Talk:Matrix representation of conic sections

??
My question is this,

"what do the eigenlines of a conic's assosiated matrix correspond to, and why?"

The ellipse 17x^2 + 12xy +8y^2 = 1

may be assosiated with (x,y) (17 6) (x) (6 8) (y)

having eigenvectors a(1,-2) and b (2,1).

My understanding of eigenvectors are that they are vectors that are not changed when transformed by the matrix.

I'm guessing there's a relationship between the focus and directrix of a conic and its eigenlines. But I can't seem to find out what that relationship is anywhere.

I'm not good enough at sums to figure it out for myself, so if you can explain this I'd be most pleased.

The eigenvectors provide the axis of the conic, or the ellipse in your case. What I find puzzling is that the matrix associated with the conic is 3x3, so you actually get 3 eigenvectors. Presumably you take the two with the largest eigenvalues or something similar. This part of the page is not very clear. Aether1999 13:11, 6 April 2007 (UTC)

My answer to you is: I believe that he is obaining the eigenvalues of the A11 matrix, not the original A matrix. I am trying to work the exact values of the center, vertices, Axes. —Preceding unsigned comment added by 68.9.67.188 (talk) 00:22, 12 May 2008 (UTC)

Need Help
I need help phrasing a certain part of the page. Under the section Axes it says: eigenvector of A, when it should say: eigenvector of A11. I was going to fix it myself but, what I thought would work didn't and I can't figure out a way to make it work. Please help.--SurrealWarrior 01:32, 20 June 2007 (UTC)

I understand what you are saying here. The article indicates that eigenvectors give the prinicpal axes. However, the equations that the eigenvectors gives, first of all, are not perpendicular, and second, are 2 lines through the origin. So, they can't be the axes. I will work some more on it. Input is appreciated. —Preceding unsigned comment added by 158.123.55.4 (talk) 13:10, 14 May 2008 (UTC)

Nonstandard notation
Wry does the author not use standard notation Ax^2 +Bxy + ...??

It is VERY hard to -use these formulas if you are using Standard reoresentation

Thanks RJ Pease —Preceding unsigned comment added by 75.166.110.225 (talk • contribs)

Examples and explanations
This article needs examples and explanations. —Preceding unsigned comment added by 188.26.60.129 (talk) 02:42, 25 May 2010 (UTC)

Determinant/Absolute value
Throughout the article the notation for absolute value is abused and used instead the one for determinant. I personally think the accepted notation for the determinant of a matrix A is det A, not |A|. 85.187.35.160 (talk) 13:34, 31 January 2011 (UTC)
 * I have changed the remaining instances to match. —Mark Dominus (talk) 14:30, 31 January 2011 (UTC)


 * Both notations for determinant are totally standard. Duoduoduo (talk) 01:55, 20 February 2011 (UTC)

What is \vec a_1?
In the "reduced equation" section of the article, the following equation appears:



T: RS(O,X,Y) \mapsto (O'=S,X',Y') \ \stackrel{\mathrm{def}}{=}\ \left\{\begin{align} \vec t &= \vec OO' = S\\ \alpha &= \operatorname{arccos} \frac{\vec a_1 \cdot {1 \choose 0}}{|\vec a_1|} \end{align} \right. $$

The expression $$\vec a_1$$ is not defined, and I cannot understand what it means. Is it intended to be the vector &lt;a11, a12, a13&gt;? If so, this abbreviation does conflicts with the recent change of notation from &lt;a11, a12, a13&gt; to &lt;A, B/2, D/2&gt;. Or is it supposed to be the eigenvector a1 from the "Axes" section? —Mark Dominus (talk) 14:53, 31 January 2011 (UTC)

And what the heck is $${1 \choose 0}$$ doing in there? $${1 \choose 0}$$=1, so why not omit it? —Mark Dominus (talk) 14:53, 31 January 2011 (UTC)

Oh, now I see. $${1 \choose 0}$$ is actually &lt;1, 0&gt; and the dot there is a scalar product. There must be a clearer way to write this. —Mark Dominus (talk) 15:00, 31 January 2011 (UTC)

Merge into here?
I disagree with the suggestion to merge Conic_section into this article. That section of Conic section is not even about the matrix representation (even though at one point it does mention a determinant as motivation for a lengthy algebraic expression). The section basically gives the conditions for the quadratic conic expression to give an ellipse, degenerate ellipse, parabola, etc. That information cannot possibly be omitted from an article on conic sections.

Or maybe the intention was to put the merge tag onto the subsequent section, Conic_section, instead of the discriminant section. But I would oppose that too — the matrix notation section is very short and the brief info in it should be there. However, I will put a Main-article-link at the top of that section to this article. Duoduoduo (talk) 21:03, 3 February 2011 (UTC)

It is a really good idea. Jackzhp (talk) 16:24, 13 February 2011 (UTC)


 * I strongly oppose merging for the following reason: Many high-school students need to refer to the conic section article, including its section on discriminant classification. If they had to wade through this article to find that discussion, they would find a great deal of material on matrices that is beyond what they have learned, and which will therefore be quite intimidating to most students.  Learning conic sections is rather hard for high-school students and it would be a bad idea to introduce further obstacles.


