# The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

## Abstract

**:**

## 1. Introduction

- (1)
- The vector addition admits scalar multiplication, giving rise to vector spaces which, in turn, form the algebraic setting for analytic Euclidean geometry. In full analogy,
- (2)
- The Einstein addition, which pertains to relativistically admissible velocities, also permits scalar multiplication, giving rise to Einstein gyrovector spaces, which, in turn, form the algebraic setting for the Klein ball model of analytic hyperbolic geometry [1,2,3,4,5]. Accordingly, the Klein model of hyperbolic geometry is also known as the relativistic model of hyperbolic geometry [6,7]. Furthermore, in full analogy,
- (3)

- (1)
- In [6], we presented the well-known proof of Ptolemy’s theorem in terms of the standard trigonometry of analytic Euclidean plane geometry. In particular, the associated law of cosines was employed.
- (2)
- In full analogy, in [6], we discovered the hyperbolic Ptolemy’s theorem in the Klein (relativistic) model of analytic hyperbolic plane geometry. The proof of the resulting hyperbolic Ptolemy’s theorem is obtained by means of the gyrotrigonometry that the Klein model of analytic hyperbolic geometry admits. In particular, the associated law of gyrocosines was employed.
- (3)
- In full analogy, in this article, we discover the hyperbolic Ptolemy’s theorem in the Poincaré ball model of analytic hyperbolic plane geometry. The proof of the resulting hyperbolic Ptolemy’s theorem is obtained by means of the gyrotrigonometry that the Poincaré model of analytic hyperbolic geometry admits. In particular, the associated law of gyrocosines is employed.

## 2. Möbius Addition and Scalar Multiplication

## 3. Gyrotrigonometry in Möbius Gyrovector Planes and Its Law of Gyrocosines

**Theorem**

**1**

**.**Let $ABC$ be a gyrotriangle in a Möbius gyrovector plane $({\mathbb{R}}_{s}^{n},\oplus ,\otimes )$ with vertices $A,B,C\in {\mathbb{R}}_{s}^{2}$, side gyrolengths $0<a,b,c<s$, and gyroangles $\alpha ,\phantom{\rule{3.33333pt}{0ex}}\beta ,\phantom{\rule{3.33333pt}{0ex}}\gamma $, as shown in Figure 2. Then, the gyrosides and gyroangles of the gyrotriangle satisfy the law of gyrocosines

## 4. Ptolemy’s theorem in the Poincaré Ball Model of Hyperbolic Geometry

**Theorem**

**2**

**.**Let $ABCD$ be a gyrocyclic gyroquadrilateral, shown in Figure 3. Then, the product of the g-modified gyrodiagonals equals the sum of the products of the g-modified opposite gyrosides; that is

## 5. Gyrodiametric Gyrotriangles

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**3**

**.**Let $ABC$ be a gyrodiametric gyrotriangle in a Möbius gyrovector space $({\mathbb{R}}_{s}^{n},\oplus ,\otimes )$, where $AC$ is the gyrodiametric gyroside. Then, the gyrotriangle gyrosides obey the Pythagorean-like equation

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**In the gyroformalism of analytic hyperbolic geometry, expressions that describe hyperbolic geometric objects take graceful forms analogous to their Euclidean counterparts. This is illustrated here, where the unique gyroline in a Möbius gyrovector plane $({\mathbb{R}}_{s}^{2},\oplus ,\otimes )$ through two given points $A,B\in {\mathbb{R}}_{s}^{2}$ is shown. When the gyroline parameter $t\phantom{\rule{-2.84544pt}{0ex}}\in \phantom{\rule{-2.84544pt}{0ex}}\mathbb{R}$ runs from $-\infty $ to ∞, the point $P\left(t\right)=A\oplus (\ominus A\oplus B)\otimes t$ runs over the gyroline ${L}_{{}_{AB}}$. In particular, at “time” $t=0$, the point is at $P\left(0\right)=A$, and, owing to the left cancellation law of Möbius addition, at “time” $t=1$, the point is at $P\left(1\right)=B$. The Möbius gyroline equation is shown in the box. The analogies it shares with the Euclidean straight line equation in the vector space approach and Euclidean geometry are obvious.

**Figure 2.**A gyrotriangle $ABC$ in the Möbius gyrovector plane $({\mathbb{R}}_{s}^{2},\oplus ,\otimes )$, along with its (i) gyrovertices $A,B,C$, (ii) gyroside gyrolengths $a,b,c$, and (iii) gyroangles $\alpha $, $\beta $, $\gamma $ and their gyrocosines.

**Figure 3.**Ptolemy’s theorem in the hyperbolic plane regulated by the Möbius gyrovector plane $({\mathbb{R}}_{s}^{2},\oplus ,\otimes )$. A gyrocyclic gyroquadrilateral $ABCD$ inscribed in its circumgyrocircle gyrocentered at its circumgyrocenter O, with gyroradius $r=\parallel \ominus O\oplus A\parallel =\parallel \ominus O\oplus B\parallel =\parallel \ominus O\oplus C\parallel =\parallel \ominus O\oplus D\parallel $ is shown. The O-gyrovertex gyroangles $\alpha $, $\beta $, $\gamma $, and $\delta $ satisfy the following equation: $\alpha +\beta +\gamma +\delta =2\pi $. The Hyperbolic Ptolemy’s theorem is fully analogous to its Euclidean counterpart, asserting that ${\left|AB\right|}_{g}{\left|CD\right|}_{g}+{\left|AD\right|}_{g}{\left|BC\right|}_{g}={\left|AC\right|}_{g}{\left|BD\right|}_{g}$, where ${\left|AB\right|}_{g}$, ect., is defined by (20) along with (6).

**Figure 4.**A gyrodiametric gyroquadrilateral in a Möbius gyrovector plane. The gyrocyclic gyroquadrilateral $ABCD$ of Figure 3 is depicted here in the special position where the two gyrodiagonals, $AC$ and $BD$, intersect at the gyroquadrilateral circumgyrocenter O, resulting in the two gyrodiametric gyrotriangles, $ABC$ and $ADC$. The common gyroside, $AC$, of these gyrotriangles is a gyrodiameter of the gyroquadrilateral circumgyrocircle. The Pythagorean-like formula that the g-modified gyroside gyrolengths of a gyrodiametric gyrotriangle obey in the Poincaré ball model of hyperbolic geometry is shown.

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**MDPI and ACS Style**

Ungar, A.A.
The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry. *Symmetry* **2023**, *15*, 1487.
https://doi.org/10.3390/sym15081487

**AMA Style**

Ungar AA.
The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry. *Symmetry*. 2023; 15(8):1487.
https://doi.org/10.3390/sym15081487

**Chicago/Turabian Style**

Ungar, Abraham A.
2023. "The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry" *Symmetry* 15, no. 8: 1487.
https://doi.org/10.3390/sym15081487