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 2.2.3. Using equations (2.8) and (2.9) the following is obtained

$${\tt M}'^\top {\bf Q}' {\tt M}' \sim {\tt M}^\top {\bf T}^\top {\bf T}^{-\top} {\bf Q} {\bf T}^{-1} {\bf T} {\tt M} = 0$$

$${\tt\Pi}'^\top {{\bf Q}^*}' {\tt\Pi}' \sim {\tt\Pi}^\top {\bf T}^{-1} {\bf T} {\bf Q}^* {\bf T}^\top {\bf T}^{-\top} {\tt\Pi} = 0$$

and thus

$${\bf Q} \mapsto {\bf Q}' \sim {\bf T}^{-\top} {\bf Q} {\bf T}^{-1}$$ (B19)

$${\bf Q}^* \mapsto {{\bf Q}^*}' \sim {\bf T} {\bf Q}^* {\bf T}^\top$$ (B20)

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