Homography and Conics

A conic is transformed through a homographic transformation $\mathbf{x}' = \mathbf{H} \mathbf{x}$ into a conic. Indeed, it follows that

$$\mathbf{x}^{\top} \mathbf{C} \mathbf{x} = \mathbf{x}'^{\top} \mathbf{H}^{-\top} \mathbf{C} \mathbf{H}^{-1} \mathbf{x}'$$ (1.85)

which is still a quadratic form $\mathbf{C}' \equiv \mathbf{H}^{-\top} \mathbf{C} \mathbf{H}^{-1}$. The use of the equivalence symbol $\equiv$ is necessary since the conic is defined up to a multiplicative factor.

This notable result allows us to demonstrate that a conic viewed in perspective is still a conic.