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arXiv · 2602.09698

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

Áron Világi·Jenő Szirmai
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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.

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Áron Világi, Jenő Szirmai. 2026-02-10. Power with Respect to Generalized Spheres and Radical Surfaces in $\mathbf{H}^n$. https://arxiv.org/abs/2602.09698

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