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WikiProject Mathematics (Rated C-class, Mid-priority)
WikiProject Mathematics
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 Field: Geometry

Why isn't there some explanation of the relationship between an ellipse and an ellipsoid? Lir 20:01 Nov 9, 2002 (UTC)

an ellipse is a 2d shape, while an ellipsoid is a 3d shape (my apologies if this has been answered already)

Missing info[edit]

hello There must be a formula for the integrated mean curvature out there. This would complete the set of invariant measures of ellipsoids. Bela Mulder 09:27, 25 Nov 2004 (UTC)

I think the oblate/prolate explanation is backwards. a>b=c seems like a cigar, while a<b=c seems like a disk.

No it's not:
OblateSpheroid.PNG ← Oblate;   Prolate → 104px
Remember a is the equatorial radius/axis:
Ellipse, showing major and minor axes
For Earth, an oblate spheroid, a = 6378.135 and b = 6356.75. So if the north and south poles were different (say, bn = 6357.0 and bs = 6356.5), then it would be triaxial, where (it would seem) a = 6378.135, b = 6357.0 and c = 6356.5, wouldn't it? ~Kaimbridge~15:43, 31 July 2006 (UTC)


The volume of an ellipsoid is 4/3 times pi times A times B times C. But what's A, what's B, and what's C? It doesn't say? Jared 19:40, 28 October 2005 (UTC)

They're the ellipsoid's axes, as used throughout the article.
Urhixidur 02:57, 29 October 2005 (UTC)

a is the length from the center of the ellipsoid to its surface at Sphereical Coordinate, in radians, (π/2, 0), which is generally the maximum y coordinate on the surface of the ellipse.

b is the length from the center of the ellipsoid to its surface at Sphereical Coordinate, in radians, (0,0), which is generally the maximum x coordinate on the surface of the ellipse.

c is the length from the center of the ellipsoid to its surface at Sphereical Coordinate, in radians, (0, π/2), which is generally the maximum z coordinate on the surface of the ellipse.

Edward Solomon 5/18/06 21:28, 18 May 2006 (UTC)

You are right about what an oblate spheroid and a prolate spheroid looks like, but the equations were reversed. If 2 of the semi axis are the same, and the third is smaller this will be an oblate spheroid, and as such when b = c, a < b will be an oblate spheroid. If however, the third is larger this will be an prolate spheroid, and as such when b = c, a > b will be an oblate spheroid.

No, No, NO, that's backwards!!!
Ellipse, showing major and minor axes
This is an oblate spheroid, like Earth: a > b! Why is this fact always being overlooked?!? P=/ ~Kaimbridge~10:39, 24 August 2006 (UTC)

Would someone like to write an elegant proof?--Adoniscik (talk) 03:47, 19 January 2008 (UTC)

Incorrect formulas?[edit]

I think I've found two errors on this page. First, the formula for calculating the eccentricity (e), seems to be backwards. It divides the larger number with the smaller and subtracts one from the result, which leads us almost always into a negative square root. I don't think this is intended. The correct form can be found from the Spheroid page, which divides the smaller with the larger, resulting in a positive number. Also it's stated on Eccentricity(mathematics) page, that the eccentricity of an ellipse is greater than zero, less than one.

Also, the formula for a prolate spheroid seems to be incorrect. This might be a result from the false eccentricity formula. After using the correct form for the eccentricity, I'm getting rather wild results:

a 0.2m b 0.2m c 0.5m

Area: 2.37

Surface area of a cylinder with the same dimensions: 1.51

There is an alternate formula on the Spheroid page, which yields an area of 1.05

I changed the formula explaining a scalene elipsoid abc to abca because the original formula could still be satisfied if a = c

Is the formula for surface area correct? mentions an error which seems to have cropped up here (E and F switched), but I'm yet to work through the elliptic integrals to decide which to believe. 16:49, 11 June 2007 (UTC)

The section on surface area is confusing and contains errors. One source of endless confusion could be avoided by demanding a .ge. b .ge. c , the ellipsoid can always be oriented in such a way without altering volume or surface. The formula for the surface area is wrong. I looked through all languages: all exept this and the chinese page got it right...-- (talk) 21:36, 25 October 2009 (UTC)

The reason for the distinction between oblate and prolate in case of the scalene surface area evades me (scalene is already defined above as the case a .ge. b .ge. c) and the symbol looks always like a misprint to me, why not expressing it through a, b and c ? And the surface formula is still wrong, just exchange E and F to get it right. -- (talk) 20:28, 23 November 2009 (UTC)
Yup, E and F were previously switched (due to their definition wording being backwards at the time: "....and , are the incomplete elliptic integrals of the first and second kind"——wording was fixed, but E and F weren't likewise reversed). Everything is correct now! P=)
In terms of a clearer understanding, it would be better to let and , then present it as
as this reduces to
for the oblate (and shows its origin)!
In fact, using throughout the article would be more illustratively correct...but that would probably be considered too OR (at least at this time!!! P=) ~Kaimbridge~ (talk) 16:37, 24 November 2009 (UTC)


I am sure this is a real ellipsoid, but it does look a lot like a prolate spheroid. Perhaps the difference in the shorter axes could be made more marked so its clearer at a glance what the deal is? Deuar 19:35, 15 June 2006 (UTC)
Now we also have what looks for all intents like an oblate spheroid (although a close look at the axes reveals different scales if you're real keen)... Deuar 12:51, 26 June 2006 (UTC)

I was just about to mention this when I saw you said the same thing. All spheroids are ellipsoids, but to avoid confusion I think it would be a good idea to add images af ellipsoids that aren't spheroids, as well as images of spheroids.

