Talk:Quadric

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Hyperbolic Parabaloid and Affine Transformations

"Geometrically, every non-degenerate quadric surface is obtained from a conic section in the plane as a surface of revolution, followed by an affine transformation."

I am skeptical. What are the conic section and affine transformation that produces a hyperbolic parabaloid?

192.249.47.195 (talk) 18:46, 28 September 2009 (UTC)[reply]

Conic section embedded in R^3

I couldn't find this in the article: Given a quadric equation in three dimensions, what are the necessary and sufficient conditions on the parameters or parameter matrix for it to describe a conic in a two-dimensional flat of R^3? And what further conditions on the matrix characterize the different types of conics? Loraof (talk) 17:59, 15 October 2014 (UTC)[reply]

I'm not clear on what exactly you are asking. Do you want the conditions for a homogeneous quadratic in three-dimensions? The coefficient matrix is then 3×3, and the condition is that it be a non-singular (strictly) semi-definite matrix (that is, with eigenvalues of both positive and negative sign). Then any plane that avoids the origin cuts that cone in a conic section. If the question is rather about a non-homogeneous quadratic polynomial in three variables, then such a thing comes from a homogeneous polynomial of four variables. Then there are two cases of the eigenvalues to consider, (++--) and (+++-). In the first case, the quadric is ruled by two families of lines (think the lines on a hyperbolic paraboloid or a hyperboloid of one sheet). As long as the plane you choose does not contain any line in either the family, it will cut out a conic. In the other case, reality becomes important (think of a plane that does not cut an elliptic paraboloid). In that case, you can restrict the quadric to the plane (say, writing everything in terms of x and y). That gives you a quadric in the plane, and the condition for that quadric to be a conic is that the coefficient matrix should be non-singular (and strictly semi-definite). Sławomir Biały (talk) 22:05, 15 October 2014 (UTC)[reply]

Thanks. Given

Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0

in scalars, or if you prefer

x^{T}Qx+x^{T}R+S=0

for 3×1 vector x, 3×3 matrix Q, 3×1 column vector R, and scalar S, under what conditions do all the points fall in a single plane?

Thanks, Loraof (talk) 01:07, 16 October 2014 (UTC).[reply]

For this to describe a plane, the matrix

{\begin{bmatrix}Q&R/2\\R^{2}/2&S\end{bmatrix}}

must have rank one. Sławomir Biały (talk) 11:17, 16 October 2014 (UTC)[reply]

Definition of a quadric (recent change)

A pair of planes in (3-space) is a quadric ! (s. recent change). --Ag2gaeh (talk) 14:05, 2 March 2017 (UTC)[reply]

Can you reliably source your assertion? A far as I know, for most authors, a quadric is an algebraic variety, and thus defined by an irreducible quadratic polynomial. A pair of planes is an algebraic set that is not an algebraic variety. Moreover, during centuries, before 20th century, quadric were considered in a purely geometric way (that is by considering equations only when they are not avoidable) and in this context, it is a non-sense to consider a pair of planes as a quartic. Also, the study of the quartics was aimed to study (and classify) the simplest non-plane surfaces; it seems absurd to qualify a pair of planes as a non-plane surface. D.Lazard (talk) 16:51, 2 March 2017 (UTC)[reply]

My calculus textbook indicates that such a set is degenerate: "In three-space . . . any second-degree equation which does not reduce to a cylinder, plane, line, or point corresponds to a surface which we call quadric. Quadric surfaces are classified into six types, and it can be shown that every second-degree equation which does not degenerate into a cylinder, a plane, etc., corresponds to one of these six types."^[1] — Anita5192 (talk) 17:55, 2 March 2017 (UTC)[reply]

This is a case where terminology is not uniformly fixed. IMO, there are two methods to find the dominant one. The first is is boring, and consists in consult many textbooks for extracting the dominant terminology. The second one is to look what is meant in the articles that use the terms without defining them. As authors of good paper use a term without defining it only if it is not ambiguous, this is a good indication of the standards. I know that, when I read "quadric" in a paper, this should mean one of the five non-degenerated quadrics (six, if imaginary ellipsoid is relevant in the context). When cones and cylinders must be taken into account, authors generally use "quadratic surface". In this case, only the context allows knowing if non-irreducible quadratic algebraic sets should be included. I have tried to take this into account in my edit, but some clarification is certainly still needed. In particular the distinction between "quadric" and "quadratic surface" deserve to be mentioned. D.Lazard (talk) 19:13, 2 March 2017 (UTC)[reply]

For example: Oswald Giering Vorlesungen über höhere Geometrie, Springer-Verlag,p. 61.

