Alas, MCOTW has been in a pretty poor state for some time, there is so much else to do in the maths articles with the grading and the push to more FA's GA thatpeople seem to have little time for COTW. Geometry guy did leave a comment on the talk page so he could do with a little push. --Salix alba (talk) 19:19, 30 April 2007 (UTC)
I'd like to do what I can to liven up COTW. My own view is that aiming for featured articles is the wrong approach: getting stubs up to GA or A-class is a much more attractive and achievable goal. Anyway, I hope to find time to have a crack at this one, but (as you probably know) I have quite a few loose ends on my plate right now! Geometry guy 20:36, 30 April 2007 (UTC)
Yes I did notice. I remember that stage in my wikipedia development where I was getting into everything. I agree about the FA goal. Yes its nice to get some maths on the main page but there are higher priorities.
I'm not at all clear myself on how Theorem should develop, apert from some cleaning up and reorgisation. Is there anything more which could be said which is not covered by proof? --Salix alba (talk) 21:11, 30 April 2007 (UTC)
I can imagine a lot of changes to this article. I hope I will have time before COTW moves on to make some edits. Geometry guy 23:39, 30 April 2007 (UTC)

Derivative

We had an edit conflict, and I took the liberty of indenting and remarking on your comment. Please feel free to reformat it as you prefer. Geometry guy 21:47, 8 May 2007 (UTC)

Mathematics CotW

Hey Salix, It took me awhile to get around to S but here we go, I am writing you to let you know that the Mathematics Collaboration of the week(soon to "of the month") is getting an overhaul of sorts and I would encourage you to participate in whatever way you can, i.e. nominate an article, contribute to an article, or sign up to be part of the project. Any help would be greatly appreciated, thanks--Cronholm144 00:08, 14 May 2007 (UTC)

Bracket (mathematics)

Hi there. I'm a bit dubious about the creation of this article without any discussion, so I figured it would be a good plan to actually ask the people in Bracket if they object. Hope you don't mind. I stuck a {{subst:prod}} on it, but I'll remove that for the moment, forthwith.--Aim Here 11:44, 14 May 2007 (UTC)

Image:Ballots and Bullets.jpg

Hello, Salix alba. An automated process has found and removed an image or media file tagged as nonfree media, and thus is being used under fair use that was in your userspace. The image (Image:Ballots and Bullets.jpg) was found at the following location: User:Salix alba/History of conflict between democracies. This image or media was attempted to be removed per criterion number 9 of our non-free content policy. The image or media was replaced with Image:NonFreeImageRemoved.svg , so your formatting of your userpage should be fine. Please find a free image or media to replace it with, and or remove the image from your userspace. User:Gnome (Bot)-talk 05:25, 15 May 2007 (UTC)

Arc vs. curvature

Hiya Salix! I saw your Eleven properties of the sphere section in the Sphere article and thought maybe you would have some insight into this question.

OK, hope you don't mind if I split your text up and modify it. My text in itallic.
Not at all. P=)

What is the difference between a (differentiable) curve, or arc, and curvature? I thought that they were just different grammatical forms of the same concept: If you pulled a string between two points on a sphere or ellipsoid, the path between the points would be an arc/curve segment along its curvature (i.e., a segment of circumference).

The distinction is more than being different gramatical forms. Curvature is a measurment of something, where an arc is a something. Consider the grove in an old LP record, as it wind in from the outside it will get tighter and tighter, that is more curved. Near the outside it will have a low curvature and close to the center it will have a high curvature. At any point on the grove we can calculate its curvature and all points will have slightly different curvatures. We can find the curvature at a point by considering the circle which closely aproximates the grove, and then finding its radius. The curvature at the point will be 1/(radius of the circle).

In particular, what I'm interested in is the radius of arc ("arcradius") vs. radius of curvature. Since radius of curvature ("RoC") is the inverse of curvature, doesn't it follow that arcradius (AR) is the inverse of arc?

${\displaystyle RoC={\frac {1}{curvature}};}$ yes
${\displaystyle \qquad \;RA={\frac {1}{arc}}?\,\!}$ means nothing, you can't divide by shapes - what is 1/triangle?.
If the arc is part of a circle, then the radius of the arc is just the radius of the circle. If the arc is not part of the circle then arc radius is not defined.
But every point along any great ellipse has an arcradius.

The principal——meridional and its perpendicular "normal"——radii are the same for each:

yes if we assume the earth is a perfect sphere - its not. We can see this by considering the line straight up (that is opposite to the direction of gravity) this direction is the normal. The Meridional can be found by cutting the earth with a plane containing the normal, so it is a normal section. From property 4 the curvtures of all normal sections are equal. If the earth is not a perfect sphere then each normals section will have slightly different curvature.
Sure they are in these cases (I'm not saying M equals N——except, of course, at the poles——but that arcradius equals radius of curvature at a given point). If you slice a small oblate spheroid down the middle, place the flat side down on a piece of paper and trace the perimeter, you will have an ellipse: At the equator, the (meridional) arc/curvature will be "pinched", providing a smaller radius of the arc/curvature, while the polar arc/curvature will be flattened, making the arc/curvature radius greater. Okay, in this particular, meridional case, what is the difference between arc and curvature? If you look at the ellipse you would say the curvature is pinched in the middle and flattened at the top and bottom and that the radius of arc/curvature at a given point equals M——right?
The zonal curves are also worth considering. Here the curves are found by slicing the earth with a plane perpendicular to the axis of rotation. Except at the equator these planes will not contain the normal, hence they will have different curvatures (in fact tighter curves - higher curvature).
Wait a minute now, here you are introducing different types of "slices": Isn't all curvature "subtended" (i.e., facing directly down), so that the other end of the slice will be antipodal (i.e., a piece of great circle)——or is THAT what arc is——or are there different "small circle" slices (like the lines of latitude on a globe tilted at an angle), angled with respect to something (e.g., direction of gravity or axis of rotation)?

The discrepancy arises with the in-between, oblique(?) values:

As the orthodromic equals the loxodromic at the differential level (i.e., for an infinitesimal distance),

false you need to consider second derivatives when examining curvature, great circles and rhumb lines will have different second derivatives and hence different curvature. Alternatively you can consider that the rhumb line is locally found by slicing with a plane which does not contain the normal, hence different curvature.
I don't know about curvature, but orthodromic and loxodromic arc and distance converge as infinitesimality is approached——or did I just not define the relationship properly?
${\displaystyle \lim _{{\mbox{arc}}\to 0}{\mbox{orthodromic}}={\mbox{loxodromic}}\,\!}$
Using the notation below, consider this example (to make things more contrasting, let a = 10000 and b = 5000; all orthodromic and loxodromic values given are equal to the precision given, with the midpoint/average latitude and azimuth used):
${\displaystyle \phi _{s}=-.0000000000000005^{\circ };\quad \phi _{m}=0;\,\!}$
${\displaystyle \phi _{f}=\Delta \lambda =+.0000000000000005^{\circ };\,\!}$
{\displaystyle {\begin{aligned}\theta &=\arccos(\sin(\phi _{s})\sin(\phi _{f})+\cos(\phi _{s})\cos(\phi _{f})\cos(\Delta \lambda )),\\&={\sqrt {(\Delta \phi )^{2}+(\cos(\phi _{m})\Delta \lambda )^{2}}},\\&=.00000000000000111803398875^{\circ };\end{aligned}}\,\!}
${\displaystyle {\widehat {\alpha }}=90^{\circ }\!-{\tilde {\alpha }}=26.565051177077989^{\circ }\quad \,(\sin({\tilde {\alpha }})^{2}=.2);\,\!}$
${\displaystyle {\tilde {\alpha }}=90^{\circ }\!-{\widehat {\alpha }}=63.434948822922011^{\circ }\quad \,(\sin({\widehat {\alpha }})^{2}=.8);\,\!}$
${\displaystyle M=2500;\quad \,N=10000;\quad \,O=5000;\quad \,{\ddot {O}}=6250;\,\!}$
{\displaystyle {\begin{aligned}{\mbox{Dx}}&={\mbox{geodetic distance}},\\&=O\cdot \theta ={\sqrt {(M\Delta \phi )^{2}+(N\cos(\phi _{m})\Delta \lambda )^{2}}},\\&=.0000000000000975668712644500;\end{aligned}}\,\!}
${\displaystyle {}_{\color {white}.}({\ddot {O}}\cdot \theta =.0000000000001219585890805625){}_{\color {white}.}\,\!}$
As you can see, the difference between O and Ö is quite pronounced! P=) (Above example adjusted for "cleaner" results)  ~Kaimbridge~18:49, 21 May 2007 (UTC)
I'll have to break for dinner here. I'm not sure how much of this applies now we have seen that the the above is incorrect. --Salix alba (talk) 18:01, 16 May 2007 (UTC)

