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Line-conic intersection

Let ${\tt m}$ and ${\tt m'}$ be two points defining a line. A point on this line can then be represented by ${\tt m} + \lambda {\tt m'}$. This point lies on a conic $S$ if and only if

\begin{displaymath}
S({\tt m}+\lambda {\tt m'}) = 0 \, ,
\end{displaymath}

which can also be written as
\begin{displaymath}
S({\tt m})+2\lambda S({\tt m},{\tt m}') + \lambda^2 S({\tt m}') =0 \, ,
\end{displaymath} (B12)

where

\begin{displaymath}
S({\tt m},{\tt m}') = {\tt m}^\top {\bf C} {\tt m}' =
S({\tt m}',{\tt m})
\end{displaymath}

This means that a line has in general two intersection points with a conic. These intersection points can be real or complex and can be obtained by solving equation (2.12).



Marc Pollefeys 2002-11-22