Invertible quadratic transformations in a projective plane

Abstract

Consider in the projective plane a curve $k$ of second order and a point $P$, which is nonsingular for $k$.

To an arbitrary point $M$, different from $P$, let correspond its polar-conjugate point $M^{\prime}$ with respect to $k$, which lies on the line $MP$. We call this transformation a generalized inversion. Also, we call quadratic transformation of the projective plane $\pi$ a map of $\pi$ into $\pi$ if the coordinates of the image $M^{\prime}$ of $M$ are homogeneous functions of second order of the coordinates of $M$. We prove that any invertible quadratic transformations $S$ allows the presentation $S=\pi_{2}J\pi_{1}$, where $\pi_{1}$ and $\pi_{2}$ are linear transformations and $J$ is a generalized inversion.