Incorret parametric form[edit]

i dont no how to change the range for the Phi and theta, can someone please change the theta value to : 0 <= theta <= 2pi, and the Phi to: 0 <= Phi <= pi. Pprrff 01:29, 2 December 2006

There are different ways to express Cartesian coordinates——the way you were reverting to is the more common (as so, I believe, more to do with its likely application, rather than correctness) "spherical coordinates", which is itself contradictory and confusing——the more recognizable (coordinate-wise) is the geographical(h=0), but the most formatically elementary is using the parametric latitude, as that is what is used in the equation of the ellipse. I'm finishing up a major rewrite of the first parts of this article (including the relationships of the parametric, planetographic and planetocentric latitudes), which should make the different Cartesian forms clear.  ~Kaimbridge~15:43, 2 December 2006 (UTC)

As applied to angular momentum[edit]

Imparting angular momentum on a sphere transforms it into an oblate spheroid. The ratio of volume to surface area of a sphere is constant (r/3), but does not generalize to other spheroids. So when one imparts angular momentum to such a sphere, does the volume change, does the surface area change, or do both change?

Put another way, will two black holes of equal mass, one spinning, one not, necessarily have either equal volume (within their horizons) or equal surface area (of the horizon itself) ?

-- 23:44, 19 February 2007 (UTC)

There cannot be a ratio of area to volume, because these two quantities use different units. It is like trying to compare kilograms with meters. In this case, the units are different only because they have different dimensions. Majopius (talk) 19:36, 8 May 2009 (UTC)

Parametric Formulas[edit]

It would seem to me that the change made on 20/05/2007 to alter the parameteric formulas was incorrect.

Although both sets of equations appear to describe an ellipsoid the change appears to have altered the meanings of the variables \beta and \lambda from their descriptions.

Can someone confirm or reject this suggestion and possible revert or edit the page?

Sardonicpresence 03:09, 30 May 2007 (UTC)

Yup, it was a crank edit——now fixed. ~Kaimbridge~09:42, 30 May 2007 (UTC)

"triaxial ellipsoid"?[edit]

I redirected the poorly worded "triaxial" article here, as it only discussed scalene ellipsoids. However, I wanted to verify the term "triaxial ellipsoid", which is not used here.

Literally, of course, all ellipsoids are triaxial, but is it true that the term "triaxial ellipsoid" used as a synonym for "scalene ellipsoid"?

Similarly, does a link to box orbit (a "triaxial orbit") belong here, or do we need to recreate and rewrite the "triaxial" article? kwami (talk) 18:40, 21 September 2008 (UTC)

Triaxial has other usage - it should go to a disambiguation page. The particular use I had in mind was "triaxal stress".

ellipsoid for kids[edit]

an easy way to imagine an ellipsoid is if you think of a football. you may want to call a football an oval spere. well it's an ellipsoid. have you ever heard of an elliptical galaxy. an elliptical galaxy is an egg shaped galaxy.

                                           by allison nelson  —Preceding unsigned comment added by (talk) 22:13, 30 January 2009 (UTC) 


If the Ball is the interior of the Sphere, then what do we call the interior of an Ellipsoid? K61824 (talk) 02:35, 3 April 2009 (UTC)

From a topological point of view, an ellipsoid is a sphere, just in different coordinates (or in an anisotropic metric). The distinction between open and closed sets is, of course, very important in analysis and topology, but less important in solid geometry. However, I too would be interested in knowing if there is a specific term for the interior. Plastikspork (talk) 14:20, 3 April 2009 (UTC)
An ellipsoid cannot necessarily be called a sphere; a sphere is a special case of an ellipsoid. Majopius (talk) 02:44, 19 May 2009 (UTC)
See Sphere#Topology. Plastikspork (talk) 05:08, 20 May 2009 (UTC)
Really? Is a square a rectangle? Please explain this concept to me and the rest of the elementary school students. (talk) 03:04, 20 May 2009 (UTC)
Yes, a square is a special case of a rectangle, but a rectangle is not necessarily a square. Majopius (talk) 00:26, 21 May 2009 (UTC)

Link to scalene[edit]

This link leads to a disambiguation page. And I do not see a way to modify the link so that it will lead to an appropriate article: there is no article "Scalene Ellipsoid", and the other options are to link to an article about a triangle. The only thing we can do about this is to create the article "Scalene Ellipsoid". Majopius (talk) 23:22, 6 May 2009 (UTC)

Seems like it should be covered here. kwami (talk) 20:31, 8 May 2009 (UTC)

contoh soal ellipsoid —Preceding unsigned comment added by (talk) 04:11, 13 March 2010 (UTC)

Matrix properties[edit]

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 (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:
Generalizing to n dimensions, I think you'd wind up having volume be proportional to but with the constant of proportionality depending on n (see N-sphere#Volume_and_surface_area). It looks like it would be
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. (talk) 16:14, 11 February 2015 (UTC)


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?[edit]

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[edit]

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: . 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 "
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 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)