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 2.2.3. Using equations (2.5) and (2.7) the following is obtained:

$${\tt m}'^\top {\bf C}' {\tt m}' \sim {\tt m}^\top {\bf H}^\top {\bf H}^{-\top} {\bf C} {\bf H}^{-1} {\bf H} {\tt m} = 0 \, ,$$

$${\tt l}'^\top {{\bf C}^*}' {\tt l}' \sim {\tt l}^\top {\bf H}^{-1} {\bf H} {\bf C}^* {\bf H}^\top {\bf H}^{-\top} {\tt l} = 0 \, ,$$

and thus

$${\bf C} \mapsto {\bf C}' \sim {\bf H}^{-\top} {\bf C} {\bf H}^{-1}$$ (B14)

$${\bf C}^* \mapsto {{\bf C}^*}' \sim {\bf H} {\bf C}^* {\bf H}^\top$$ (B15)

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