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

TL;DR

This paper develops a unified theory of point power relative to generalized spheres in hyperbolic space, extending classical theorems to facilitate the study of hyperball packings and power diagrams.

Contribution

It introduces a new formula for hyperspheres and proves the radical surface of two non-concentric generalized spheres is a hyperplane, extending classical geometric results.

Findings

Derived a novel hypersphere power formula

Proved radical surface of two non-concentric generalized spheres is a hyperplane

Provided tools for hyperball packing and power diagram construction

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.