Talk:Matrix representation of conic sections

message: 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)
thread_title: ??

message: 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)
thread_title:  Need Help 

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

message: 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)
replies: text: I have changed the remaining instances to match. —Mark Dominus (talk) 14:30, 31 January 2011 (UTC)
replies: text: Both notations for determinant are totally standard.  Duoduoduo (talk) 01:55, 20 February 2011 (UTC)
thread_title:  Determinant/Absolute value 

message: 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 <a11, a12, a13>?  If so, this abbreviation does conflicts with the recent change of notation from <a11, a12, a13> to <A, B/2, D/2>.  Or is it supposed to be the eigenvector a1 from the "Axes" section? —Mark Dominus (talk) 14:53, 31 January 2011 (UTC)
thread_title:  What is \vec a_1? 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title: Point ellipse

message: 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)
thread_title:  Vertices section confusing 

message: 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)
thread_title:  Reduced Equation Section confusing 

message: 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)
thread_title:  Classification of Degenerate Conics 

message: 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)
replies: text: 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)
thread_title:  Confusing notation 

message: 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)
thread_title:  Foci 

message: 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)
thread_title:  Examples? 

message: 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)
thread_title:  Confused about this part in Axes