Let

${\displaystyle \phi _{s}={\mbox{a standpoint geodetic/geographical latitude}};\,\!}$
${\displaystyle {\widehat {\alpha }}_{s}={\mbox{globoidal/spherical azimuth at }}\phi _{s};\,\!}$
${\displaystyle {\tilde {\alpha }}_{s}={\tilde {\alpha }}(\phi _{s})={\mbox{geodetic/elliptical azimuth at }}\phi _{s};\,\!}$
{\displaystyle {\begin{aligned}O&={\mbox{Oblique arcradius at }}\phi _{s},\\&={\sqrt {(M\cos({\widehat {\alpha }}_{s}))^{2}+(N\sin({\widehat {\alpha }}_{s}))^{2}}}={\frac {1}{\sqrt {\left({\frac {\cos({\tilde {\alpha }}(\phi _{s}))}{M}}\right)^{2}+\left({\frac {\sin({\tilde {\alpha }}(\phi _{s}))}{N}}\right)^{2}}}}\end{aligned}};\,\!}
{\displaystyle {\begin{aligned}{\ddot {O}}&={\mbox{Oblique radius of curvature at }}\phi _{s},\\&={\frac {(M\cos({\widehat {\alpha }}_{s}))^{2}+(N\sin({\widehat {\alpha }}_{s}))^{2}}{M\cos({\widehat {\alpha }}_{s})^{2}+N\sin({\widehat {\alpha }}_{s})^{2}}}={\frac {1}{{\frac {\cos({\tilde {\alpha }}_{s})^{2}}{M}}+{\frac {\sin({\tilde {\alpha }}_{s})^{2}}{N}}}}\end{aligned}};\,\!}

Is there a more established name identification for Ö? The clearest distinctive definition I've been able to find is "the radius of curvature in the normal section", though that can create confusion with N! Visually, what is the difference between O and Ö? If you look at an ellipsoid, the "surface radius" of a north-south line equals M, an east-west line equals N and a point along any great circle/ellipse equals O: So what does Ö show as? Or, is Ö an abstract quantity, in the same way that ${\displaystyle \scriptstyle {\cos(\phi _{s})MN}\,\!}$ is the radius2 for surface area? In fact, would a good analogy be "curvature is to a curve what area is to a line"?
To me, O seems much more important and relevant than Ö (as it is what is used for loxodromic calculations and——albeit, indirectly——for the classical geodesic, as well as directly for the far lesser known, geographically delineated geodesic——what I refer to as the "parageodesic": Is there a more proper name?), yet there seems to be nothing written about it (I plan to correct that injustice P=), but I would like to find some sources that define and formulate it, like M and N). ~Kaimbridge~14:49, 16 May 2007 (UTC)

Yes there are different notations of curvature, see Frenet-Serret formulas for one chaterisation for a curve in space. We can apply this to a curve on the sphere. Now we need to distinguish between the frenet normal to the curve ${\displaystyle N_{f}}$ say and the normal to the sphere ${\displaystyle N_{s}}$ say. When the curve is a great circle then the two normals will coinside. For other curves you can get a notion of curvature by considering the plane which the curve lies in (at least locally). You could get different curvatures by projecting the curve onto different planes, say the tangent plane, or the plane containing the great circle with the same bearing. In general these curvature will be different.
I'm a little confused as to the various quantities you mention, a, b, the difference between spherical and elliptical azimuth. Ideally a picture with the different things on would be great.
a and b are just the equatorial and polar radii/semi-axes; spherical azimuth uses ${\displaystyle \phi \,\!}$ and ${\displaystyle \lambda \,\!}$(i.e., geographical or globoidal azimuth), while the elliptical ("geodetic") uses the parametric latitude, ${\displaystyle \beta \,\!}$, and ellipsoidal/auxiliary longitude, ${\displaystyle \omega \,\!}$ (though a lot of geodetic formularies——like Vincenty's classic paper (PDF)——use ${\displaystyle \lambda \,\!}$ for ${\displaystyle \Delta \omega \,\!}$). I don't know what geodetic formula you use (if any), but I prefer straight forward, numerical (Gaussian) quadrature (using UBasic)——though I started out with Vincenty's and Sodano's needlessly drawn out series expansion formulations, then got ahold of Richard Rapp's Geometric Geodesy and its formula analysis and compendium,which straightened me out in terms of the basic concepts. P=)  ~Kaimbridge~16:58, 22 May 2007 (UTC)
I had intended to try to get my head round this, but got diverted into redoing the formula at Rhumb line. --Salix alba (talk) 22:18, 21 May 2007 (UTC)
I don't know if this image will help. It shows a sphere cut by two planes, the intersections defines two curves. The curvatures of these curves can be easily found, they are just the recriprocals of the radii of the two circles. The important point to note is that both curves have the same baring, in this case 90°. From this we can conclude that the baring alone is not sufficient to define the curvature. The same argument follows for all surfaces and for all barings.
I've been looking at your calculations and I'm not sure where your formula for Oblique arcradius is coming from. --Salix alba (talk) 08:21, 31 May 2007 (UTC)

Formulation

Okay, let me attempt a more thorough formulaic analysis, as I do believe this is either an ignored or overlooked (but quite fundamental) concept/property. (you may want to move this whole topic to a subpage, as I am attempting to cram in about six articles worth of information P=); also, between doing a fair amount of cut-and-paste and having only one working eye——with an advancing cataract, no less P=(——if something seems out of place——such as an "n" for "m"——it very well could just be a typo...though I do believe I've caught everything, particularly with the formulation!)

There are two distinct derivations for O: The previously noted loxodromic, as well as a composite function (line integral?) based orthodromic one that——as I will show——also applies to M.

Loxodromic approach

${\displaystyle C\Delta \lambda \approx \cos(\phi _{m})\Delta \lambda ;\,\!}$ (i.e., the inverse Gudermannian function)
{\displaystyle {\begin{aligned}\mathrm {H} &=\sec({\widehat {\bar {\alpha }}})\Delta \phi =\Delta \phi {\color {white}{\frac {\Big |}{}}\!\!\!}{\sqrt {1+\left({\frac {C\Delta \lambda }{\Delta \phi }}\right)^{2}}},\\&={\sqrt {(\Delta \phi )^{2}+(C\Delta \lambda )^{2}}},\\&={\sqrt {\left(\cos({\widehat {\bar {\alpha }}})\mathrm {H} \right)^{2}+\left(\sin({\widehat {\bar {\alpha }}})\mathrm {H} \right)^{2}}};\end{aligned}}\,\!}
(to distinguish between orthodromic and loxodromic,
loxodromic azimuth will be denoted as ${\displaystyle {\bar {\alpha }}{\color {white}{\big |}}\,\!}$)
${\displaystyle {\widehat {\bar {\alpha }}}=\arctan \left({\frac {C\Delta \lambda }{\Delta \phi }}\right);\,\!}$
${\displaystyle {\tilde {\bar {\alpha }}}(\phi )=\arctan \left({\frac {N(\phi )C\Delta \lambda }{M(\phi )\Delta \phi }}\right)=\arctan \left({\frac {N(\phi )}{M(\phi )}}\tan({\widehat {\bar {\alpha }}})\right);\,\!}$