Is a conic section a quadric ? In German text-books there are pairs of lines (planes in space) quadrics. Is there a cultural gap ?--Ag2gaeh (talk) 19:15, 2 March 2017 (UTC)[reply]

The English definition of a quadric NOW does not comply with all (?) other WIKIs. And there is a contradiction: The definition excludes pairs of planes, the list of quadrics below contains such objects. There are two major subclasses of quadrics: nondegenerate and degenerate. Both are quadrics. --Ag2gaeh (talk) 09:32, 3 March 2017 (UTC)[reply]

References

^ Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042

The numbers don't add up

The section Quadric#Euclidean space starts out by giving some equations and permissible epsilon values, for a total of 17 possibilities. Then it mentions "these 17 normal forms", and mentions 7 of them that are imaginary only, single-real-point, or otherwise trivial. That leaves 10 remaining. Then the next paragraph begins "Thus, there are nine true quadrics", and specifies these as 4 degenerate quadrics and 5 non-degenerate ones, "which are detailed in the following table". Then a paragraph says "there are 17 such normal forms. Of these 16 forms, ...". Then a table is given, consisting of a non-degenerate section containing 10 and a degenerate section containing 6.

I hope that someone familiar with these can clarify how these different passages relate to each other numerically, and which ones in one passage coincide with which ones in each other passage. Loraof (talk) 23:25, 17 October 2017 (UTC)[reply]

I have added the lacking case (

\varepsilon _{3}=0

). I have also replaced the last paragraph before the table by a summary of the classification. In fact, this paragraph classified implicitly the imaginary ellipsoid among the degenerate quadrics, which contradicts the definition of "degenerate" given in Degeneracy (mathematics). Also, it is unclear whether the decomposed quadrics should be, or not, included in the degenerated one. It is therefore better to have a formulation that avoids having to decide.

About the table: If one omits the "special cases", the table gives 5 non-degenerate and 3 degenerate quadrics. It could be less confusing to move the special cases in one or two specific sections of the table. D.Lazard (talk) 10:04, 18 October 2017 (UTC)[reply]

Great job – thanks! Loraof (talk) 14:59, 18 October 2017 (UTC)[reply]

Color Mismatch

The colors in the image in the Euclidean Plane section are mismatched from the description of the image. 209.6.73.123 (talk) 22:49, 4 September 2018 (UTC)[reply]

It is fixed now. Thank you for pointing this out.—Anita5192 (talk) 23:28, 4 September 2018 (UTC)[reply]

Perhaps the color "goldenrod" is better:

\color {goldenrod}{abc}

. Please try or look at Help:Displaying a formula. --Ag2gaeh (talk) 07:24, 5 September 2018 (UTC)[reply]

Goldenrod is less difficult to see, but still difficult. Actually I don’t think the image needs a color-coordinated caption. If the colors are completely removed from the caption, the specific eccentricities clarify the correspondence. I suggest we make the caption text entirely black.—Anita5192 (talk) 16:05, 5 September 2018 (UTC)[reply]

Revision of section "Projective quadrics over fields"

I tried to improve section Projective quadrics over fields. If there is no objection, I shall remove the tag on "multiple issues".--Ag2gaeh (talk) 09:56, 2 March 2019 (UTC)[reply]

Grammar and/or Spelling

In the current version of the article I read the sentence: "This define affine quadrics." Either grammatically or spelling-wise there seems to be an error in this sentence, which obscures its meaning to me. What did the author intend to write?

Was it "This defines affine quadrics."?

Or was it "These define affine quadrics."?

