Power with Respect to Generalized Spheres and Radical Surfaces in $\mathbf{H}^n$

Abstract

This paper presents a unified theory for the power of a point with respect to generalized spheres (spheres, horospheres, and hyperspheres) in $n$-dimensional hyperbolic space $\mathbf{H}^n$. By extending the classical secant theorem, we derive a novel formula for hyperspheres and also prove that the radical surface of any two non-concentric generalized spheres is a hyperplane. These results provide tools for constructing power diagrams and studying hyperball packings.