Transcendental Equations for Nonlinear Optimization in Hyperbolic Space ^†

Abstract

We present a novel application of transcendental equations for nonlinear distance optimization in hyperbolic space. Through asymptotic approximations using Fourier and Taylor series expansions, we obtain approximations for the transcendental equations with non-zero real values on the boundary λ. The series expansion of the logarithmic form of our equations around two arbitrary points P₁ and P₂ can be used to find values close to definite coordinates on λ. Applying principles from the Poincaré hyperbolic disk—a non-Euclidean space with constant negative curvature—we construct optimization methods following λ of our transcendental equations.

1. Introduction

In “The Klein, Hilbert and Poincaré metrics of a domain” by A.F. Beardon, Beardon discusses Poincaré distance in the hyperbolic plane, geodesics, and the Hilbert metric in the real coordinate space. Specifically, Beardon noted that for larger distances in the hyperbolic plane, the shape resembles a geodesic. Moreover, in the convex domain on the hyperbolic plane, Beardon determined several key important features with the Hilbert metric, notably, a distance formula using hyperbolic connections between two points in the Euclidean space with a subsequent parameter restriction. On a similar note, the paper ”A Euclidean Model for Euclidean Geometry” by Adolf Mader presents a model of Euclidean geometry. This model is a realization of the Euclidean axioms in a non-Euclidean space and shows that the parallel postulate is not a necessary axiom of Euclidean geometry. The connection between the Mader model and the Poincaré hyperbolic disk model (as detailed in ”Bounded Models of the Euclidean Plane” by R. L. Swain, David Gans, and K. O. May) shows that both models can be used to represent the same set of geometric axioms.

2. Theories

2.1. Hyperbolic Space

Hyperbolic space is a n-dimension non-Euclidean metric space (denoted by ${ℍ}^n$, where n ∈ $\mathbb{N}$ and n ≥ 2) of constant negative (sectional) curvature denoted by K, where K < 0. The n-dimensional unit sphere in Euclidean space can be defined as the set ($x\in {ℝ}^{\left(n+1\right)} :x·x=1$). Thus, ${ℍ}^n$ is the set $x\in {ℝ}^{\left(n+1\right)} :x·x=1$.

2.2. Hyperbolic Trigonometry

The inverse hyperbolic functions, denoted as arccosh, arcsinh, and arctanh [1,2], are the function that maps real numbers to hyperbolic angles. For this research, we are interested in the properties of the inverse hyperbolic cosine function (arccosh) defined as

$$\mathrm{arccosh}\left(x\right)=\ln \left(x+\sqrt{x^2-1}\right),·\left\{x\in ℝ:x\ge 1\right\}$$

(1)

2.3. Hyperbolic Geometry and Distances

In hyperbolic geometry, distances, angles, and geometric properties are based on the hyperbolic metric which defines the distance function between any two points in ${ℍ}^n$. The hyperbolic metric satisfies the properties of a metric space, including non-negativity, the identity of indiscernibles, symmetry, and triangle inequality. Nonlinear distance optimization in hyperbolic space is constrained by boundary conditions, such as the unit disk of the Poincaré model, limiting the range of possible distances. The unit disk models of the hyperbolic space (Poincaré and Beltrami–Klein) [3] provide no direct comparison with the Euclidean space limited by the key issue of distance preservation. Additionally, the boundary conditions of the hyperbolic trigonometric functions constrain arccosh($x$)’s $x\in ℝ :x\ge 1$ boundary.

