Intersection between conic and line in homogeneous space In homogeneous space (so 3 coordinates for each point) I have:


*

*A conic C, defined by a symmetric 3x3 matrix of real values. The conic actually should have only imaginary points (don't know if this is important).

*A line l, defined by a vector of 3 real values


How do I find the intersection of the two?
I think I shoud be able to find the intersection (I expect two complex solutions), but I'm having troubles doing it.
Solving it with the classic pen&paper leads me to a solution X=(x, y, 1) with x and y complex such that, when I try to verify that the point belongs to C (by cheching if XCX' = 0, where the ' stands for transposed), it seems that it does NOT belong to C.
For those who know computer vision stuff: C is actually the image of the absolute conic, estimated from a picture, while l is a vanishing line of a plane. I'm trying to intersect the two in order to find the circular points, and then do a metric rectification of the plane in the image.
 A: The matrix
$$L=\hat l=\begin{pmatrix}0&-l_3&l_2\\l_3&0&-l_1\\-l_2&l_1&0\end{pmatrix}$$
can be used to describe a cross product with $l$: $Lg=l\times g$.
Now consider $D = L^T\cdot C\cdot L$. It is a degenerate conic which you best interpret dually as a pair of points, namely the points of intersection. A line $g$ is tangent to that conic if its intersection with $l$, which can be computed as $l\times g=Lg$, lies on $C$, i.e. if $(l\times g)^TC(l\times g)=0$.
All you have to do is decompose that conic $D$ into its components. You already have $\operatorname{rank}(D)=2$. Now consider $P=D+\lambda L$. For some suitable $\lambda$, this matrix will have rank $1$. Simply look at any $2\times 2$ subdeterminant, and choose $\lambda$ in such a way that it becomes zero. You will have two possible choices which only differ by sign. Either one will do.
Once you have this, $P=pq^T$ is a matrix of rank $1$. You can choose any non-zero column and call that $p$, and any non-zero row will be $q$ (up to scalar multiples). So I'd look for the greatest absolute value in that matrix and choose its row and column as the two points of intersection.