{\displaystyle {\begin{aligned}{\rm {Globoidal\;Dx}}&=r\mathrm {H} ={\color {white}{\bigg |}\!\!}{\sqrt {(r\cos({\widehat {\bar {\alpha }}})\mathrm {H} )^{2}+(r\sin({\widehat {\bar {\alpha }}})\mathrm {H} )^{2}}},\\&=r\sec({\widehat {\bar {\alpha }}})d\phi ;\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}{\rm {Elliptical\;Dx}}&=\int _{\phi _{s}}^{\phi _{f}}{\color {white}{\bigg |}\!\!}{\sqrt {(M(\phi )\cos({\widehat {\bar {\alpha }}})\mathrm {H} )^{2}+(N(\phi )\sin({\widehat {\bar {\alpha }}})\mathrm {H} )^{2}}}{\frac {d\phi }{\Delta \phi }},\\&={\frac {\mathrm {H} }{\Delta \phi }}\int _{\phi _{s}}^{\phi _{f}}{\color {white}{\widehat {\color {black}{\sqrt {(M(\phi )\cos({\widehat {\bar {\alpha }}}))^{2}+(N(\phi )\sin({\widehat {\bar {\alpha }}}))^{2}}}}}}d\phi ,\\&=\sec({\widehat {\bar {\alpha }}})\int _{\phi _{s}}^{\phi _{f}}O({\widehat {\bar {\alpha }}},\phi )d\phi ;\end{aligned}}\,\!}

Elliptic elements/properties

${\displaystyle o\!\varepsilon =\arccos \left({\frac {b}{a}}\right)\!\!:\;\;e^{2}=\sin(o\!\varepsilon )^{2}={\frac {a^{2}-b^{2}}{a^{2}}};\,\!}$ (angular eccentricity)

${\displaystyle \beta =\beta (\phi )=\arctan(\cos(o\!\varepsilon )\tan(\phi ));\,\!}$
${\displaystyle {\color {white}{\bigg |}\!\!}\beta '=\beta '(\phi )={\frac {d\beta (\phi )}{d\phi }}={\sqrt {m'(\phi )n'(\phi )}}=\cos(o\!\varepsilon )n'(\phi )^{2}=\sec(o\!\varepsilon )C'(\beta )^{2};\,\!}$

${\displaystyle \phi =\phi (\beta )=\arctan(\sec(o\!\varepsilon )\tan(\beta ));\,\!}$
${\displaystyle \phi '=\phi '(\beta )={\frac {d\phi (\beta )}{d\beta }}={\frac {\cos(o\!\varepsilon )}{C'(\beta )^{2}}}=\sec(o\!\varepsilon )E'(\phi )^{2};\,\!}$

${\displaystyle E'(\phi )={\sqrt {1-(\sin(\phi )\sin(o\!\varepsilon ))^{2}}}={\sqrt {\cos(o\!\varepsilon )^{2}+(\cos(\phi )\sin(o\!\varepsilon ))^{2}}}={\frac {\cos(o\!\varepsilon )}{C'(\beta )}};\,\!}$
${\displaystyle C'(\beta )={\sqrt {1-(\cos(\beta )\sin(o\!\varepsilon ))^{2}}}={\sqrt {\cos(o\!\varepsilon )^{2}+(\sin(\beta )\sin(o\!\varepsilon ))^{2}}}={\frac {\cos(o\!\varepsilon )}{E'(\phi )}};\,\!}$
${\displaystyle \cos(o\!\varepsilon )=E'(\phi )C'(\beta );\,\!}$

${\displaystyle m'(\phi )={\frac {\cos(o\!\varepsilon )^{2}}{E'(\phi )^{3}}}=\cos(o\!\varepsilon )^{2}n'(\phi )^{3}=\sec(o\!\varepsilon )C'(\beta )^{3};\,\!}$
${\displaystyle n'(\phi )={\frac {1}{E'(\phi )}}=\sec(o\!\varepsilon )C'(\beta );\,\!}$
${\displaystyle M=M(\phi )=a\,m'(\phi );\quad \;N=N(\phi )=a\,n'(\phi );\,\!}$

${\displaystyle x=a\cos(\beta )=N\cos(\phi );\quad \,y=b\sin(\beta )=\cos(o\!\varepsilon )^{2}N\sin(\phi );\,\!}$
${\displaystyle {\color {white}{\bigg |}\!}R=R(\phi )={\sqrt {x^{2}+y^{2}}}=a\,E'(\beta )=a\,n'(\phi ){\sqrt {\cos(\phi )^{2}+\sin(\phi )^{2}\cos(o\!\varepsilon )^{4}}};\,\!}$
${\displaystyle {\acute {x}}={\acute {a}}\cos(\beta )=b\cos(\beta )\quad \,{\acute {y}}={\acute {b}}\sin(\beta )=a\sin(\beta );\,\!}$
${\displaystyle {\color {white}{\bigg |}\!\!}{\acute {R}}={\acute {R}}(\beta )={\sqrt {{\acute {x}}^{2}+{\acute {y}}^{2}}}=a\,C'(\beta )=\cos(o\!\varepsilon )N(\phi );\,\!}$

Meridional Case

${\displaystyle {\rm {{Meridional\;Dx}=\int _{\phi _{s}}^{\phi _{f}}M(\phi )d\phi =\int _{\beta _{s}}^{\beta _{f}}{\acute {R}}(\beta )d\beta ;\,\!}}}$

So, how is this possible, you ask?

{\displaystyle {\begin{aligned}{\acute {R}}(\beta )d\beta &={\acute {R}}(\beta ){\frac {d\beta }{d\phi }}d\phi ={\acute {R}}(\beta ){\frac {d\beta (\phi )}{d\phi }}d\phi ={\acute {R}}(\beta )\cdot \beta '(\phi )d\phi ,\\&=\cos(o\!\varepsilon )N(\phi )\cdot \beta '(\phi )d\phi ,\\&=a\cos(o\!\varepsilon )n'(\phi )\cdot \cos(o\!\varepsilon )n'(\phi )^{2}d\phi ,\\&=a\cos(o\!\varepsilon )^{2}n'(\phi )^{3}d\phi =M(\phi )d\phi ;\end{aligned}}\,\!}

Conversely,

{\displaystyle {\begin{aligned}M(\phi )d\phi &=M(\phi ){\frac {d\phi (\beta )}{d\beta }}d\beta =M(\phi )\cdot \phi '(\beta )d\beta ,\\&=a\sec(o\!\varepsilon )C'(\beta )^{3}\cdot {\frac {cos(o\!\varepsilon )}{C'(\beta )^{2}}}d\beta ,\\&=a\,C'(\beta )d\beta ={\acute {R}}(\beta )d\beta ;\end{aligned}}\,\!}

Orthodromic

${\displaystyle \scriptstyle {\widehat {\mathrm {A} }}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {{\widehat {\alpha }}_{s}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {{\widehat {\alpha }}_{f}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {{\widehat {\sigma }}_{s}}{\color {white}{\big |}}\,\!}$ and ${\displaystyle \scriptstyle {{\widehat {\sigma }}_{f}}{\color {white}{\big |}}\,\!}$ are found with ${\displaystyle \scriptstyle {\phi _{s}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {\phi _{f}}{\color {white}{\big |}}\,\!}$ and ${\displaystyle \scriptstyle {\Delta \lambda }{\color {white}{\big |}}\,\!}$, comprising the central angle (see Transverse and Oblique Projections):

