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The tone of the article, especially in the introduction, is problematic.

The phrase rarely acknowledged and underutilized, but quite practical is not in an appropriate formal tone and gives the impression that the concept which is the subject of this article is the fruit of original research, whose non-recognition the author laments. Of course, this is an established concept but that is the impression given.

Likewise the phrases eccentricity pressing into the circle and squashing it, or have fallen into general obscurity, have no place in an encyclopedia. Ceroklis 17:43, 9 October 2007 (UTC) reply

A description of the versine, vercosine, coversine and coversine has no place in this article, since these functions are not exclusively used in this context. This description should be integrated into the article on trigonometric functions, or maybe a page describing those should be created and linked from there. Ceroklis 18:00, 9 October 2007 (UTC) reply

I concur. -- Chetvorno ^TALK 04:07, 29 May 2009 (UTC) reply

I have moved everything that concerned the definition of the versine and related functions to Versine. Furthermore, I managed to simplify the article so that there is no reference to these (dated) functions left. Barsamin ( talk) 11:14, 11 October 2009 (UTC) reply

There is some redundancy between different articles on ellipses. Why should this article repeat the definition of linear eccentricity or Flattening ? There is also quite a bit that is already in the ellipse article. Maybe it would be best to keep this article a short definition of angular eccentricity (a bit like the article on Flattening) and to move the discussion on the different ways to represent the shape of an ellipse to the ellipse article, where it truly belongs. However the ellipse article is already long and these concepts also apply to ellipsoids, not just ellipses. So perhaps a separate article on "concepts of shape for ellipses and ellipsoids" should host this discussion. In fact I am basically advocating a split of the definition of angular eccentricity and the section on "elliptic parameters". Ceroklis 18:05, 9 October 2007 (UTC) reply

(1) There is the suggestion that angular eccentricity should be regarded as the fundamental measure of eccentricity from which others are derived; see, for example, the use of the word "ultimately" in the introduction. The is hardly a "neutral point of view". Surely, it's more accurate to say that all the measures are inter-related and can be expressed in terms of each other. In point of fact, the angular eccentricity is rarely encountered in the literature; much more frequently used is one of eccentricities or one of the flattenings.

Right, but this article's primary purpose is about the angle itself, not its functions 173.9.95.230 ( talk) 18:06, 21 October 2010 (UTC) reply

(2) The notation "oe" is non-standard and is likely not be acceptable to most journals. I recommend sticking with alpha. In the few articles where there's a conflict with another use of alpha, then use epsilon or varepsilon instead.

That would be even more ambiguous since that is used for eccentricity itself! 173.9.95.230 ( talk) 18:06, 21 October 2010 (UTC) reply

(3) The "Applications" section attempts to establish the importance of the angular eccentricity. But the examples given fail in this. The quantity sqrt(1 - e^2 * sin(phi)^2) can be written as

 sqrt(1 + 2*n*cos(2*phi) + n^2)/(1+n)

where n = (a-b)/(a+b). (The article has f' in place of n. But this is incorrect.) This gives a rapidly convergent series, but more importantly it's easy to see *why* the series converges rapidly--because of the symmetry of n when a and b are interchanged. What's the reader supposed to learn from that sequence of 11 equations? I'm not sure what to make of the statement that using the angular eccentricy "provides a more illustrative--if not even its definitely mathematical--origin".

(4) Finally, a big problem is that nearly all the articles on ellipsoids have had the normal eccentricities replaced by the angular eccentricity. This is contrary to standard usage and merely serves to obfusticate otherwise familiar equations. For example, the article on Latitude, includes the subexpression

 ln(sec(alpha) * (1+sin(alpha)))

in the formula for the authalic latitude. What a complicated mess! Much simpler and much clearer would be to substitute

 atanh(e)

Cffk ( talk) 02:43, 23 September 2010 (UTC) reply

I would have to agree with the original article's idea that α is a poor choice of symbol for a.e. Not only because the original article says that α symbolizes azimuth (although I don't think that is a problem in this article) but also because α looks too much like radius a ${\displaystyle \alpha \;\;a}$ 74.10.197.201 ( talk) —Preceding undated comment added 17:12, 21 October 2010 (UTC). reply

