Talk:Quasi-sphere

Hyperspseudosphere

Can quasi-sphere also be called hyperpseudosphere? It is a hypersurface, thus the "hyper" prefix. In a pseudo-Euclidean space, the pseudo-Euclidean metric of the square of a vector is the pseudo-Euclidean squared radius of a pseudo-sphere. If the space is Euclidean, then you just get a hypersphere. In a space-time metric space, you get a pseudosphere that looks like a sphere in space that is changing in size with time, or it looks like a unit hyperboloid (pseudosphere) in two space dimensions and the time dimension, which can also (awkwardly) be called a hyperhyperboloid. Twy2008 (talk) 09:39, 18 July 2017 (UTC)

I would say that (as defined in this article) a quasi-sphere is more general as a metric hypersurface: it includes normal spheres, whereas pseudospheres do not. One does find the term (e.g. here), but its use does not seem to necessarily imply an embedding in a space that the quasi-sphere of this article does. Unsettled (new) terminology like this generally seems to be a bit clumsy. —Quondum 04:48, 19 July 2017 (UTC)

To amplify on the generality a bit further: an n-sphere is a double cover of an elliptic geometry, an n-pseudosphere is a double cover of a hyperbolic geometry, but quasi-spheres not only include these, but also (double covers of) all deSitter and anti-deSitter geometries (each embedded in a flat space of one dimension higher in a way that inherits the metric tensor). —Quondum 05:17, 19 July 2017 (UTC)

"Unsettled (new) terminology" is right. Looking for usage in Mathscinet and Google produced little. Add refs for the double covers of deSitter cosmo as it may be that hyperboloid will serve better. — Rgdboer (talk) 23:22, 19 July 2017 (UTC)

Finding any references for these terms is difficult, as you say. The only references that I've found on the topic that I consider reasonable have been cited. Though de Sitter space and anti-de Sitter space may be viewed as quasi-spheres when thought of as embedded in pseudo-Euclidean spaces of one dimension greater, without these embeddings they are not quasi-spheres in the sense of this article. I don't think that these should be mentioned other than as links in § See also. My only point in my post above is that the term quasi-sphere covers a larger class of objects than hypersphere, hyperpseudospere, or even their union as classes. Hyperboloid is similarly less general. My post above is explanatory only, an argument against considering the term hyperpseudosphere as a possible synonym for quasi-sphere, i.e., it is original research and is probably not suitable for use in the article. —Quondum 01:36, 20 July 2017 (UTC)

Notation

Does x⋅x mean Q(x) ? The notation is used as follows:

ax ⋅ x + b ⋅ x + c = 0,

Now I see it: B = symmetric bilinear form obtained from polarization identity. Yes.

Part of the problem using "sphere" for this concept is indicated in the addition concerning counter-sphere. Q(x) = 1 is a sphere if Q is positive definite, but for other Q hyperbola or hyperboloid is more suggestive. — Rgdboer (talk) 21:11, 23 July 2017 (UTC) Ammended. — Rgdboer (talk) 21:16, 23 July 2017 (UTC)

The notation x ⋅ y is from the sources. This is associated with the dot product for many people, so we could replace it with B(x, y) or ⟨x, y⟩ if that makes it easier for readers.

This article does not use the term sphere except with its Euclidean meaning. It defines the term quasi-sphere as per the sources; a quasi-sphere is not always a sphere (although any sphere is a quasi-sphere). With reference to the term counter-sphere, we can see from the extract of the article

"... in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radius squared, those with negative radius squared, those with zero radius squared" (with a footnote for the hyperplanes).

and from the defining equation that quasi-sphere already includes this. Indeed, the distinction is meaningless, and depends purely on a convention. It is insisting on two separate terms for the two hyperbolas x² − y² = 1 and y² − x² = 1. The "counter-sphere" would just be a name for one of these three sets. There is no geometric distinction between the two "non-null" sets other than that they are separate. The problem with making this distinction becomes even more pronounced when geometries over other fields (such as finite fields or complex numbers) are considered; the concept generalizes without change to these, and is equally useful in these contexts.