{\displaystyle {\begin{aligned}{\widehat {\sigma }}_{q}&={}_{\rm {{(Globoidal)\;transverse\;colatitude/angular\;distance\;along\,{\widehat {\mathrm {A} }}\;from\;the\;equator\;to\;{\mathit {\phi _{q}}}},}}\\&={\widehat {\sigma }}(\phi _{q})=\arcsin {\big (}\sec({\widehat {\mathrm {A} }})\sin(\phi _{q}){\big )}=\arccos \left({\frac {\tan({\widehat {\mathrm {A} }})}{\tan({\widehat {\alpha }}_{q})}}\right),\\&\qquad \qquad \!\!=\arctan {\big (}\sec({\widehat {\alpha }}_{q})\tan(\phi _{q}){\big )};\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}\phi _{q}=\phi ({\widehat {\sigma }}_{q})&=\arcsin {\big (}\cos({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}_{q}){\big )},\\&=\arccos \left({\sqrt {\cos({\widehat {\sigma }}_{q})^{2}+(\sin({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}_{q}))^{2}}}\right),\\&=\arctan {\big (}\cos({\widehat {\alpha }}_{q})\tan({\widehat {\sigma }}_{q}){\big )};\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}{\widehat {\mathrm {A} }}&={}_{\rm {{(Globoidal)\;arc\;path/transverse\;meridian/{\mathit {\widehat {\alpha }}}\;at\;the\;equator},}}\\&=\arcsin {\big (}\cos(\phi _{q})\sin({\widehat {\alpha }}_{q}){\big )}=\arccos \left({\frac {\sin(\phi _{q})}{\sin({\widehat {\sigma }}_{q})}}\right),\\&=\arctan {\big (}\cos({\widehat {\sigma }}_{q})\tan({\widehat {\alpha }}_{q}){\big )}=\arctan {\big (}\sin({\widehat {\alpha }}_{q})\sin({\widehat {\sigma }}_{q})\cot(\phi _{q}){\big )};\end{aligned}}\,\!}(Clairault's Constant)
${\displaystyle {\widehat {\alpha }}_{q}=\arcsin {\big (}\sec(\phi _{q})\sin({\widehat {\mathrm {A} }}){\big )};\,\!}$

The ellipsoidal ${\displaystyle \scriptstyle {\tilde {\mathrm {A} }}{\color {white}{\big |}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {{\tilde {\alpha }}_{s}}{\color {white}{\big |}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {{\tilde {\alpha }}_{f}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {{\tilde {\sigma }}_{s}}{\color {white}{\big |}}\,\!}$ and ${\displaystyle \scriptstyle {{\tilde {\sigma }}_{f}}{\color {white}{\big |}}\,\!}$ are found using the exact same equations, only substituting ${\displaystyle \scriptstyle {\phi _{s}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {\phi _{f}}{\color {white}{\big |}}\,\!}$ and ${\displaystyle \scriptstyle {\Delta \lambda }{\color {white}{\big |}}\,\!}$ with ${\displaystyle \scriptstyle {\beta _{s}}{\color {white}{\big |}}\,\!}$, ${\displaystyle \scriptstyle {\beta _{f}}{\color {white}{\big |}}\,\!}$ and ${\displaystyle \scriptstyle {\Delta \omega }{\color {white}{\big |}}\,\!}$, where ${\displaystyle \scriptstyle {\Delta \omega }{\color {white}{\big |}}\,\!}$ is the auxiliary ${\displaystyle \scriptstyle {\Delta \lambda }{\color {white}{\big |}}\,\!}$.

As ${\displaystyle \scriptstyle {\sin(\phi )=\cos({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }})}{\color {white}{\big |}}\,\!}$, thus ${\displaystyle \scriptstyle {\sin(\beta )=\cos({\tilde {\mathrm {A} }})\sin({\tilde {\sigma }})}{\color {white}{\big |}}\,\!}$, therefore,

{\displaystyle {\begin{aligned}E'(\phi )&=E'(0,\phi )=E'({\widehat {\mathrm {A} }},{\widehat {\sigma }})={\frac {\cos(o\!\varepsilon )}{C'({\tilde {\mathrm {A} }},{\tilde {\sigma }})}},\\&={\sqrt {1-(\cos({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }})\sin(o\!\varepsilon ))^{2}}};\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}C'(\beta )&=C'(0,\beta )=C'({\tilde {\mathrm {A} }},{\tilde {\sigma }})={\frac {\cos(o\!\varepsilon )}{E'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}},\\&={\sqrt {\cos(o\!\varepsilon )^{2}+(\cos({\tilde {\mathrm {A} }})\sin({\tilde {\sigma }})\sin(o\!\varepsilon ))^{2}}};\end{aligned}}\,\!}
${\displaystyle m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})=m'(\phi )=\sec(o\!\varepsilon )C'({\tilde {\mathrm {A} }},{\tilde {\sigma }})^{3};\qquad \,{\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }})={\acute {R}}(\beta );\quad \ldots {\mbox{etc.}}\,\!}$

${\displaystyle {\tilde {\mathrm {A} }}=\arcsin {\big (}\cos(\beta )\sin({\tilde {\alpha }}){\big )}=\arccos \left({\frac {\sin(\beta )}{\sin({\tilde {\sigma }})}}\right);\,\!}$
{\displaystyle {\begin{aligned}\beta &=\beta ({\widehat {\mathrm {A} }},{\widehat {\sigma }})=\arcsin {\big (}\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}){\big )},\\&=\beta ({\tilde {\mathrm {A} }},{\tilde {\sigma }})=\arcsin {\big (}\cos({\tilde {\mathrm {A} }})\sin({\tilde {\sigma }}){\big )}=\arctan {\big (}\cos({\tilde {\alpha }})\tan({\tilde {\sigma }}){\big )};\end{aligned}}\,\!}

{\displaystyle {\begin{aligned}{\tilde {\sigma }}={\tilde {\sigma }}({\tilde {\mathrm {A} }},{\widehat {\sigma }})&=\arcsin \left({\frac {\sin(\beta ({\widehat {\mathrm {A} }},{\widehat {\sigma }}))}{\cos({\tilde {\mathrm {A} }})}}\right),\\&=\arcsin \left(\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }}){\frac {\cos({\widehat {\mathrm {A} }})}{\cos({\tilde {\mathrm {A} }})}}\sin({\widehat {\sigma }})\right);\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}{\tilde {\sigma }}({\widehat {\sigma }})&=\arcsin \left({\frac {\sin(\beta ({\widehat {\mathrm {A} }},{\widehat {\sigma }}))}{\cos({\tilde {\mathrm {A} }}({\widehat {\sigma }}))}}\right),\\&=\arcsin \left(\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }}){\frac {\cos({\widehat {\mathrm {A} }})}{\cos({\tilde {\mathrm {A} }}({\widehat {\sigma }}))}}\sin({\widehat {\sigma }})\right),\\&=\arccos \left({\frac {m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}{\frac {\cos({\widehat {\mathrm {A} }})}{\cos({\tilde {\mathrm {A} }}({\widehat {\sigma }}))}}\cos({\widehat {\sigma }})\right),\\&=\arctan \left(\cos(o\!\varepsilon ){\frac {o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}\tan({\widehat {\sigma }})\right);\end{aligned}}\,\!}

