Talk:Ellipsoid

On notation and brackets
I was amused to see that 'artanh' is a recommended notation. By whom? Please provide examples in professional papers that can be consulted online. I have not met it in either texts that I used in teaching mathematics or in research papers that I read. I prefer the widespread arctanh notation as found in (reference 3) the Handbook of Mathematical Functions produced by American National Institute of Science and Technology (and its predecessor, Abromowitz and Stegun). Both of these books may be considered as fairly authoritative.

As for brackets, they are indeed a matter of taste and most mathematical texts simply write 'sin z' for aesthetic reasons. Both of the above handbooks do not use brackets for simple arguments of the trig functions. Engineering books tend to be a little more pedantic and less aesthetic in this matter.

I should declare an interest in that I added the section on area some five years ago -- without brackets! Further comments please. Peter Mercator (talk) 15:35, 30 June 2017 (UTC)
 * For notation fo inverse hyperbolic functions, see artanh and the notes 1,2,3 of this section, which contain quotations of reliable sources on the matter.
 * For brackets, they may be omitted only in the unambiguous case. Here, omitting brackets is slightly ambiguous, as $$\cos a \sin b$$ may be interpreted as $$\cos( a \sin b)$$ was well as $$\cos (a) \sin (b).$$ I admit that the first interpretation is unusual and that this is not really ambiguous for an experimented mathematician, but I am not sure that the notation without parentheses is absolutely clear for every reader. In the case of this article, there are products of three trigonometric functions. This is immediately clear for the reader when there are brackets, while, without brackets, one has to read the formulas for understanding their structure. As aesthetic and readability are very close for math formulas, this is a strong reason for using brackets in the case. D.Lazard (talk) 16:38, 30 June 2017 (UTC)

Plane sections
An IP user has set a question in section "Plane sections" of the article, about the proof that planes sections of an ellipsoid are circles, single points of empty point. I have put it in the field "reason=" of a template clarification needed. As I am not sure wether a detailed proof deserves to appear in the article. I'll answer here.

Let $E$ be an ellipsoid, $P$ be a plane, $f$ be an affine transformation that maps the unit sphere onto $E$, and $g$ be the inverse transformation. The image by $g$ of the intersection of $E$ and $P$ is the intersection $C$ of the unit sphere $g(E)$ and the plane $g(P)$. It is thus either a circle, a point, or the empty set. Thus the intersection of $E$ and $P$ is $f(C)$, that is an ellipse, a point or the empty circle.

I do not know if such a proof deserve to be put in the article. Therefore, I leave to the community to decide what should be done. D.Lazard (talk) 14:57, 18 September 2017 (UTC)


 * This is good, but my immediate impression is that it is a bit too verbose and far afield for the main text of this article. Perhaps it could be encapsulated into the note that you just inserted, or inserted as an example in Affine transformation and linked from this article.—Anita5192 (talk) 19:54, 18 September 2017 (UTC)


 * I agree with Anita. I'll also add a reference for the result in this article. --Bill Cherowitzo (talk) 04:47, 19 September 2017 (UTC)

Important link, explanation
the edit 858602570 is correct, Geoid, an important object modeled by an elipsoid. It is not an elipsoid, it is modeled as an elipsoid (instead a sphere). The most used elipsoid-Goid model in nowadays, is the WGS84 elipsoid. --Krauss (talk)
 * First of all, "important" is your personal opinion. I agree that the geoid is an important concept and that working with the geoid requires to know about ellipsoids. However this is a mathematical article, not an article about geodesy, and importance should be estimated relatively to readers of this article. As they are probably interested primarily to mathematics and specifically to geometry, a link to Geoid seems unimportant for them. In any case, the link will not help them to better understand the article and its context.
 * Also, the body of the article provides links to Earth ellipsoid and Reference ellipsoid, which, for this article, are more appropriate than Geoid. Thus, adding Geoid in See also section, would be against the recommendations of WP:NOTSEEALSO: As a general rule, the "See also" section should not repeat links that appear in the article's body or its navigation boxes. D.Lazard (talk) 11:19, 8 September 2018 (UTC)
 * I agree. This existing links to Earth ellipsoid and Reference ellipsoid suffice for readers to make the connection.  There's no need to add a link for geoid; this is more likely than not to be confusing. cffk (talk) 12:16, 8 September 2018 (UTC)
 * I disagree. The presented arguments justify adding the link to geoid. cffk (talk)'s confusion is his own private problem. Cocorrector (talk) 13:19, 19 November 2018 (UTC)


