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Transformation of a conic/dual conic

The transformation equations for conics and dual conics under a homography ${\bf H}$ can be obtained in a similar way to Section 2.1.3. Using equations (2.5) and (2.7) the following is obtained:
$\displaystyle {\tt m}'^\top {\bf C}' {\tt m}'$ $\textstyle \sim$ $\displaystyle {\tt m}^\top {\bf H}^\top {\bf H}^{-\top} {\bf C} {\bf H}^{-1} {\bf H} {\tt m}
= 0 \, ,$  
$\displaystyle {\tt l}'^\top {{\bf C}^*}' {\tt l}'$ $\textstyle \sim$ $\displaystyle {\tt l}^\top {\bf H}^{-1} {\bf H} {\bf C}^* {\bf H}^\top {\bf H}^{-\top} {\tt l}
= 0 \, ,$  

and thus
$\displaystyle {\bf C}$ $\textstyle \mapsto$ $\displaystyle {\bf C}' \sim {\bf H}^{-\top} {\bf C} {\bf H}^{-1}$ (B14)
$\displaystyle {\bf C}^*$ $\textstyle \mapsto$ $\displaystyle {{\bf C}^*}' \sim {\bf H} {\bf C}^* {\bf H}^\top$ (B15)

Observe that (2.14) and (2.15) also imply that $({\bf C}')^*=({\bf C}^*)'$.

Marc Pollefeys 2002-11-22