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I just came across the document "Calculating with Conics" (PDF) (Chapter 11 of a larger work) and I became interested with the method presented here, on how to solve the coordinates of the point of intersection of two conics using their matrix representation (Section 11.4).

However, I am confused about what really happens in the part where the values $\alpha$, $\beta$, $\gamma$, and $\delta$ are to be obtained, and I don't know if they are single numbers or matrices.

From page 191:

$$\begin{align} \alpha &= [A_1, A_2, A_3] \\ \beta &= [A_1, A_2, B_3]+[A_1, B_2, A_3]+[B_1, A_2, A_3] \\ \gamma &= [A_1, B_2, B_3]+[B_1, A_2, B_3]+[B_1, B_2, A_3] \\ \delta &= [B_1, B_2, B_3]\end{align}$$ where the $A_i$ and $B_i$ are the column vectors of matrices $A$ and $B$ representing two conics.

This is not a homework. I am just a high school student interested in other ways to solve systems of conics.

Can someone simply explain to me how to get the 4 values from the matrix representation of the conics?

Blue
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Pahi Raminem Ail
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  • Link of the document: https://www-m10.ma.tum.de/foswiki/pub/Lehre/WS0809/GeometrieKalkueleWS0809/ch11.pdf – Pahi Raminem Ail Sep 13 '19 at 11:27
  • I've added information from the document to provide some context. Please re-edit if I have inadvertently altered the intent of your question, but also to provide even more clarity about the source of your confusion. (Have you checked earlier chapters to see how the notation has been defined?) – Blue Sep 13 '19 at 13:12
  • That’s the problem with grabbing an isolated chapter of a much longer work—it might use notation that was defined elsewhere in the book. The bracket notation here is introduced back in chapter 4 (“Lines And Cross-Ratios”). It stands for the determinant of the matrix with those columns. You will also need to know some basic properties of determinants. – amd Sep 13 '19 at 19:32
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    MvG gives a nice summary of this algorithm in this answer. You might want to have a look at it, too, as well as the things he links to in that answer. – amd Sep 13 '19 at 19:37
  • So, knowing that these bracketed terms are determinants, can you follow this derivation on your own or do you still need details. – amd Sep 13 '19 at 20:14
  • Ohh, I should have checked the last chapters. It's okay, I can follow. Thanks a lot to all of you! – Pahi Raminem Ail Sep 13 '19 at 23:42

1 Answers1

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This bracket notation is introduced in Chapter 4 of the book. It denotes the determinant of the matrix with columns given by the expressions within the brackets.

amd
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