{\displaystyle {\begin{aligned}{\tilde {\sigma }}'&={\frac {d\arcsin \left({\frac {\sin(\beta ({\widehat {\mathrm {A} }},{\widehat {\sigma }}))}{\cos({\tilde {\mathrm {A} }})}}\right)}{d{\widehat {\sigma }}}}={\frac {d\arcsin \left(\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }}){\frac {\cos({\widehat {\mathrm {A} }})}{\cos({\tilde {\mathrm {A} }})}}\sin({\widehat {\sigma }})\right)}{d{\widehat {\sigma }}}},\\&={\tilde {\sigma }}'({\tilde {\mathrm {A} }},{\widehat {\sigma }})=n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}\,{\frac {\tan({\tilde {\sigma }}({\tilde {\mathrm {A} }},{\widehat {\sigma }}))}{\tan({\widehat {\sigma }})}};\end{aligned}}\,\!}
${\displaystyle {\color {white}{\frac {\bigg |}{\bigg |}}\!}{\Bigg (}{\widehat {\sigma }}'={\frac {1}{{\tilde {\sigma }}'}}={\widehat {\sigma }}'({\widehat {\mathrm {A} }},{\tilde {\sigma }})=\left({\frac {\cos(o\!\varepsilon )}{C'({\tilde {\mathrm {A} }},{\tilde {\sigma }})}}\right)^{2}{\frac {\tan({\widehat {\sigma }}({\widehat {\mathrm {A} }},{\tilde {\sigma }})}{\tan({\tilde {\sigma }})}}{\Bigg )}{\color {white}.}\,\!}$
{\displaystyle {\begin{aligned}{\tilde {\sigma }}'({\widehat {\bar {\alpha }}},\phi )&={\sqrt {(\beta '(\phi )\cos({\widehat {\bar {\alpha }}}))^{2}+(\sec(o\!\varepsilon )\sin({\widehat {\bar {\alpha }}}))^{2}}},{\color {white}{\Bigg |}}\\&={\sqrt {m'(\phi )n'(\phi )\cos({\widehat {\bar {\alpha }}})^{2}+(\sec(o\!\varepsilon )\sin({\widehat {\bar {\alpha }}}))^{2}}},\\&{\color {white}=}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{\mbox{and for the local (i.e.,}}\lim _{\Delta {\widehat {\sigma }}\to 0}{\mbox{), shifting}}\;{\widehat {\bar {\alpha }}}\to {\widehat {\alpha }}{\mbox{ equals}}\\{\tilde {\sigma }}'({\widehat {\sigma }})&={\sqrt {m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\left(1-\left({\frac {\sin({\widehat {\mathrm {A} }})}{\cos(\phi )}}\right)^{2}\right)+\left(\sec(o\!\varepsilon ){\frac {\sin({\widehat {\mathrm {A} }})}{\cos(\phi )}}\right)^{2}}},\\&={\sqrt {m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})+{\frac {\sin({\widehat {\mathrm {A} }})^{2}(\sec(o\!\varepsilon )^{2}-m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }}))}{\cos({\widehat {\sigma }})^{2}+(\sin({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}))^{2}}}}},\\&=n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}\,{\frac {\tan({\tilde {\sigma }}({\widehat {\sigma }}))}{\tan({\widehat {\sigma }})}}={\frac {o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}};\end{aligned}}\,\!}

Ellipsoidal distance

Where,

{\displaystyle {\begin{aligned}\Delta \lambda &={\Big [}\arctan {\big (}\sin({\widehat {\mathrm {A} }})\tan({\widehat {\sigma }}_{f}){\big )}+C{\Big ]}-{\Big [}\arctan {\big (}\sin({\widehat {\mathrm {A} }})\tan({\widehat {\sigma }}_{s}){\big )}+C{\Big ]},{\color {white}{\bigg |}}\\&=\int _{{\widehat {\sigma }}_{s}}^{{\widehat {\sigma }}_{f}}{\frac {\sin({\widehat {\mathrm {A} }})}{\cos(\phi ({\widehat {\sigma }}))^{2}}}d{\widehat {\sigma }}=\sin({\widehat {\mathrm {A} }})\int _{{\widehat {\sigma }}_{s}}^{{\widehat {\sigma }}_{f}}{\frac {d{\widehat {\sigma }}}{\cos({\widehat {\sigma }})^{2}+(\sin({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}))^{2}}};\end{aligned}}\,\!}

{\displaystyle {\begin{aligned}{\stackrel {{}_{\;\;\circ }}{\mathcal {A}}}(\Delta \omega )&=\int _{{\tilde {\sigma }}_{s}}^{{\tilde {\sigma }}_{f}}{\frac {\sin({\tilde {\mathrm {A} }})}{\cos(\beta ({\tilde {\mathrm {A} }},{\tilde {\sigma }}))^{2}}}\cdot {\Big (}1-C'({\tilde {\mathrm {A} }},{\tilde {\sigma }}){\Big )}d{\tilde {\sigma }},\\&=\sin({\tilde {\mathrm {A} }})\int _{{\tilde {\sigma }}_{s}}^{{\tilde {\sigma }}_{f}}{\frac {\sin(o\!\varepsilon )^{2}}{1+C'({\tilde {\mathrm {A} }},{\tilde {\sigma }})}}d{\tilde {\sigma }}=\sin({\tilde {\mathrm {A} }})\int _{{\tilde {\sigma }}_{s}}^{{\tilde {\sigma }}_{f}}{\frac {a^{2}-b^{2}}{a^{2}+a{\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }})}}d{\tilde {\sigma }},\\&=\sin({\tilde {\mathrm {A} }})\int _{{\tilde {\sigma }}_{s}}^{{\tilde {\sigma }}_{f}}{\frac {a^{2}-b^{2}}{a^{2}+bN(\phi (\beta ({\tilde {\mathrm {A} }},{\tilde {\sigma }})))}}d{\tilde {\sigma }};\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}\Delta \omega &=\int _{{\widehat {\sigma }}_{s}}^{{\widehat {\sigma }}_{f}}{\frac {\sin({\widehat {\mathrm {A} }})}{\cos(\phi ({\widehat {\sigma }}))^{2}}}d{\widehat {\sigma }}+\int _{{\tilde {\sigma }}_{s}}^{{\tilde {\sigma }}_{f}}{\frac {\sin({\tilde {\mathrm {A} }})}{\cos(\beta ({\tilde {\mathrm {A} }},{\tilde {\sigma }}))^{2}}}\cdot {\Big (}1-C'({\tilde {\mathrm {A} }},{\tilde {\sigma }}){\Big )}d{\tilde {\sigma }},\\&=\Delta \lambda +{\stackrel {{}_{\;\;\circ }}{\mathcal {A}}}(\Delta \omega );\end{aligned}}\,\!}
${\displaystyle {\Big (}{\tilde {\mathrm {A} }},\,{\tilde {\sigma }}_{s},\,{\tilde {\sigma }}_{f}{\mbox{ and }}\Delta \omega {\mbox{ are reiterated until }}\Delta \omega _{n}=\Delta \omega _{n-1}{\Big )}\,\!}$

then

${\displaystyle {\rm {{Ellipsoidal\;Dx}=\int _{{\tilde {\sigma }}_{s}}^{{\tilde {\sigma }}_{f}}{\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }})d{\tilde {\sigma }};\,\!}}}$

and, using the same process as the meridional case,

{\displaystyle {\begin{aligned}{\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }})d{\tilde {\sigma }}&={\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }}){\frac {d{\tilde {\sigma }}}{d{\widehat {\sigma }}}}d{\widehat {\sigma }}={\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }}({\widehat {\sigma }})){\frac {d{\tilde {\sigma }}({\widehat {\sigma }})}{d{\widehat {\sigma }}}}d{\widehat {\sigma }},\\&=O({\widehat {\mathrm {A} }},{\widehat {\sigma }})d{\widehat {\sigma }}={\acute {R}}({\tilde {\mathrm {A} }}({\widehat {\sigma }}),{\tilde {\sigma }}({\widehat {\sigma }}))\cdot {\tilde {\sigma }}'({\widehat {\sigma }})d{\widehat {\sigma }},\\&=\cos(o\!\varepsilon )N({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cdot {\scriptstyle {\sqrt {m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})+{\frac {\sin({\widehat {\mathrm {A} }})^{2}\left(\sec(o\!\varepsilon )^{2}-m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\right)}{\cos({\widehat {\sigma }})^{2}+(\sin({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}))^{2}}}}}}d{\widehat {\sigma }},\\&={\sqrt {M({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}+{\frac {\sin({\widehat {\mathrm {A} }})^{2}{\Big (}N({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}-M({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}{\Big )}}{\cos({\widehat {\sigma }})^{2}+(\sin({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}))^{2}}}}}d{\widehat {\sigma }};\end{aligned}}\,\!}

thus,

{\displaystyle {\begin{aligned}o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})&={\sqrt {m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}+{\frac {\sin({\widehat {\mathrm {A} }})^{2}{\Big (}n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}-m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}{\Big )}}{\cos({\widehat {\sigma }})^{2}+(\sin({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}))^{2}}}}};{\color {white}{\frac {\bigg |}{}}}\\O&=O({\widehat {\mathrm {A} }},{\widehat {\sigma }})=ao'({\widehat {\mathrm {A} }},{\widehat {\sigma }});\end{aligned}}\,\!}