Or anything else?Redav (talk) 09:14, 26 April 2020 (UTC)[reply]

The first interpretation is the correct one. However, as "This" is ambiguous, I have moved the definition at the beginning of the section, and added explanations for clarifying the end of the section. D.Lazard (talk) 09:48, 26 April 2020 (UTC)[reply]

Rational Parametrization

I'm not 100% sure that the recent efforts qualify as an improvement yet, but thank you for the work ... let's keep at it! Some questions:

I have no problem assuming that $t_{0}=1$ for points not at infinity, but why is it okay to assume $t_{n}=1$ ? That is, won't that miss some not-at-infinity points of the quadric if we disallow $T_{n}=0$ ? (Alternatively, if we somehow know that $T_{n}=0$ always gives something that is tangent to the quadric at $A$ , let's explain that.)
"This pametrization establishes a bijection between a projective conic section and the projective line" -- Each point of the quadric (not in the tangent hyperplane of $A$ ) is one-to-one with a line through the origin, e.g., a point in projective space, right?
Is "projective conic section" the same as "quadric"? If so, let's consistently use the latter.
In Quadric § Intersection with a line, cases b and c are not both called "tangent" but they are in this newly rewritten section. Let's say something about this difference of usage, or otherwise make things more consistent.
Do you have literature that refers to this as "rational parametrization"? Given that it is a mapping from (most of) the quadric to a projective space, the natural name for it (to me) is a "projection".

— $Q$ uantling (talk | contribs) 00:47, 28 November 2022 (UTC)[reply]

There is no $t_{0}$ and no $T_{0}$ Howeverm I have added some clarification. Normally, this clarification should belong to another article, such as Parametric equation (not explained there) or Rational variety (explained there, in a much too technical way). This is explained in a special case in Möbius transformation.
This is a bijection between the points of the projective conic section and the points of the projective line. The fact that the points of projective spaces are often defined as vector lines has nothing to to here.
A quadric may have any dimension. A conic section is a quadric of dimension one.
The last sentence of § Intersection with a line contradicts the beginning of the section. So, this is this section that must be edited. Neverttheless I have added a clarification.
"Rational parametrization" is a standard terminology that is used in several Wikipedia articles. So, I have created the redirect Rational parametrization which redirects to the first occurence of the phrase in Rational variety. Also "rational parametric equation" occurs in Parametric equation. So, the target of the new redirect could be changed to Parametric equation after having edited this article. I have linked the section to this redirect.

D.Lazard (talk) 10:55, 28 November 2022 (UTC)[reply]

@D.Lazard: thank you for your replies and article edits. Regarding Question 1, elsewhere in the article we number the non-projective dimensions as

1

to

n

and then use the index of

0

for the additional dimension used in projective geometry, whose value is typically

1

for points not at infinity and is

0

for points at infinity. I admit to confusion for the present case ... is the use of index

n

in the present section different from this use of

0

elsewhere ... or should it be changed to

0

for consistency with the rest of the article? I apologize for not quite following this well enough that I can answer my own question.

Regarding Question 2, with "This pametrization establishes a bijection between a projective conic section and the projective line", might we instead say more generally that "This pametrization establishes a bijection between a quadric and a projective space"? Also, what should we do about the caveat that the points in the tangent hyperplane are excluded? —

Q

uantling (talk | contribs) 17:29, 28 November 2022 (UTC)[reply]

Point 1: the

t_{i}

are the coordinates of a direction vector in the space that contains the quadric. So the

t_{i}

must be indexed in the same way as the space coordinates

x_{i}.

As the directions form a projective space of dimension

n-1,

one must equal one of the

t_{i}

to 1 for having affine parameters. It is

t_{n}

that has been equated to 1, since otherwise one would not have an indexing by the first positive integers.

Point 2:Your second question is the answer to the first one: in general, the parametrization does not cover the whole quadric; that is, the points of the intersection of the quadric and its tangent hyperplane at

A

are not parametrized. However, I have fixed an error: not every point of the projective space ot the parameters gives a point of the quadric: the directions contained in the tangent hyperplane at

A

do not give any point of the quadric. D.Lazard (talk) 10:39, 29 November 2022 (UTC)[reply]

I have added examples as a new subsection. This should be a clarification for many readers. D.Lazard (talk) 16:52, 29 November 2022 (UTC)[reply]

Thank you, this is coming along nicely. Another question (to demonstrate my ignorance): if

A

is a non-singular point of the quadric, is it impossible that the line

a + λt

is completely contained within the quadric? —

Q

uantling (talk | contribs) 23:33, 29 November 2022 (UTC)[reply]

Consider the hyperbolic paraboloid

z=xy

and

A=(0,0,0).

the two lines

(\lambda ,0,0)

and

(0,\lambda ,0)

are contained in the quadric. More generally, for every point of a hyperbolic paraboloid or a one-sheet hyperboloid, there are there are two line passing through the point and contained in the quadric. The same is true for every non-degenerate quadric surface, if one considers nonreal complex points. D.Lazard (talk) 09:33, 30 November 2022 (UTC)[reply]

