Journal for Geometry and Graphics 28 (2024), No. 1, 055--072
Copyright by the authors licensed under CC BY SA 4.0
Double Contact Conics in Involution
George Lefkaditis
Patras University, Patras, Greece, Patras, Greece
glef@upatras.gr
Anastasia Taouktsoglou
Democritus University of Thrace, Xanthi, Greece, Xanthi, Greece
ataoukts@pme.duth.gr
[Abstract-pdf]
Three coplanar line segments $OA$, $OB$, $OC$ are given and three concentric ellipses $C_1$, $C_2$, $C_3$ are defined, so that every two of the segments are conjugate semi-diameters of one ellipse. In previous studies we proved using Analytic Plane Geometry that the problem of finding an ellipse circumscribed to $C_1$, $C_2$, $C_3$ has at most two solutions. The \emph{primary solution $T_1$} is always an ellipse. The \emph{secondary solution $T_2$} (if it exists) is an ellipse or a hyperbola. We also constructed $T_1$ using Synthetic Projective Plane Geometry.\par This study investigates the existence and the construction of $T_2$ with Synthetic Projective Geometry, particularly Theory of Involution. We prove that the common diameters of every couple of $C_1$, $C_2$, $C_3$ correspond through an involution $f$. Criteria of Synthetic Projective Geometry determine whether $f$ is hyperbolic or elliptic. If $f$ is hyperbolic, exactly two double contact conics $T_1$, $T_2$ exist circumscribed to $C_1$, $C_2$, $C_3$. $T_1$ is always an ellipse. $T_2$ is an ellipse, a hyperbola or a degenerate parabola. The common diameters of $T_1$, $T_2$ define the double lines of $f$. If $f$ is elliptic, still two double contact conics $T_1$, $T_2$ exist. Now $T_1$ is an ellipse circumscribed and $T_2$ an ellipse inscribed to $C_1$, $C_2$, $C_3$. Regardless of whether $f$ is hyperbolic or elliptic, we construct $T_2$ using the already constructed ellipse $T_1$ and the involution $f$.
Keywords: Mutually conjugate ellipses, double contact conic, elliptic/hyperbolic involution, double rays, Fregier point.
MSC: 51N15; 51N20, 68U05.
[ Fulltext-pdf (9151 KB)]