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Let $P(V)$ be a projective space, and $V$ its underlying $n$-dimensional $\mathbb{K}$-vector space. I know that the algebraic equation of a conic section in $V$ is of the general form:

$x^TAx+2Bx+c=0$, where $x$ is a $n$-dimensional column vector, $A$ a $n \times n$-matrix of a quadratic form $q$, $B$ a $1 \times n$- matrix of a linear map $g$, and $c \in \mathbb{K}$.

How does this give you the equation of a projective conic section $Q=\lbrace $vect$ \lbrace \vec{v} \rbrace \in P(V)|q(\vec{v})=0 \rbrace$ associated with $q$?

Note: vect$\lbrace\vec{v} \rbrace$ is the $1$-dimensional subspace of $V$ generated by $\vec{v}$

1 Answers1

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Conventionally you homogenize by setting the last coordinate equal to $1$. So in the projective plane, the inhomogeneous equation,

$$(x,y)\cdot A\cdot\begin{pmatrix}x\\y\end{pmatrix}+ 2\cdot B\cdot\begin{pmatrix}x\\y\end{pmatrix} + c=0$$

would translate to its homogeneous counterpart

$$(x,y,1)\cdot\left(\begin{array}{ccc|c} &&&\\ &A&&B^T\\ &&&\\\hline &B&&c \end{array}\right)\cdot\begin{pmatrix}x\\y\\1\end{pmatrix}=0$$

MvG
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