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Transformation of a quadric/dual quadric

The transformation equations for quadrics and dual quadrics under a homography ${\bf T}$ can be obtained in a similar way to Section 2.1.3. Using equations (2.8) and (2.9) the following is obtained
$\displaystyle {\tt M}'^\top {\bf Q}' {\tt M}'$ $\textstyle \sim$ $\displaystyle {\tt M}^\top {\bf T}^\top {\bf T}^{-\top} {\bf Q} {\bf T}^{-1} {\bf T} {\tt M}
= 0$  
$\displaystyle {\tt\Pi}'^\top {{\bf Q}^*}' {\tt\Pi}'$ $\textstyle \sim$ $\displaystyle {\tt\Pi}^\top {\bf T}^{-1} {\bf T} {\bf Q}^* {\bf T}^\top {\bf T}^{-\top} {\tt\Pi}
= 0$  

and thus
$\displaystyle {\bf Q}$ $\textstyle \mapsto$ $\displaystyle {\bf Q}' \sim {\bf T}^{-\top} {\bf Q} {\bf T}^{-1}$ (B19)
$\displaystyle {\bf Q}^*$ $\textstyle \mapsto$ $\displaystyle {{\bf Q}^*}' \sim {\bf T} {\bf Q}^* {\bf T}^\top$ (B20)

Observe again that $({\bf Q}')^*=({\bf Q}^*)'$.



Marc Pollefeys 2002-11-22