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Tangent to a conic

The two intersection points of a line with a conic coincide if the discriminant of equation (2.12) is zero. This can be written as

\begin{displaymath}
S({\tt m},{\tt m}')-S({\tt m})S({\tt m}')=0 \enspace .
\end{displaymath}

If the point ${\tt m}$ is considered fixed, this forms a quadratic equation in the coordinates of ${\tt m}'$ which represents the two tangents from ${\tt m}$ to the conic. If ${\tt m}$ belongs to the conic, $S({\tt m})=0$ and the equation of the tangents becomes

\begin{displaymath}
S({\tt m},{\tt m}')= {\tt m}^\top {\bf C} {\tt m}' = 0 \enspace ,
\end{displaymath}

which is linear in the coefficients of ${\tt m}'$. This means that there is only one tangent to the conic at a point of the conic. This tangent ${\tt l}$ is thus represented by :
\begin{displaymath}
{\tt l} \sim {\bf C}^\top {\tt m} = {\bf C} {\tt m}
\end{displaymath} (B13)



Marc Pollefeys 2002-11-22