 * Hi thanks. Now I see that Earth ellipsoid and Reference ellipsoid are links in the article, that is good (Earth model is important and is comtemplated). I see also that Geoid article is not citating the term ellipsoid... It is because in my mind I do a "jump" over the intermediary concept, that is Geodetic datum... Now the "see also" section is correct. PS: of course, the 3 concepts are similar, perhaps one day some articles will be merget to the "most important"...  The "importance of the article" is not personal opinion, is a collective behaviour of pageviews, see |Earth_ellipsoid|Geodetic_datum|Geoid statistics. Geoid and Geodetic datum are the most "important" in this objective facet of the Wikipedia's articles.  Krauss (talk) 13:53, 19 November 2018 (UTC)

Mac Cullagh ellipsoid
An editor as inserted in the list item about Poinsot's ellipsoid a comment about a so called "Mac Cullagh ellipsoid". I have reverted this edit, and I'll revert it again for the following reasons:
 * The term "Mac Cullagh ellipsoid" is not used in the literature. A Scholar Google search on these words results only in Mac Cullagh's articles and some articles of 19th century citing Mac Cullagh work. The searches of "Mac Cullagh ellipsoid", "MacCullagh ellipsoid", "McCullagh ellipsoid" (between quotes) provides only two hits. Thus this terminology is definitively not notable and not reliably sourced. Therefore it does not belong to Wikipedia per the policy WP:OR.
 * From the given description, it seems that the so-called Mac Cullagh ellipsoid is exactly the Poinsot's ellipsoid. As Poinsot's work is earlier that Mac Cullagh's, there is no for mentioning Mac Cullagh in this article. This would give WP:UNDUE weight to this work, which deserves only a mention in the history section of Poinsot's ellipsoid.

Thus I'll revert again the mention of "Mac Cullagh ellipsoid" in this article. If you disagree, please, read carefully WP:BRD and do not start WP:Edit warring.

I'll also revert the insertion of Geoid in section "See also", as the linked article does not contain the word "ellipsoid", and is linked in other articles appearing in this see also section. It thus not useful for any reader to link this article. D.Lazard (talk) 19:11, 16 November 2018 (UTC)


 * I agree, even a novice reading of Poinsot article shows it is the same formula (it uses T instead of 2E). Maybe you should put a mention there of the alternative name. Geoid instead of being a "see also" should probably be in this use list.Spitzak (talk) 19:54, 16 November 2018 (UTC)


 * Glad to see D.Lazard and Spitzak clarifying the problem. Let me further clarify it to them both: МасCullagh ellipsoid IS NOT ths same as Poinsot's ellipsoid. Not to worry though since there are many things that would be obvious to novices although they are not true. So, a formula using 2E instead of T IS NOT the same. Cocorrector (talk) 12:21, 19 November 2018 (UTC)

MacCullagh ellipsoid is now at WP:AfD. D.Lazard (talk) 12:38, 19 November 2018 (UTC)


 * With some luck Spitzak can learn the difference between Poinsot's ellipsoid and MacCullagh ellipsoid in Zhuravlev Foundations of Theoretical Mechanics (Fizmatlit, Moscow, 2008)) [in Russian], whereas D.Lazard displays little ability to do the same. Our sorrow for him should not preclude us from neutralizing him. By the way, his objection to Geoid seems to be consistently stupid. Cocorrector (talk) 12:49, 19 November 2018 (UTC)

e_1
I think the formula in this section for e_1 is incorrect by setting its z-coordinate to 0. If so, after inverse affine transformation, the corresponding z-coodinate is still 0, which is not necessary. — Preceding unsigned comment added by 24.188.214.97 (talk) 07:27, 3 December 2018 (UTC)

Incorrect surface normal parametric form
The equation for the surface normal in the Parametric Representation section appears to be incorrect for non-spherical ellipsoids. When I try to replicate, the normals point roughly outward but do not match the surface curvature except at the poles and the equator. It is as if the map from the isotropic sphere to the anisotropic ellipsoid is not taken into account. 98.69.156.214 (talk) 03:20, 10 April 2020 (UTC)


 * The parametric representation is regular only for $$\theta\ne\pi/2$$. I added this restriction. It is $$\vec x_\varphi \times \vec x_\theta= \vec n \cos\theta$$.--Ag2gaeh (talk) 07:58, 10 April 2020 (UTC)

The circumscribed box
The volume of an ellipsoid is 2/3 the volume of a circumscribed elliptic cylinder, and π/6 the volume of the circumscribed box.