Heronian triangles

Quantling has recently added the following paragraph in section § Rational points:

A Heronian triangle is a triangle whose side lengths and area are integers. The interior angles of a Heronian triangle have half-angle tangents, $x_{1},$ $x_{2},$ and $x_{3},$ that are rational and satisfy the quadratic relationship $x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3}=1.$ Thus rational parametrization can be used to generate all such triples of rational tangents. These in turn give rational side lengths ${\frac {2x_{1}}{1+x_{1}^{2}}}\,,$ ${\frac {2x_{2}}{1+x_{2}^{2}}}\,,$ and ${\frac {2x_{3}}{1+x_{3}^{2}}}$ and a rational area ${\frac {4x_{1}x_{2}x_{3}}{(1+x_{1}^{2})(1+x_{2}^{2})(1+x_{3}^{2})}}$ for a triangle with those half-angle tangents. This triangle can be scaled to a similar triangle whose side lengths and area are integers.

I'll remove this paragraph because of the following issues, which, all together, make that this paragraph does not belong to this section.

The section is about rational points of quadrics, and the primitive Heronian triangles are in one to one correspondence with the rational points of the quartic obtained by squaring Heron's formula. As a quartic is not a quadric, the relationship between Heronian triangles and rational points (the subject of this section) does not belong to this article.
The assertions contained in the paragraph are true, but not evident (I had to think about a while for finding a proof). So they must be sourced, proved, linked, or easy to find in Heron triangle. This is not the case, and the paragraph does follows the policy of WP:Verifiability.
It is not said that the relation $x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3}=1$ is satisfied by half-angle tangents of every triangle.
Parametrizing this quadric does not requires the method described in this article, as the parametrization ${\textstyle x_{1}=t,x_{2}=u,x_{3}={\frac {uv}{u+v}}}$ is evident.
Several rational parametrizations of the "Heronian quartic" are given in Heronian triangle, but it is unclear whether any of them results from a paramatrization of the quadric $x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3}=1.$ So this way of parametrizing heronian triangles seems WP:OR.

D.Lazard (talk) 15:49, 2 December 2022 (UTC)[reply]

I agree that if it isn't WP:CALC then an editor can request that there be a citation. Go ahead and remove it if you wish. It's been a while since I've had my algebraic varieties class, so odds are long that I will find a Wikipedia-quality citation, but I'll keep my eyes open. (Yes, we could clarify that

x_{1}x_{2}+x_{2}x_{3}+x_{1}x_{3}=1

is satisfied by every triangle; it leads to a Heronian triangles if and only if each of the variables is positive and rational. Yes, there is a way to frame Heronian triangles as using a quartic equation, but that doesn't make the quadratic approach any less valuable. Yes, there are other ways to solve this quadratic, but there are other ways to solve the circle too; e.g.,

x_{1}=u,

x_{2}=\pm {\sqrt {1-u^{2}}}

) —

Q

uantling (talk | contribs) 16:21, 2 December 2022 (UTC)[reply]

I hope someone adds information about the standard quadric

The standard quadric in complex (n+1)-space C^(n+1) is defined as the set of points (z_0, ..., z_n) in C^(n+1) such that z_0^2 + ... + z_n^2 = 0.

Likewise since z_0^2 + ... + z_n^2 is a homogeneous polynomial, the standard quadric may alternatively be seen as a complex projective hypersurface in complex projective (n+1)-space CP^(n+1).

This is an important construction in mathematics and should not be omitted from the article. 2601:200:C082:2EA0:50AA:AEB:9A5D:EB80 (talk) 03:41, 18 February 2023 (UTC)[reply]

For adding such information, one requires

A reliable source attesting that "standard quadric" is a common name forthis concept (Looking on Scholar Google, I did not find any source using "standard quadric" in this sense)
Notable properties of these hypersurfaces that deserve to be added in this article

As far as I know, the main property of what you call "standard quadric" is already in the article, stated as In complex projective space all of the nondegenerate quadrics become indistinguishable from each other. This means that, up to complex projective transformations, the standard quadrics are exactly the nondegenerate quadrics D.Lazard (talk) 10:39, 18 February 2023 (UTC)[reply]

[1] Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042

[1]