2.4. Transcendental Equation and λ Boundary Conditions in Poincaré and Mader Models

2.4.1. Poincaré Model

The Poincaré disk model [4] represents the hyperbolic plane using a unit disk, $\mathbb{D}$, in the complex plane $\mathbb{C}$. A point, $z\in \mathbb{D}$, is represented by its complex coordinates $z=x+yi$, where x and y are real numbers, satisfying $x^2+y^2<1$. Applying principles from the Poincaré hyperbolic disk, we construct a novel optimization model consisting of the transcendental arcosh equation bound by the λ boundary function, denoted as λ(r) to represent real values in the complex hyperbolic plane. The equation is defined as

$$\mathrm{arcosh}\left(x+y\right)=\ln \left[\left(x+y\right)+\sqrt{{\left(x+y\right)}^2-1}\right]$$

(2)

where x and y are coordinates of points belonging to the Poincaré disk $\mathbb{D}$. The domain for the function is x + y > 1, and the range is the set of all real numbers. The boundary λ to which this function is constrained is given by the following function:

$$\mathsf{\lambda }\left(r\right)={\displaystyle \frac{1-r^2}{1+r^2}}$$

(3)

where $r=\sqrt{x^2+y^2}$ is the Euclidean norm of the point z in the disk. This function λ provides a mapping between the Poincaré disk and Mader models representing the boundary of the unit disk $\mathbb{D}$ where the real values of the function arccosh(x + y) lie.

2.4.2. Mader Model

The λ boundary function, denoted as λ(r), connects the Poincaré disk model to the Mader model of Euclidean space [5]. The λ boundary function acts as a bijective mapping function $\mathbb{D}$ → $\mathbb{F}$ where $\mathbb{F}$ is a model in Euclidean space. The Mader model, denoted as $\mathbb{F}$, is defined as the interior of a unit disk ω in the standard Euclidean plane. Points inside the disk are represented by coordinates (x,y). The boundary of the disk (the circle) represents points where $x^2+y^2=1$. The λ boundary function preserves distances between the Euclidean plane $\mathbb{E}$ and the Poincaré disk $\mathbb{D}$ via the bijections $\mathbb{E}$ → $\mathbb{F}$ and $\mathbb{D}$ → $\mathbb{F}$.

3. Novel Model for Hyperbolic Transformation

Our novel model acts as a unified space that correlates the geodesic pathways among Mader, Poincaré, and Euclidean geometries by employing transcendental arccosh equations, boundary conditions λ, and series approximations.

Given the Euclidean, Mader, and Poincaré distance metrics $d_E$, $d_M$, and $d_P$, respectively, we establish bijective transformations that allow us to navigate between these geometric spaces [6].

$${\mathsf{\Psi }}_E:{ℝ}^2\to E,{\mathsf{\Psi }}_M:{ℝ}^2\to M,\mathrm{and}{\mathsf{\Psi }}_P:{ℝ}^2\to P$$

The model employs Taylor series approximations in the transcendental arccosh function and the λ boundary conditions. Specifically, we use the Taylor series expansion of the natural logarithm form of the transcendental equations to provide approximations for values near the expansion points, P₁ and P₂.

3.1. Poincaré Model, λ Boundary Transformation, and Transcendental Arccosh

In the Poincaré disk model, each point within the unit disk $\mathbb{D}$, in the complex plane $\mathbb{C}$, is characterized by complex coordinates $z=x+yi$, where x, y ∈ $\mathbb{R}$ and $x^2+y^2=1$.

$$\mathrm{arcosh}\left(x+y\right)=\ln \left[\left(x+y\right)+\sqrt{{\left(x+y\right)}^2-1}\right]$$

(4)

Proof. 

Let y = cosh(x):

$$2y=e^x+e^{-x},$$

$$2y=e^x+e^{-x},$$

Substituting $u=e^x$, we obtain a quadratic equation in u:

$$2yu-u^2-1=0.$$

Solving the equation using the quadratic formula:

$$\begin{array}{cc}u\hfill & ={\displaystyle \frac{-2y\pm \sqrt{{\left(2y\right)}^2-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}}\hfill \\ \hfill & =y\pm \sqrt{y^2-1}.\hfill \end{array}$$

