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.
Fields of Study & Subjects
Mathematics and Applications
Geometric and Algebraic Topology
Point processes and geometric inequalities
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 10 February 2026
- Language
- EN
- Document type
- Preprint