Talk:Ellipsoid

Matrix properties

In the introduction, the general formula for an ellipsoid is mentioned which includes matrix A. Many of the subsequent properties are not related to this matrix anymore, which is a shame since I'm working with an ellipsoid formula of this form.

More specifically, I have the idea that the volume of the ellipsoid is proportional to the square root of the determinant of A, which is the sort of information which is missing. I would appreciate it if somebody could fill in some of these properties (I'm unsure on the exact factors). — Preceding unsigned comment added by 131.155.70.54 (talk) 12:42, 10 August 2011 (UTC)

You are right. The positive-definite matrix A describes an ellipsoid. If you eigendecompose A, the eigenvectors give you the principal axes of the ellipsoid and the eigenvalues give you the square of the length of the principal radii. So the determinant (which is also the product of the eigenvalues) gives you the square of the volume of a rectangular solid aligned with the axes of the ellipsoid, with one corner in the center of the ellipsoid and stretched so it reaches the edges of the ellipsoid. Thus the square root of the determinant is the square root of the product of the eigenvalues and so the product of the square root of the eigenvalues and so is the product of the lengths and so is the volume of that rectangular solid. So in the case of a unit sphere at the origin, A is the identity, the square root of the determinant is 1. In 3D, then, I think the volume of the ellipsoid is just:

$V={\frac {4}{3}}\pi {\sqrt {\det(A)}}$

Generalizing to n dimensions, I think you'd wind up having volume be proportional to ${\sqrt {\det(A)}}$ but with the constant of proportionality depending on n (see N-sphere#Volume_and_surface_area). It looks like it would be

$V_{n}={\frac {\pi ^{\frac {n}{2}}}{\Gamma ({\frac {n}{2}}+1)}}{\sqrt {\det(A)}}$

I hope that helps. —Ben FrantzDale (talk) 13:23, 10 August 2011 (UTC)

Now the matrix isn't even mentioned in the introduction at all, but the volume is still given in one form using the determinant of its inverse (which is different to what you've stated here, not sure which is correct). For now I'm just going to remove it, if someone wants to use the matrix treatment they can put it through the whole article consistently. 130.63.110.250 (talk) 16:14, 11 February 2015 (UTC)

Jacobi

Is a Jacobi ellipsoid a special case of scalene? It is frequently mentioned in hydrodynamics, so we should mention or rd. here. — kwami (talk) 21:56, 17 January 2012 (UTC)

Yes. Jacobi ellipsoids are rotating tri-axial ellipsoids. ('Scalene' is rarely used in the literature). The terminology is used in the discussion of the equilibrium form of a rotating self-gravitating fluid. As the angular momentum increases the equilibrium form changes from an axially symmetric oblate ellipsoid (a Maclaurin ellipsoid) to the tri-axial (rotating) Jacobi ellipsoid. An internet search reveals an extensive literature. Perhaps this should be mentioned very briefly in the 'Dynamics' section with links to other Wiki pages, the web or standard texts (of which the classic is by Chandresekhar). But not too many details on this page please. Please read around the subject first.  Peter Mercator (talk) 22:51, 20 January 2012 (UTC)

I'd read around the subject, but didn't know if I was missing something that the sources thought was too obvious to mention, so I didn't want to just give them as synonyms. I've added a line to that section.

If 'scalene' is rare, I can see that 'Jacobi' might be preferable, since spheres are technically triaxial. — kwami (talk) 23:22, 20 January 2012 (UTC)

What are higher-dimensional ellipsoids called?

The introduction states that "an ellipsoid is [..] a three dimensional analogue of an ellipse". A later section states that "One can also define ellipsoids in higher dimensions". While the math is clear to me, I'm confused about the standard terminology: does "ellipsoid" now just refer to the 3-dimensional thingy, or does the term apply in all dimensions >2? Thanks! - Saibod (talk) 16:29, 19 October 2012 (UTC)

My instinct says all dimensions, just as sphere is used (at least by some writers) for all dimensions. But that's merely me. —Tamfang (talk) 16:51, 19 October 2012 (UTC)

If you wish to be precise, you can use hyperellipsoid or n-hyperellipsoid. Sometimes hyphenated. There are a number of hits on GBooks.

Schwatzman (1996) The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English says, "A hyperellipse is the same as a hyperellipsoid, provided that a hyperellipse isn't construed as a three-dimensional ellipsoid".