Since $u=e^x>0$, choosing the positive root:

$$e^x=y+\sqrt{y^2-1}.$$

Taking the natural logarithm of both sides:

$$x=\ln \left(y+\sqrt{y^2-1}\right).$$

Since y = cosh(x) and arccosh(y) = x:

$$\mathrm{arcosh}\left(z\right)=\ln \left(z+\sqrt{z^2-1}\right).$$

Substituting $z=x+y$:

$$\mathrm{arcosh}\left(x+y\right)=\ln \left(\left(x+y\right)+\sqrt{{\left(x+y\right)}^2-1}\right),$$

for all x and y such that $x+y\ge 1$, which is the domain of the arcosh function.

$$r=\sqrt{x^2+y^2}$$

$$\begin{gathered} \mathsf{\lambda }\left(r\right)=\ln \left[r+\sqrt{r^2-1}\right] \\ \therefore \mathrm{arcosh}\left(x+y\right)=\mathsf{\lambda }\left(r\right)=\ln \left[\left(x+y\right)+\sqrt{{\left(x+y\right)}^2-1}\right] \end{gathered}$$

The function λ(r) sets the boundary conditions, effectively constricting the solution space of the transcendental equation within the constraints of the Poincaré disk model. □

3.2. Mader to Euclidean Transformation, Distance Function $d_e$

The Mader model is integrated by transformation function Ψ that bridges the distance along the diameter (denoted by r) with the Euclidean distance $d_e$.

$$r=\mathsf{\Psi }\left(d_e\right)={\displaystyle \frac{d_e}{d_e+\sqrt{d_e^2+1}}}$$

(5)

Proof. 

Given the Poincaré metric:

$$ds={\displaystyle \frac{2 dx}{1-x^2}}$$

To find the hyperbolic distance $r$ from 0 to $d_e$, we integrate:

$$r={{\displaystyle \int }}_0^{d_e}{\displaystyle \frac{2 dx}{1-x^2}}$$

Using trigonometric substitution:

Let $x=\tan \mathrm{h}\left(u\right)$ imply $dx={\sec \mathrm{h}}^2\left(u\right) du$.

When $x=0$, $u=0$ and $x=d_e$, $u=\mathrm{artanh}\left(d_e\right)$.

Thus,

$$r={{\displaystyle \int }}_0^{\mathrm{artanh}\left(d_e\right)}2{\sec \mathrm{h}}^2\left(u\right) du$$

$$r=2u{|}_0^{\mathrm{artanh}\left(d_e\right)}=2\mathrm{artanh}\left(d_e\right)$$

Using the identity for the inverse hyperbolic tangent,

$$\mathrm{arctanh}\left(x\right)={\displaystyle \frac{1}{2}}\log \left({\displaystyle \frac{1+x}{1-x}}\right)$$

$$r=\log \left({\displaystyle \frac{1+d_e}{1-d_e}}\right)$$

$$e^r={\displaystyle \frac{1+d_e}{1-d_e}}$$

$$d_e={\displaystyle \frac{1-e^r}{e^r+1}}$$

Multiplying both the numerator and denominator by $e^{-{\displaystyle \frac{r}{2}}}$:

$$d_e={\displaystyle \frac{e^{-\frac{r}{2}}-e^{\frac{r}{2}}}{e^{-\frac{r}{2}}+e^{\frac{r}{2}}}}$$

Using the identity $e^r=\cos \mathrm{h}\left(r\right)+\sin \mathrm{h}\left(r\right)$:

$$d_e={\displaystyle \frac{\sin \mathrm{h}\left(\frac{r}{2}\right)}{\cos \mathrm{h}\left(\frac{r}{2}\right)}}$$

Using the hyperbolic identities:

$$\sin \mathrm{h}\left(x\right)={\displaystyle \frac{e^x-e^{-x}}{2}}$$

$$\cos \mathrm{h}\left(x\right)={\displaystyle \frac{e^x+e^{-x}}{2}}$$

$$d_e={\displaystyle \frac{e^{\frac{r}{2}}-e^{-\frac{r}{2}}}{e^{\frac{r}{2}}+e^{-\frac{r}{2}}}}$$