As even the globoidal orthodromic azimuth (${\displaystyle \scriptstyle {\widehat {\alpha }}\,\!}$) changes at each point along an arc path——equaling ${\displaystyle \scriptstyle {\widehat {\mathrm {A} }}\,\!}$ at the common equator and 90° at the transverse equator——the local elliptic azimuth, ${\displaystyle \scriptstyle {{\tilde {\alpha }}({\widehat {\sigma }})}\,\!}$, can be found from O/o':

${\displaystyle \sin({\tilde {\alpha }}({\widehat {\sigma }}))={\frac {n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}\sin({\widehat {\alpha }});\quad \cos({\tilde {\alpha }}({\widehat {\sigma }}))={\frac {m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}\cos({\widehat {\alpha }});\,\!}$
${\displaystyle {\tilde {\alpha }}({\widehat {\sigma }})=\arctan \left({\frac {n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\sin({\widehat {\alpha }})}{m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos({\widehat {\alpha }})}}\right)=\arctan \left({\frac {n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}\tan({\widehat {\alpha }})\right);\,\!}$

And, as ${\displaystyle \scriptstyle {\sin({\tilde {\mathrm {A} }})=\cos(\beta )\sin({\tilde {\alpha }})}\,\!}$. then

{\displaystyle {\begin{aligned}{\tilde {\mathrm {A} }}({\widehat {\sigma }})&=\arcsin {\big (}\cos(\beta )\sin({\tilde {\alpha }}){\big )}=\arcsin {\big (}\cos(\beta )\cdot \sin({\tilde {\alpha }}({\widehat {\sigma }})){\big )},\\&=\arcsin \left(n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos(\phi )\cdot {\frac {n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}\sin({\widehat {\alpha }})\right),\\&=\arcsin \left({\frac {n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})^{2}}{o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}\sin({\widehat {\mathrm {A} }})\right),\\&=\arccos \left(\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }}){\frac {\sin({\widehat {\sigma }})}{\sin({\tilde {\sigma }}({\widehat {\sigma }}))}}\cos({\widehat {\mathrm {A} }})\right),\\&=\arctan \left(\sec(o\!\varepsilon ){\frac {n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}}{\frac {\csc({\widehat {\sigma }})}{\csc({\tilde {\sigma }}({\widehat {\sigma }}))}}\tan({\widehat {\mathrm {A} }})\right);\end{aligned}}\,\!}

Along with the established relationships between ${\displaystyle \scriptstyle {\beta }{\color {white}{\big |}}\,\!}$ and ${\displaystyle \scriptstyle {\phi }{\color {white}{\big |}}\,\!}$,

{\displaystyle {\begin{aligned}\sin(\beta )&=\qquad \qquad \qquad \;\,\cos({\tilde {\mathrm {A} }})\sin({\tilde {\sigma }}),\\&=\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }})=\cos(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\sin(\phi );\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}\cos(\beta )&=\qquad \quad \;{\sqrt {\cos({\tilde {\sigma }})^{2}+(\sin({\tilde {\mathrm {A} }})\sin({\tilde {\sigma }}))^{2}}},\\&=n'({\widehat {\mathrm {A} }},{\widehat {\sigma }}){\sqrt {\cos({\widehat {\sigma }})^{2}+(\sin({\widehat {\mathrm {A} }})\sin({\widehat {\sigma }}))^{2}}}=n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos(\phi );\end{aligned}}\,\!}

other key relationships with ${\displaystyle \scriptstyle {o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})}{\color {white}{\big |}}\,\!}$ become apparent:

${\displaystyle o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\sin({\tilde {\alpha }}({\widehat {\sigma }}))\,=n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\sin({\widehat {\alpha }});\,\!}$
${\displaystyle o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos({\tilde {\alpha }}({\widehat {\sigma }}))=m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos({\widehat {\alpha }});\,\!}$
${\displaystyle {}_{\color {white}|}o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cot({\tilde {\sigma }}({\widehat {\sigma }}))=\sec(o\!\varepsilon )m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cot({\widehat {\sigma }});\,\!}$
${\displaystyle o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\csc({\tilde {\sigma }}({\widehat {\sigma }}))\tan({\tilde {\mathrm {A} }}({\widehat {\sigma }}))=\sec(o\!\varepsilon )n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\csc({\widehat {\sigma }})\tan({\widehat {\mathrm {A} }});\,\!}$
{\displaystyle {}_{\color {white}|}{\begin{aligned}o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos(\beta )\cos({\tilde {\alpha }}({\widehat {\sigma }}))&=o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos({\tilde {\alpha }}({\widehat {\sigma }}))\cos({\tilde {\sigma }}({\widehat {\sigma }})),\\&=m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos({\widehat {\mathrm {A} }})\cos({\widehat {\sigma }}),\\&=m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\cos(\phi )\cos({\widehat {\alpha }});\end{aligned}}\,\!}
{\displaystyle {\begin{aligned}o'({\widehat {\mathrm {A} }},{\widehat {\sigma }})&=m'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\sec(o\!\varepsilon ){\frac {\cot({\widehat {\sigma }})}{\cot({\tilde {\sigma }}({\widehat {\sigma }}))}}=m'({\widehat {\mathrm {A} }},{\widehat {\sigma }}){\frac {\cos({\widehat {\alpha }})}{\cos({\tilde {\alpha }}({\widehat {\sigma }}))}},\\&=n'({\widehat {\mathrm {A} }},{\widehat {\sigma }})\sec(o\!\varepsilon ){\frac {\csc({\widehat {\sigma }})}{\csc({\tilde {\sigma }}({\widehat {\sigma }}))}}{\frac {\tan({\widehat {\mathrm {A} }})}{\tan({\tilde {\mathrm {A} }}({\widehat {\sigma }}))}}=n'({\widehat {\mathrm {A} }},{\widehat {\sigma }}){\frac {\sin({\widehat {\alpha }})}{\sin({\tilde {\alpha }}({\widehat {\sigma }}))}};\end{aligned}}\,\!}

Finally,

{\displaystyle {\begin{aligned}O&={\frac {O}{\sqrt {\cos({\widehat {\alpha }})^{2}+\sin({\widehat {\alpha }})^{2}}}}={\frac {O}{\sqrt {\left({\frac {O}{M}}\cos({\tilde {\alpha }}({\widehat {\sigma }}))\right)^{2}+\left({\frac {O}{N}}\sin({\tilde {\alpha }}({\widehat {\sigma }}))\right)^{2}}}},\\&={\frac {1}{\sqrt {\left({\frac {\cos({\tilde {\alpha }}({\widehat {\sigma }}))}{M}}\right)^{2}+\left({\frac {\sin({\tilde {\alpha }}({\widehat {\sigma }}))}{N}}\right)^{2}}}};\end{aligned}}\,\!}

In casual examination, it would seem that ${\displaystyle \scriptstyle {\int _{{\tilde {\sigma }}_{s}}^{{\tilde {\sigma }}_{f}}{\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }})d{\tilde {\sigma }}=\int _{{\widehat {\sigma }}_{s}}^{{\widehat {\sigma }}_{f}}O({\widehat {\mathrm {A} }},{\widehat {\sigma }})d{\widehat {\sigma }}}\,\!}$.
And it would, except for one key property (where ${\displaystyle {\color {white}{\big |}}\scriptstyle {{\widehat {\sigma }}_{m}}{\color {white}{\big |}}\,\!}$ is the midpoint between ${\displaystyle {\color {white}{\big |}}\scriptstyle {{\widehat {\sigma }}_{s}}{\color {white}{\big |}}\,\!}$ and ${\displaystyle {\color {white}{\big |}}\scriptstyle {{\widehat {\sigma }}_{f}}{\color {white}{\big |}}\,\!}$):