Yes, I would also have to agree that ${\displaystyle o\!\varepsilon =\arccos \left({b \over a}\right)}$ looks more appropriate and less confusing than ${\displaystyle \alpha =\arccos \left({b \over a}\right)}$ 173.9.95.230 ( talk) 18:06, 21 October 2010 (UTC) reply

We shouldn't be making things up. Anyway by the same argument it looks too much like aε which gives the distance from the origin to a focus. There are books on Elliptic functions which use things like θ or γ instead for modular angle but I don't see the problem with just leaving α alone. How many people are going to mix up an angle with a length? This is done all the time with triangles where the sides are a,b,c and the angles are α,β,γ Dmcq ( talk) 12:29, 27 April 2011 (UTC) reply

I have recently asked several professionals in the field about angular eccentricity. All are agreed that this parameter is not used in modern literature. The inclusion of this parameter as a central feature of so many wiki articles is a distortion of this subject area and as such it is contrary to wiki guidelines. Bearing all this in mind, and failing to find any genuine applications, I have reduced the article to neutral definitions.

1. Removed contentious claims that AE (angular eccentricity) is more fundamental than other parameters.
2. Added some simplified formulae.
3. Removed references to Spheroids. The areas are treated in those articles using conventional notation.
4. Removed the "applications". Note that it is possible to get from the first to last equation in two lines using the definitions of eccentricity and flattening in terms of the semi-axes. AE is not required at all. This is high quality obfuscation. The point about convergence is already made in the Meridian arc.
5. Removed the 'See Other' refs. Garfield does show AE in one table but never uses it. The Map projections article is essentially a counter example because it includes several articles which avoid AE!
6. Removed inaccessible German and Russian references are.
7. Removed the Bessel ref which is treated in Meridian arc.

  Peter Mercator ( talk) 22:47, 10 January 2012 (UTC) reply

I have changed ${\displaystyle sin^{-1}}$ and ${\displaystyle cos^{-1}}$ in the definition with ${\displaystyle arcsin}$ and ${\displaystyle arccos}$ respectively. These are the standard names of these functions.

The old notation is commonplace but ambiguous. This is discussed in the second paragraph of Inverse trigonometric functions (note how they use arcsin in all the formulas).

Basically the problem is that ${\displaystyle sin^{-1}(x)}$ can equally mean ${\displaystyle arcsin(x)}$ or ${\displaystyle {\frac {1}{sin(x)}}}$. To know which meaning is intended you are forced to check the table further down. With this change, the definition is 100% unambiguous. This is what we should strive for in a encyclopedia. Bomazi ( talk) 22:23, 18 May 2012 (UTC) reply

Standard? Says who? As far as I know, there is no "standard" other than acceptance, and the notation ${\displaystyle sin^{-1}}$ is certainly widely accepted: indeed, you yourself say that it is "commonplace", which as far as I can see is another way of saying the same thing. Ambiguous? In principal it could mean 1/sin(x), but in practice it never does. The notation ${\displaystyle f^{-1}}$ is standard for inverse functions, and there is no good reason for treating trigonometric functions differently. There is, on the other hand, a good case for avoiding the notation ${\displaystyle sin^{2}x}$ to mean ${\displaystyle (sin(x))^{2}}$, as logically it should mean ${\displaystyle sin(sin(x))}$, as that is how ${\displaystyle f^{2}(x)}$ is used. It is possible that usage has shifted in recent years, but from my experience I should say that ${\displaystyle sin^{-1}}$ is by far the more common notation in English, which is what we should consider, not what someone thinks logically should be the notation, quite apart from the fact that your view of what it logically should be is debatable. The editor who uses the pseudonym " JamesBWatson" ( talk) 16:16, 28 February 2014 (UTC) reply

I concur with the previous editor. "-1" for inverse function is more fundamental than "-1" for inverse power. It is quite possible to live with the inverse function and arcsin notations: many professionals do. On the other hand I know no examples where ${\displaystyle sin^{2}x}$ should be construed as ${\displaystyle sin(sin(x))}$ but that perhaps is a blind alley we shouldn't explore. Usage is more important than opinion.   Peter Mercator ( talk) 23:32, 28 February 2014 (UTC) reply

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I added parens around. arcsin argument for clarity. But it was reverted. I think the parens help readability because the -1 to sin makes the e really far away and a quick look can result in misreading the equation without the ()s in place. What do others think? WilliamKF ( talk) 21:09, 21 December 2017 (UTC) reply

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