Multiplying the numerator and denominator by $e^{-{\displaystyle \frac{r}{2}}}$:

$$d_e={\displaystyle \frac{1-e^{-r}}{1+e^{-r}}}$$

$$r=\ln \left({\displaystyle \frac{d_e+1}{d_e}}\right)$$

Squaring both sides:

$$r^2={\ln }^2\left({\displaystyle \frac{d_e+1}{d_e}}\right)$$

Taking the positive square root (since $r>0$ for $d_e>0$):

$$r={\displaystyle \frac{d_e}{\sqrt{d_e^2+1}}}$$

We assume that

$$d_e\ll \sqrt{d_e^2+1}$$

$$\therefore r={\displaystyle \frac{d_e}{d_e+\sqrt{d_e^2+1}}}$$

$$\mathsf{\Psi }\left(R\right)={\displaystyle \frac{R}{R+\sqrt{R^2+1}}}$$

This transformation function $\mathsf{\Psi }$ enables a bijection between Mader and Euclidean geometries, connecting these two models using our model. □

3.3. Hyperbolic, Euclidean Distance and Mapping Function

We map the hyperbolic distance, $d_h$, in the Poincaré disk model through the Euclidean norm of $z$, $\left|\left|z\right|\right|$, using the mapping function $\mathsf{\Phi }$.

$$d_h=\mathsf{\Phi }\left(\left|\left|z\right|\right|\right)=\mathrm{arccosh}\left(1+{\displaystyle \frac{2{\left|\left|z\right|\right|}^2}{1-{\left|\left|z\right|\right|}^2}}\right)$$

(6)

Proof. 

The distance formula in the Poincaré disk model for two points, $z_1$ and $z_2$, is

$$d_h\left(z_1,z_2\right)=\mathrm{arccosh}\left(1+{\displaystyle \frac{2{\left|\left|z_1-z_2\right|\right|}^2}{\left(1-\left|\right|z_1|{|}^2\right)\left(1-\left|\right|z_2|{|}^2\right)}}\right)$$

For the distance between the origin $0$ and a point $z$:

$$d_h\left(0,z\right)=\mathrm{arccosh}\left(1+{\displaystyle \frac{2{\left|\left|z\right|\right|}^2}{1-{\left|\left|z\right|\right|}^2}}\right)$$

Using the Euclidean norm for $z=x+yi$:

$$\left|\left|z\right|\right|=\sqrt{x^2+y^2}$$

$$d_h=\mathrm{arccosh}\left(1+{\displaystyle \frac{2\left(x^2+y^2\right)}{1-\left(x^2+y^2\right)}}\right)$$

$$\therefore d_h=\mathsf{\Phi }\left(\left|\left|z\right|\right|\right)=\mathrm{arccosh}\left(1+{\displaystyle \frac{2{\left|\left|z\right|\right|}^2}{1-{\left|\left|z\right|\right|}^2}}\right)$$

 □

3.4. Mapping Function

Consider an arbitrary point $P$ in the Poincaré disk model $\mathbb{D}$. Let the coordinates of $P$ be $\left(x,y\right)$, where $x^2+y^2<1$. We define a mapping $\mathsf{\Psi }:\mathbb{D}\to \mathbb{M}$, such that a point $P$ in $\mathbb{D}$ is mapped to a point $P^{\prime }$ in $\mathbb{M}$, the Mader model.

$$\mathsf{\Psi }\left(x,y\right)=\left(h\left(x\right),h\left(y\right)\right)$$

(7)

$$h\left(z\right)=\mathrm{arccosh}\left(\ln \left(1+z\right)\right)$$

(8)

For $z$ in $\mathbb{D}$, the function $h$ effectively translates each coordinate from $\mathbb{D}$ to a corresponding coordinate in $\mathbb{M}$.

