Homography and Conics

Since a homographic transformation is invertible, it also preserves the degeneracy of a conic. Therefore, a nondegenerate conic $\mathbf{x}' \sim \mathbf{H} \mathbf{x}$ is transformed into a nondegenerate conic.

Indeed, consider a conic represented by the symmetric matrix $\mathbf{C}$ and described by the equation

$$\mathbf{x}^{\top} \mathbf{C} \mathbf{x} = 0,$$ (1.124)

Substituting $\mathbf{x} \sim \mathbf{H}^{-1}\mathbf{x}'$ yields

$$\mathbf{x}'^{\top} \mathbf{H}^{-\top} \mathbf{C} \mathbf{H}^{-1} \mathbf{x}' = 0,$$ (1.125)

which is still a quadratic form

$$\mathbf{C}' \equiv \mathbf{H}^{-\top} \mathbf{C} \mathbf{H}^{-1}.$$ (1.126)

The equivalence symbol $\equiv$ is necessary because the conic is known up to a multiplicative factor.

This notable result makes it possible to prove that a conic viewed in perspective is still a conic.