These do not circumscribe uniquely; our preferential embedding in at least the box case can likely be specified as the one having minimum volume. &mdash; MaxEnt 00:45, 26 May 2020 (UTC)

Affine vs linear image of sphere : unnecessary pedantic distinction
The top of article describes ellipsoid as an affine image of a sphere. Since the position of the sphere is not fixed, linear transformations (if understood as homogeneous degree 1 maps) suffice. This is better for introduction as not everyone shares the mathematics terminology that linear transformations must be homogeneous or involve a choice of origin, and "linear" is more intuitive. "Affine" is a more math-specific term and the Wiki link to a page about transformations preserving "an affine structure" is ridiculous as an explanation for people who may not know what is an ellipsoid. It is more of a definition for purists that can be elaborated inside the article. 73.89.25.252 (talk) 04:39, 14 June 2020 (UTC)
 * Ellipsoids are considered in the Euclidean space. The concept of a linear transformation is not defined in a Euclidean space. So, it is wrong to use "linear transformation" here. It is not pedantry to use the only existing correct term. Nevertheless, I agree that the first sentence of Affine transformation was much too technical. I have fixed it. D.Lazard (talk) 14:31, 14 June 2020 (UTC)

Incomplete statement in the "As a quadric" section
In the "as a quadric" section, the following is stated:


 * an arbitrarily-oriented ellipsoid, centered at $v$, is defined by the solutions $x$ to the equation
 * $$(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1,$$
 * where $A$ is a positive definite matrix and $x$, $v$ are vectors.


 * The eigenvectors of $A$ define the principal axes of the ellipsoid and the eigenvalues of $A$ are the reciprocals of the squares of the semi-axes: $a^{−2}$, $b^{−2}$ and $c^{−2}$.

I am not a mathematician, but am a scientist and I tried using this fact to find the size of an ellipsoid. I suspect that the statement of the eigenvalues being the reciprocal squares of the semi-axes is only true when the matrix A is real (or maybe when the eigenvectors of the matrix are real). Hopefully someone here can check this requirement. I can show a simple example proving that some condition is missing in the statement. If you start with a complex positive definite matrix $A$, the expression
 * $$(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{A}\, (\mathbf{x}-\mathbf{v}) = 1,$$

is identical to the alternative expression
 * $$(\mathbf{x}-\mathbf{v})^\mathsf{T}\! \boldsymbol{Re[A]}\, (\mathbf{x}-\mathbf{v}) = 1,$$

because the imaginary components cancel out on the left hand side, when doing the matrix multiplication, if the matrix A is Hermitian. Therefore, both matrices $A$ and B = Re[A] are positive-definite matrices, and they both describe the SAME ellipsoid, because the equation is identical when matrix-vector multiplication is carried out. However, in general, the two matrices have different eigenvalues! So the statement that its eigenvalues are equal to the inverse squares of the semi-axes of the ellipsoid cannot be simultaneously a correct statement for both matrices. By calculating an explicit example, I found that when I take Re[A], then its eigenvalues DO coincide with the inverse squares of the semi-axes, but not when I keep A complex. Long story sort: some condition is needed before stating that the eigenvalues are the inverse squared semi-axes. Probably requirement of real eigenvectors or similar. El pak (talk) 16:55, 1 July 2021 (UTC)


 * I think that the intended assumption is that the matrix A is real and positive-definite. We might as well assume also that A is symmetric. (Actually it doesn't have to be, but nothing is gained by allowing an anti-symmetric part.) Therefore A is diagonalizable, via an orthogonal change of basis, with only real eigenvalues, by the spectral theorem. And those real eigenvalues are positive because of the positive-definiteness assumption. Do you have any objections to my altering the article accordingly?


 * To answer your question, the volume is a b c 4 &pi; / 3, where a-2, b-2, c-2 are the eigenvalues of A, as in the text that you cited. Mgnbar (talk) 17:28, 1 July 2021 (UTC)


 * I am not an expert but that sounds perfectly sensible and I would be very pleased if you alter the article with that. Thanks! El pak (talk) 16:35, 7 July 2021 (UTC)

A problem with the interpretation of the parameterization.
In Kreyszig, Advanced Engineering Mathematics, 4th ed, on p. 431 there is a parametric representation of a sphere and one is given as part of a problem for an ellipsoid but there is no interpretation of it in terms of the spheroid. In the parameterization θ is not an angle between the equator and a point on the ellipsoid but rather it is a parameter similar to Kepler's eccentric anomaly. ~ Jbergquist (talk) 23:26, 19 June 2023 (UTC)


 * The angles θ and φ used in the parameterization of the ellipsoid are associated with a point on a sphere of radius a which is different than that on the ellipsoid. Jbergquist (talk) 23:39, 19 June 2023 (UTC)


 * I don't have access to the Kreyszig source right now. Are you saying that this article is mis-representing it?
 * If instead you are asking about the content, the parametrization given is simply spherical coordinates stretched by factors of a, b, and c. So, if Kreyszig is an inappropriate source, then we can instead cite any spherical coordinates source combined with Routine calculation. Mgnbar (talk) 03:27, 20 June 2023 (UTC)