Pencils of Frégier Conics

Abstract

For each point P on a conic c, the involution of right angles at P induces an elliptic involution on c whose center F is called the Frégier point of P. Replacing the right angles at P between assigned pairs of lines with an arbitrary angle phi yields a projective mapping of lines in the pencil about P, and thus, on c. The lines joining corresponding points on c do no longer pass through a single point and envelop a conic f which can be seen as the generalization of the Frégier point and shall be called a generalized Frégier conic. By varying the angle, we obtain a pencil of generalized Frégier conics which is a pencil of the third kind. We shall study the thus defined conics and discover, among other objects, general Poncelet triangle families.