But then, no-one ever says *hypersquare or *hypercircle, so "hyperellipse" would be odd usage.

There's also n-ellipsoid for generic usage (not just n > 3), by analogy with n-sphere, n-cube, and n-rectangular parallelepiped. But n-ellipse without the -oid has a different meaning. — kwami (talk) 21:18, 19 October 2012 (UTC)

Remarks to the lead

A simple geometrical definition of an ellipsoid, similar to an ellipse, does not exist. So the lead should use the formal definition as a quadric via its normalform as made in many other WIKIs: ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1$. From this equation many properties can be seen easily. By the way: the sentence An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry,... seems to me wrong/misunderstanding. The article Axial symmetry means rotational symmetry.--Ag2gaeh (talk) 13:14, 27 February 2017 (UTC)

What is wrong with "an ellipsoid is a quadric surface such that every planar cross section is either an ellipse, or is empty, or is reduced to a single point"? And with "an ellipsoid is a quadric surface that is bounded, which means that it may be enclosed in a sufficiently large sphere"? I agree that these definitions refer to quadric surfaces, but the definition by normal form refers also to a quadric equation. IMO, starting with the normal form is misleading, as suggesting that the axes of the ellipse must be parallel to the coordinate axes. Also, MOS:INTRO states explicitly that formulas must be avoided in the lead, when possible.

You are wrong when stating that "a simple geometrical definition of an ellipsoid does not exist". There are several. If you do not like those that I have given, you could consider "an ellipse is the image of a sphere by an affine transformation". Although simple and geometrical, this seems too technical for the lead. A more elementary variant could be used in the lead such that "an ellipsoid is obtained from a sphere by scaling it in three pairwise perpendicular directions; in other words an ellipsoid is the image of a sphere by a transformation $x\to ax,y\to by,z\to cz.$"

Axial symmetry is a stub that uses a unusual terminology and should be merged with Circular symmetry, which is about exactly the same topic. As far as I know, axial symmetry, is a common name for a rotational symmetry in 3D. I'll fix the target of the pipe in the article. D.Lazard (talk) 16:54, 27 February 2017 (UTC)

To the symmetry: But, a general ellipsoid has no rotational symmetry. To the definition in the lead: You are more experienced.

I would like to insert two chapters of the German article: a) Ebene Schnitte (plane sections) and b) Ellipsoid in beliebiger Lage/Parameterdarstellung (ellipsoid as affine image of the unit sphere). Would You agree ? --Ag2gaeh (talk) 17:38, 27 February 2017 (UTC)

A general ellipsoid has a rotational symmetry of angle π around each of its principal axes: it is obtained from the normal form equation by changing the signs of any two of the variables. By the way, a general ellipsoid has also three reflection symmetries around the planes defined by two of the principal axes of the ellipsoid., making a symmetry group of order 8, isomorphic to $(\mathbb {Z} /2\mathbb {Z} )^{3}.$ This is a fundamental property that is lacking in both English and German articles.

I would agree to import some material from the German article (as well as from the French article). However these articles are also very incomplete, and it would better to look to classical textbooks (for example Berger's Geometry). Examples of lacking content: Symmetry group; how recognizing an ellipsoid (among other quadrics) from its implicit equation; how computing the center, the principal axes and their length; the fact that, in Euclidean geometry, two ellipsoids are congruent (that is, are isometric) if and only if they have the same normal form equation, while, in affine geometry, any two ellipsoids are congruent (that is, one is the image of the other through an affine transformation), ... D.Lazard (talk) 22:39, 27 February 2017 (UTC)

Sorry, I had Your previous target of axial symmetry in mind, which suggested any angle is allowed. You should insert Your explanations given here into the lead. Your explanations would be easy to understand, if the normalform would be available in the lead. Yes, much more material should be inserted.--Ag2gaeh (talk) 08:35, 28 February 2017 (UTC)

Section 'Plane sections' is hardly readable

The section 'Plane sections' is, in it's current form, hardly readable.

It starts "It's easy to check:" ... WHAT is easy to check? And from there on it doesn't get any better.

Several phrases that are not explained nor linked to a Wiki article for further reading. Greetings, RagingR2 (talk) 12:59, 24 May 2017 (UTC)

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#Notation 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

@D.Lazard: 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 @D.Lazard: 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 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)