${\displaystyle {\tilde {\mathrm {A} }}=\lim _{{\widehat {\sigma }}_{s}\to {\widehat {\sigma }}_{f}}{\tilde {\mathrm {A} }}({\widehat {\sigma }}_{m});\qquad {\tilde {\sigma }}=\lim _{{\widehat {\sigma }}_{s}\to {\widehat {\sigma }}_{f}}{\tilde {\sigma }}({\widehat {\sigma }}_{m});\,\!}$

This is because ${\displaystyle \scriptstyle {{\tilde {\mathrm {A} }}({\widehat {\sigma }})}\,\!}$ and ${\displaystyle \scriptstyle {{\tilde {\sigma }}({\widehat {\sigma }})}\,\!}$ are based on the immediate ("local"), elliptic condition of ${\displaystyle \scriptstyle {\phi }\,\!}$ at ${\displaystyle \scriptstyle {\widehat {\sigma }}\,\!}$ along ${\displaystyle \scriptstyle {\widehat {\mathrm {A} }}\,\!}$——it is "static", meaning it does not consider "ellipsoidal fluidity": If, with a sphere, one takes a piece of string and pulls along——say, ${\displaystyle \scriptstyle {{\widehat {\mathrm {A} }}=73^{\circ }}\,\!}$——the string will stay on that 73° arc path, all the way over to the equator on the other side (i.e., its antipode). On an oblate ellipsoid, however, the string will slightly drift towards the pole, with the drift increasing to a dramatic shift, as both ${\displaystyle \scriptstyle {\Delta \omega }\,\!}$ and ${\displaystyle \scriptstyle {\Delta {\tilde {\sigma }}}\,\!}$ approach 180° (since the north-south path is the shortest circumference on an oblate spheroid). Thus ${\displaystyle \scriptstyle {{\tilde {\mathrm {A} }}({\widehat {\sigma }})}\,\!}$ and ${\displaystyle \scriptstyle {{\tilde {\sigma }}({\widehat {\sigma }})}\,\!}$ are the particular ellipsoidal, geodetic ${\displaystyle \scriptstyle {{\tilde {\mathrm {A} }},\,{\tilde {\sigma }}}\,\!}$ values at ${\displaystyle \scriptstyle {{\widehat {\mathrm {A} }},\,{\widehat {\sigma }}}\,\!}$. If one splits that antipodal distance into two equal halves,

${\displaystyle {}_{\color {white}.}\phi _{s:1}=0,\quad \phi _{f:1}=17^{\circ },\,\Delta \lambda _{1}=90^{\circ }\;({\widehat {\alpha }}_{s:1}={\widehat {\mathrm {A} }}=73^{\circ });\,\!}$
${\displaystyle \phi _{s:2}=17^{\circ },\,\phi _{f:2}=0,\quad \Delta \lambda _{2}=90^{\circ }\;({\widehat {\alpha }}_{s:2}=90^{\circ });\,\!}$

each segment would obviously equate. However, the sum of the two, true, ${\displaystyle \scriptstyle {{\acute {R}}({\tilde {\mathrm {A} }},{\tilde {\sigma }})}\,\!}$ based, ellipsoidal geodetic distances, DxCs, do not equal the singular, whole DxC. In contrast, the sum of the two, ${\displaystyle \scriptstyle {O({\widehat {\mathrm {A} }},{\widehat {\sigma }})}\,\!}$ based, elliptical "parageodetic" distances, DxEs, do equal the whole DxE. If the DxCs and DxEs are split into smaller and smaller segments, as the number of segments increase——and the size of each segment decreases——not only do the sums of all of the DxE sets continue to equate, but the sums of all of the DxC sets begin to merge into a single value——that of the DxE sum, until the segments are infinitesimal in length, when each DxC segment equals the DxE segment!
Of course, this ellipsoidal fluidity (is there a more established name?) works both ways. Given the two known sets of endpoints and using the reverse localization——i.e., ${\displaystyle \scriptstyle {{\widehat {\mathrm {A} }}({\tilde {\sigma }}),\,{\widehat {\sigma }}({\tilde {\sigma }})}\,\!}$——one can plot ${\displaystyle \scriptstyle {\phi }\,\!}$ along ${\displaystyle \scriptstyle {\tilde {\mathrm {A} }}\,\!}$, noting the change in ${\displaystyle \scriptstyle {\widehat {\mathrm {A} }}\,\!}$ and ${\displaystyle \scriptstyle {\widehat {\sigma }}\,\!}$ throughout the ellipsoidally defined arc segment.
Is geodetic distance considered polygonal in nature? If not, would the ${\displaystyle \scriptstyle {O({\widehat {\mathrm {A} }},{\widehat {\sigma }})}\,\!}$ based, "parageodetic" distance be, in actuality, the "polygonal geodetic" distance (which would seem to be a more direct, more accurate calculation than——if I understand its basic formulation——the loxodromical "geodetic triangulation network" method)?

If that's not the case, could the "traditional" geodesic be the "vectored geodesic" and my "parageodesic", the "scalar geodesic"?  ~Kaimbridge~16:47, 16 July 2007 (UTC)

Returning to the original question, besides being the arcradius, couldn't O also be considered a radius of curvature?: Just as M is the radius of curvature along the common meridian, since O is aligned along arc paths, or transverse meridians, shouldn't O also be rightfully classified as the "transverse meridional radius of curvature"? ~Kaimbridge~19:32, 1 July 2007 (UTC)

References

Hey, I got your E-mail, did you get mine in return? My E-mail is acting up and I am not sure if it went through or not. Thanks--Cronholm144 20:48, 16 May 2007 (UTC)

Nope, nothing has appeared. --Salix alba (talk) 20:59, 16 May 2007 (UTC)
The files were so large that the E-mail took a couple minutes to load. You should have the first five now--Cronholm144 21
17, 16 May 2007 (UTC)
got them, thanks muchly. Coxeters a good read. --Salix alba (talk) 22:24, 16 May 2007 (UTC)

Image:Never at War - book cover.jpg

Hello Salix alba, an automated process has found an image or media file tagged as nonfree media, such as fair use. The image (Image:Never at War - book cover.jpg) was found at the following location: User:Salix alba/History of conflict between democracies. This image or media will be removed per statement number 9 of our non-free content policy. The image or media will be replaced with Image:NonFreeImageRemoved.svg , so your formatting of your userpage should be fine. The image that was replaced will not be automatically deleted, but it could be deleted at a later date. Articles using the same image should not be affected by my edits. I ask you to please not readd the image to your userpage and could consider finding a replacement image licensed under either the Creative Commons or GFDL license or released to the public domain. Thanks for your attention and cooperation. User:Gnome (Bot)-talk 02:06, 17 May 2007 (UTC)

On the page edit

Hello, I noticed you edit one of my talk pages [3]. Thank you for taking care of that. You may be interested in knowing this page is no longer really mantained. The "real" page is level of detail. I have considered spreading the modify myself but I'm not really aware of what it does.
MaxDZ8 talk 20:08, 20 May 2007 (UTC)

Sortdates

Just to say: good work on the sorting pre 1000AD/CE mathematicians by date! Geometry guy 20:19, 20 May 2007 (UTC)

It was a very strange bug with the previous version, and I don't really understand why it did not work. I asked on WP:VPT which boggled some people. --Salix alba (talk) 20:28, 20 May 2007 (UTC)

Iteratively re-weighted least squares

Oops, my bad. I copy pasted the text from the AfC page without thinking about capitalising. Thanks for that. G1ggy! 08:17, 28 May 2007 (UTC)

No probs. First time I've done anything with AFC I should have read the instructions on what you do with an article first. --Salix alba (talk) 15:45, 28 May 2007 (UTC)

Georg Cantor

Hi Salix alba,

Of course I remember you from when I used to get into screechy arguments on WP:GA/R. Sorry about that... I'm still very pro-inline-cites, but I'm trying to calm down my act a little...

