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

TL;DR

The paper extends the power-of-a-point concept to generalized spheres in hyperbolic space $\mathbf{H}^n$, including hyperspheres with two branches, by developing a hyperbolic secant framework and relying on synthetic proofs in the Poincaré disk model. It proves a Power of a Point theorem for hypercycle branches using $\tanh$ and $\coth$ distances, and shows that the radical surface of two non-concentric generalized spheres is a hyperplane, enabling hyperbolic power diagrams. The results unify incidence theory for circles, horocycles, and hyperspheres, and have direct implications for hyperball packing/covering decompositions via radical hyperplanes. The approach provides a coordinate-free, geometric toolkit for constructing and analyzing hyperbolic power diagrams and contributes to understanding optimal packing densities in $\mathbf{H}^n$. Overall, the work bridges classical Euclidean/spherical power theory with hyperbolic geometry, offering scalable methods for higher-dimensional hyperbolic packings and related geometric structures.

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.

Figures (9)

• Figure 1: Figure 1: Power of a point theorem
• Figure 2: Figure 2: Illustration of Lemma \ref{['lem:important1']}
• Figure 3: Figure 3: Illustration of Lemma \ref{['lem:ortogcirc']}
• Figure 4: Figure 4: Hyperbolic secant theorem in the Poincaré disk model
• Figure 5: Figure 5: Illustration of Lemma \ref{['lem:hypc1']}

Theorems & Definitions (26)

• Theorem 2.1: Power of a point theorem in $\mathbf{E}^2$
• Remark
• Theorem 2.2: Power of a point theorem in $\mathbf{S}^2$
• Definition 3.1
• Lemma 3.1
• Remark
• Lemma 3.2
• Lemma 3.3
• Theorem 4.1: Power of a point theorem for circles in $\mathbf{H}^2$
• Remark