Reflection Spaces Related to Hyperbolic Planes

This research investigates the algebraic and geometric properties of reflection spaces within the framework of general hyperbolic planes. While the theory of reflection spaces, as established by Karzel and Taherian, has been applied extensively to elliptic planes, its application to hyperbolic geometry has remained largely unexplored. We provide a comprehensive analysis of the involution-based structures generated by point reflections, line reflections, and their collective union within the motion group of a hyperbolic plane. Our findings reveal a tiered structural hierarchy in which point reflections alone constitute a polar-free reflection space that maintains an isomorphism with the plane's original incidence structure. In contrast, the set of line reflections yields a reducible, polar reflection space that recovers the dual geometric structure. Most significantly, we demonstrate that the union of point and line reflections results in a maximal reducible polar reflection space. Under the condition that the hyperbolic plane is complete, we extend the incidence structure of this maximal space and prove its embeddability into a projective plane, where the boundary set of ends forms a geometric oval.