Real-locus compatibility of the birational transformation

Determine whether the birational transformation between the incidence curve and the elliptic curve preserves the real parts, and characterize the topology of the real part of the associated elliptic curve.

Background

The hyperbolic Kepler billiard is reduced to a Poncelet configuration involving the foci-circle and foci-caustic circle. After complexification, these conics define an incidence curve of genus one and a birationally equivalent elliptic curve given by $y^2=-\det(tQ_0+Q_1)$. The subsequent dynamical analysis relies on understanding the real locus of this elliptic curve, because the physical billiard dynamics is real.

The paper later discusses real structures and the topology of the real elliptic curve, but the margin note explicitly identifies the compatibility of the birational transformation with real parts—and the resulting topology—as an issue requiring verification.