Central Conics in ${\mathbb{H}}^2$ Are Fibers over the Group of Steiner Conics

We provide an intrinsic construction of the central conics in the real hyperbolic plane ${\mathbb{H}}^2$, whereby each conic C is the composition of a unique pair of Steiner conics (those generated by collineations). The composition is achieved by elliptic curve addition on intersection points of the two components with their orthogonal trajectories, which have a natural representation as genus 1 curves in any inversive model of ${\mathbb{H}}^2$. The central Steiner conics that have a focal axis L are identified with the subgroup $G\left(L\right)$ of collineations generated by reflections in the lines perpendicular to L. We obtain a $G\left(L\right)$-equivariant partition of the central conics by defining the fiber over $g\in G\left(L\right)$ to be the set of compositions C such that $\pi \left(C\right)=g$. Here, $\pi \left(C\right)$ is the unique Steiner conic tangent to C at the points on L, and is the product of the two elements in $G\left(L\right)$ that represent the components of C. We use the terminology of fibers strictly in an incidence-geometric sense.