Central Conics in H2 Are Fibers over the Group of Steiner Conics

We provide an intrinsic construction of the central conics in the real hyperbolic plane H2, 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 H2. The central Steiner conics that have a focal axis L are identified with the subgroup GL of collineations generated by reflections in the lines perpendicular to L. We obtain a GL-equivariant partition of the central conics by defining the fiber over g∈GL to be the set of compositions C such that πC=g. Here, πC is the unique Steiner conic tangent to C at the points on L, and is the product of the two elements in GL that represent the components of C. We use the terminology of fibers strictly in an incidence-geometric sense.