Invertible quadratic transformations in a projective plane (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 1996)
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 $M^{′}$ 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 $π$ a map of $π$ into $π$ if the coordinates of the image $M^{′}$ of $M$ are homogeneous functions of second order of the coordinates of $M$. We prove that any invertible quadratic transformations $S$ allows the presentation $S=π_{2}Jπ_{1}$, where $π_{1}$ and $π_{2}$ are linear transformations and $J$ is a generalized inversion.