Anyhow, I see you were the editor who nominated Georg Cantor for GA. It just passed; I'm talking about sending it to WP:FAC (in my mind, skipping Peer review). There's a thread on the article's talk.. I would be honored if you would join... and in fact, if a consensus arises to send the article to FAC, I hope Math people will be the ones who nominate it (instead of me, tho I'm the one who initiated the discussion). Thanks! Ling.Nut 19:36, 2 June 2007 (UTC)

Nice to see the work you have done on Georg Cantor, and nice to know its passed GA, now all we need is to get some non-bio maths articles through GA. --Salix alba (talk) 21:35, 3 June 2007 (UTC)
Hey! :-) We've argued about discussed that topic before. I'm gonna repeat myself from way back when. :-)Here's the deal: please try to step outside yourself and imagine that you're not a Math Guy. For example, imagine that you're (ahem!) a Linguistics guy. You're reviewing GA candidates, you see a math or science article, and the article contains tons of assertions such as (I'll just make up some gobbledygook):
Frankly, it is your responsibility as a reviewer to make sure this stuff is not WP:OR! I hope you'll re-read that sentence; it is the Key to All. But... how can a non-science person do this? It can be very difficult. I got lucky with a statement in Georg Cantor (saying his theorems are not "naive") and found a citation.. but as I said, I got lucky. So as a reviewer, you really want inline cites to provide a place where this can be tracked down.... BUT ... OK so Science Folks want to skip (largely) skip the cites, as described in detail in Wikipedia:Scientific citation guidelines. This is a dilemma for the reviewer! Cites aren't always needed... but... how can I as a reviewer (hypothetically speaking) do the responsible thing & protect Wikipedia against WP:OR?
Speaking for myself, Ling.Nut, I see only one answer: each "Hard Sciences" wikiproject should have at least and preferably a couple or three gatekeepers (under another name.. that name sounds corny) who are ratified by consensus at that wikiproject to check for WP:OR, and who sign off (probably in the article's Talk and definitely supplying a diff of a canonical version) on the statement that the assertions are common knowledge & are not WP:OR. To me, this is eminently reasonable, doable, win–win, etc etc etc.
That's all. :-)
If you can help in any way with Georg Cantor, that would be a service to all mankind. :-)
Later!
Ling.Nut 00:04, 4 June 2007 (UTC)

Klee and GA vs Start

Hi Salix alba - thanks for listing the Klee article (which I rated as Start-Class). Do you have any comments on WT:WPM#GA and math ratings? Geometry guy 21:03, 3 June 2007 (UTC)

Thanks for the page revert

Your revert of the vandalism to my user page has been noted and I greatly appreciate it. I'm busy with changes in my life so I haven't been doing much editing lately. Nothing exceptional or overly traumatic, just adjustments to aspects of my life. (Is that vague enough?) Hopefully things will settle down in the next week or two and I'll be back to more regular editing on Wikipedia. Thanks again. Pigman 00:39, 11 June 2007 (UTC)

Curious extra space

Hi there. I saw your recent edits to the Knot theory page. In the "history" section, toward the bottom, there are two references next to each other, one to Kontsevich, and one to Bar-Natan. Inside the left parenthesis, I see a little space before Kontsevich's name, but when I look at the markup, there seems to be no reason for it to appear. Is this just my browser or do you see it too? VectorPosse 22:12, 30 June 2007 (UTC)

Yes I do see it, I suspect a bug with the {{Harvnb}} template. --Salix alba (talk) 22:15, 30 June 2007 (UTC)

Bracket algebra

I created the article you requested. It's a stub so feel free to boost it to start with some context and applications. Cheers--Cronholm144 10:32, 11 July 2007 (UTC)

Parametric Surface

Added a subsection to your Parametric surface article on Surface Area. If you wouldn't mind taking a quick look at it, I'd appreciate it. The second part (after the formula) may be weasel wordy or possibly entirely out of place. Anyway, if you don't mind, take a quick look and tell me what you think here. Thanx. --Plynch22 23:24, 18 July 2007 (UTC)

esher

I believe that these images are not subject to the template. the article in question has no critical commentary about the painting in question. The text contains bare bone statistics, and the image usage is purely illustrative. If you start applying the "non-free-2D" in this way, then by this logic you may upload EVERYTHING ever drawn into wikipedia ant add to a painter's gallery supplying them with 1-2 lines taken from a catalogue, like "Painter X.Yzzy painted it in 1959, oil on canvas, sold for a breadcrumb, and then dropped dead". 'Míkka 16:18, 20 July 2007 (UTC)

I realize that my POV may be tilting over the gray area into dark too much, so I posted a message Wikipedia_talk:Non-free_content#dubious_fair_use about the user who massively uploads images in such way. But I got no comments yet. 'Míkka 16:24, 20 July 2007 (UTC)

Math Question

Hello. You offered some input on the following Help Reference Desk -- Wikipedia:Reference desk/Mathematics -- where I asked about calculating the time elapsed between a starting date and an ending date. I asked a follow-up question on that page, if you would be so kind to take a look there. Thanks. (Joseph A. Spadaro 17:03, 9 August 2007 (UTC))

Ignore

Ignore this, someone was vandalizing your archives and for some reason when I reverted and warned with VP, it warned your page instead of the vandal's. Very strange... Dreadstar 09:56, 14 August 2007 (UTC)

Image:Sunflower spiral.png

Hi Salix The point about the golden ratio angle of 137.5o is that it covers the available space most efficiently, leaving no blank areas, so that developing flowerbuds all have roughly the same leg-room. To show this more objectively, I think the points of the spirals on the left and right should be covered with 'buds' the same size as that of the middle. cheers Raasgat 13:05, 25 August 2007 (UTC)

Missing

I can't believe you mathematics people do not have an article for Transverse axis (mathematics)! Right when I need it, I find there is no such article. Humm. Is there a place where I can request it? Regards, Mattisse 17:00, 1 September 2007 (UTC)

Well there is Wikipedia:Requested articles/Mathematics which is a sea of red. It is an odd case this one as is not a term used inside mathematics, (I've not heard it used). It meaning can be deduced from the dictdef for transverse[4], that is side to side, axis just meaning a line or direction. I've fixed Transverse axis so it has a little explanation. What was the context for your query?
Hum, I've found some references to Transverse axis of a Hyperbola, watch this space. --Salix alba (talk) 15:12, 6 September 2007 (UTC)
PS you got a thanks for your work on Orion (mythology) here. --Salix alba (talk) 21:10, 6 September 2007 (UTC)
Thanks for pointing out the thanks. (I never would have known otherwise.) The transverse issue was for Bao'en Temple. The Transversality explanation did the trick. The Chinese were very concerned about such issues. --Mattisse 22:51, 6 September 2007 (UTC)

Harpenden

I am sorry about putting the false information about Harpenden on there. This flowery language I later realise makes it sound like I am being mean about Harpenden, where I have close connections, and would say "Harpenden is a town in the City and District of St Albans of Hertfordshire in the East of England. It lies on the A1081, between Luton and St Albans. It smelled of cockroaches, with rats all over and that there is no sewage system and the people do not have anything -- no arms, no legs, no eyes". Sometimes people make mistakes. I certainly did. —Preceding unsigned comment added by Michaeldrayson (talkcontribs)

double redirects, etc.

When you move an article, it's a good idea to quickly fix any double redirects thus created. I've done this with the mathematical quilts article. Michael Hardy 22:23, 26 September 2007 (UTC)

Yes I know. I had intended to do it but I ran out of time. Thanks for fixing that. --Salix alba (talk) 08:13, 27 September 2007 (UTC)