I have an issue with the terms hyperbola and hyperboloid. These terms may regarded as already having well-defined meanings in any 2- or 3-dimensional affine space (but their definition is independent of any quadratic form associated with the space). Their definition completely clashes with that of the quasi-sphere: many objects that are hyperboloids are not quasi-spheres, and vice versa. —Quondum 22:20, 23 July 2017 (UTC)

In that case a null cone is a quasi-sphere. That would put this term up against numerous sources preferring null cone. — Rgdboer (talk) 22:33, 23 July 2017 (UTC)

I don't see why. Null cones are quasi-spheres, but the latter is a broader class. If one wished to refer to only those quasi-spheres that are null cones, then the term to use would be 'null cone'. Just as one often prefers the term 'dog' to 'animal' when the context restricts the animal in question to a dog. —Quondum 02:19, 24 July 2017 (UTC)

Evidently if X is a real quadratic space, by the trichotomy (mathematics), there are three quasi-spheres A, B, C, such that

$\forall x\in X\ \ \exists r>0\ \ (rx\in A\ \lor \ rx\in B\ \lor \ rx\in C).$

Perhaps this proposition is preferable to the "counter-sphere" contribution. — Rgdboer (talk) 20:37, 24 July 2017 (UTC)

This belongs in the section Quasi-sphere § Partitioning, which informally states this already. Feel free to formalize this and to reference trichotomy, if you wish, though remember that hyperplanes need special care (they partition into these three sets as well). This is also generalizes to all fields, suggesting that it is a robust principle. (The generalization takes the form of the value of the quadratic form applied to a radius being either a zero, a square, or a nonsquare, with care again taken with hyperplanes. For a field like the complex numbers and a standard quadratic form –not a sesquilinear form– there are clearly only two partitions under this partitioning.) —Quondum 21:38, 24 July 2017 (UTC)

Yes, the scope of the article could be expanded by setting the context as a quadratic space of an isotropic form instead of the pseudo-Euclidean context which is limited to real spaces. So much is learned from finite fields that a general context would be helpful. Here one approaches quadratic set being used in finite geometry. Certainly a quasi-sphere is a quadric. — Rgdboer (talk) 22:14, 25 July 2017 (UTC)

A glance makes me think the following:

• A quasi-sphere is a quadratic set. But not necessarily the reverse: a quasi-sphere is defined in the context of a quadratic space (which can be considered to be a projective space), with reference to the form. A quadric set is defined in the context of a projective space, with respect to any quadratic form. Thus, any elliptic cone is a quadratic set, but of these only light cones are quasispheres.
• A quadric is a quadric set. But degenerate quadratic sets are not generally called quadrics.

At least, that is how I understand it. So there is a strong connection, but they are all distinct.

On the quadratic space of a (nondegenerate) quadratic form, the pseudo-Euclidean space as defined here is just a real quadratic (affine) space. It might make sense to change the terminology in the article.

On the generalization to arbitrary fields, I think that this is a good idea. However, I do not have references. I'd be surprised if the literature does not cover this, though. As with many articles, one might want to start with the real case, and then present generalizations in a section. (Alternatively, one might present the general case, and have a section devoted to the specialization to the real case for those readers who need something more familiar, less abstract. This latter seems reasonable, guessing at the typical reader.) —Quondum 00:18, 26 July 2017 (UTC)

Deletion

I hate to be that guy, but I think this article should be deleted. This is not a standard concept and terminology. Moreover I would argue that the prefix "quasi-" is a poor choice of terminology, when "pseudo-" is available and obviously more natural. The two references cited in the article are anecdotal, they each talk about "quasi-spheres" for one page or less, and these sections are not cited elsewhere.