 * On the other hand, if including discriminant classification into this article seems appropriate, there is no reason that an abbreviated version of the corresponding section of the conic section article could not be included in this one. (Just as a very abbreviated and elementary mention of this article is included in the conic section article -- headed by the phrase: "Main article: Matrix representation of conic sections".)Daqu (talk) 20:19, 27 July 2011 (UTC)

Point ellipse
I am changing the assertion that a degenerate ellipse is the empty set -- actually it is a point ellipse. Here's an example: $$x^2+xy+y^2+2x+2y+\frac{4}{3}=0$$ meets the two criteria for a degenerate ellipse; it has the single real solution $$x=\frac{-2}{3}=y.$$ Duoduoduo (talk) 19:06, 19 February 2011 (UTC)


 * I think that both are correct. In general, isometric transformations will place the ellipse so that its center at the origin and its axes on the coordinate axes; the ellipse then has the equation (x/a)² + (y/b)² = c.  If c is zero, the solution is a single point; if c is negative, there is no solution.  If a=b, the ellipse is circular, and the degenerate cases abovr correspond to a circle of zero radius and to a circle of negative radius, respectively.  I think that the classification in article is incorrect in these cases.  I will cross-check more carefully later.  —Mark Dominus (talk) 20:01, 19 February 2011 (UTC)


 * If c is negative, then there are an infinitude of solutions, all imaginary. This is not called a degenerate case -- it's called an imaginary ellipse. The classifications in the article are standard; in particular, it's standard that a degenerate ellipse is a point ellipse. I'll put in the condition for an imaginary ellipse soon, along with references.  Duoduoduo (talk) 01:27, 20 February 2011 (UTC)

Vertices section confusing
In the Vertices section, it describes in text that the intersection of the conic's axes with itself are the vertices. Then there is an equation that says V = {e, Q}. I don't see 'e' appear anywhere in the section above (about axes), and I don't actually understand how this expression of V = {two equations} is actually helpful (or correct)? Perhaps an actual expression for the computation of the vertices should go here?

daviddoria (talk) 19:17, 27 February 2013 (UTC)

Reduced Equation Section confusing
I see that if you divide det(A_Q)/det(A_{33}) by itself, you will get the 1 (after moving it to the right hand side of the equation), but I don't see how \lambda_1 x'^2 divided by det(A_Q)/det(A_{33}) produces x'^2/a^2 ?

Also, in the equation for the transformation coordinates, \vec{a_1} is used, but I don't see it defined anywhere?

daviddoria (talk) 19:22, 27 February 2013 (UTC)

Classification of Degenerate Conics
I think there is an error in the Classification section for degernerate conics when det(AQ)=det(A33)=0. I think that the discriminant used in this case should actually be D2+E2−4(A+C)F instead of E2−4CF. The current formula is not symmetric with respect to x and y and so cannot be correct. A counter example is x2=1 which consists of two parallel lines at x=±1 but which the stated criterion implies should be a single line. — Preceding unsigned comment added by Pantherspride (talk • contribs) 19:33, 27 March 2013 (UTC)

Confusing notation
Expanding on the comment of daviddoria on 27 Feb 2013 above, I can't make heads or tails out of the notation in the sections "Center", "Axes", and "Vertices". Could someone look at it and either explain it there or correct it? Thanks! Loraof (talk) 20:12, 10 May 2016 (UTC)

I've removed the puzzling and pointless notation from the sections "Center" and "Vertices". The section "Axes" still needs work.


 * Sorry that I didn't get to this sooner. I'll be happy to help. It seems that the Ayoub paper was used heavily here, but the confusing notation was not part of that paper. Bill Cherowitzo (talk) 00:07, 12 May 2016 (UTC)

Foci
The lead says that the matrix representation makes it easy to find the foci, but I don't think the article says anything about that. Can someone put something in about that? Thanks. Loraof (talk) 04:02, 13 May 2016 (UTC)

Examples?
I think the work on this page is going well. I intend on adding a section on parabolas and reworking the section on tangents. I am wondering if we should add a worked example to the page. There are several worked examples in the citations that we could use (but for some reason they all turn out to be ellipses!) and the article does seem to be a little short on how this material can be used in practice. Bill Cherowitzo (talk) 18:18, 16 May 2016 (UTC)

Confused about this part in Axes
Says here: "The eigenvector having the smallest eigenvalue (in absolute value) corresponds to the major axis.[11]" I assume from the context that this means something like "the eigenvector determined from the eigenvalue of the matrix with the smallest absolute value corresponds to the major axis". The problem is, this part doesn't seem to be correct!

Consider the hyperbola $$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$$ with $$A=-4,B=-6,C=4,D=14,E=-2,F=0$$, the last three determined by it passing thru origin and its central point being $$(1,1)$$. Calculating the eigenvalues of the $$A_{33}$$ matrix, I get $$5$$ and $$-5$$. Their absolute value's are equal, so which eigenvector corresponds to the axis with the foci? And if I set $$C=4\pm0.2$$ this moves the foci very little but flips the eigenvector from one axis to the other. What am I doing wrong? 85.48.66.168 (talk) 19:47, 10 May 2022 (UTC)