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 el-
liptic 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 G(L) of collineations generated by reections
in the lines perpendicular to L. We define the fiber over g 2 G(L) to be the
set of compositions C such that Pi (C) = g. Here, Pi (C) is the unique Steiner
conic tangent to C at the points on L, and we show that Pi (C) is the product
of the two elements in G(L) that represent the components of C. The central
conics are partitioned into these fibers, which are acted upon transitively by
G(L). The geometry and algebra of the fiber bundle are emphasized, without
topological considerations.