3.5. Distance Function

Given two points, $P_1^{\prime }$ and $P_2^{\prime }$, in $\mathbb{M}$, which correspond to $P_1$ and $P_2$ in $\mathbb{D}$, the distance between $P_1^{\prime }$ and $P_2^{\prime }$ in the Mader model, $d_{\mathbb{M}}\left(P_1\prime ,P_2\prime \right)$, is defined in terms of the distance between $P_1$ and $P_2$ in the Poincaré disk model, $d_{\mathbb{D}}\left(P_1,P_2\right)$, as follows:

$$d_M\left(\mathsf{\Psi }\left(x_1,y_1\right),\mathsf{\Psi }\left(x_2,y_2\right)\right)=h\left(d_D\left(\left(x_1,y_1\right),\left(x_2,y_2\right)\right)\right)$$

(9)

The transformation $\mathsf{\Psi }$ offers a one-to-one mapping between the Poincaré disk $\mathbb{D}$ and Mader model $\mathbb{M}$ using the function $h$ defined in (7). By making use of the function $h$ and the transformation $\mathsf{\Psi }$, this continuous mapping preserves the underlying geometric structures to compare the geometric properties between $\mathbb{D}$ and $\mathbb{M}$.

4. Series Expansions for Approximations

Understanding the behavior of the function λ(r) and the hyperbolic distance $d_h$ under different geometrical transformations is critical. To address this, we employ Taylor series expansion to approximate these functions around the point $P$.

$$f\left(x+y\right)=\ln [\left(x+y\right)+\sqrt{{\left(x+y\right)}^2-1}$$

The Taylor series expansion of a function $f$ about a point $P$ is represented as

$$\begin{array}{c}f\left(x+y\right)\approx f\left(P\right)+f^{\prime \left(P\right)\left(x+y-P\right)}+{\displaystyle \frac{f^{″\left(P\right)}}{2!}}{\left(x+y-P\right)}^2+{\displaystyle \frac{f^{‴\left(P\right)}}{3!}}{\left(x+y-P\right)}^3+\cdots \end{array}$$

$$f\left(P\right)=\ln \left[P+\sqrt{P^2-1}\right]$$

(10)

The hyperbolic distance function $d_h$ can be approximated using a series expansion around the point $P$ in the Poincaré disk model.

$$\begin{array}{cc}{d}_h& =\mathsf{\Phi }\left(\left|\left|z\right|\right|\right)=\mathrm{arccosh}\left(1+{\displaystyle \frac{2{\left|\left|z\right|\right|}^2}{1-{\left|\left|z\right|\right|}^2}}\right)\\ \hfill & \hfill \end{array}$$

(11)

$$\approx \mathsf{\Phi }\left(P\right)+\left(\left|\left|z\right|\right|-P\right){\mathsf{\Phi }}^{\prime }\left(P\right)+{\displaystyle \frac{{\left(\left|\left|z\right|\right|-P\right)}^2}{2}}{\mathsf{\Phi }}^{″}\left(P\right)+\cdots$$

(12)

The general term for the series expansion about the point $P$ is

$${\displaystyle \frac{{\mathsf{\Phi }}^n\left[P\right]}{n!}}{\left[\left|\left|z\right|\right|-P\right]}^n$$

(13)

where ${\mathsf{\Phi }}^{\left(n\right)}\left[P\right]$ represents the nth derivative of $\mathsf{\Phi }$ concerning $\left|\left|z\right|\right|$, evaluated at $P$.

5. Applications

Simple Distance Optimization Application

In the Mader model, $P_1$ and $P_2$ are transformed into points within the unit disk. We calculate the Euclidean distances, denoted as $d_{e1}$ and $d_{e2}$, of these points from the origin. Using the mapping function $\mathsf{\Psi }$ from the Mader model to the Poincaré model, these points are translated to the Poincaré disk.

$$\mathsf{\Psi }\left(r\right)={\displaystyle \frac{2r}{1+r^2}}$$

(14)

We calculate the hyperbolic distances $d_{h1}$ and $d_{h2}$ for $P_1$ and $P_2$, respectively, in the Poincaré disk model.

$$d_{h1}=\mathrm{arccosh}\left(1+{\displaystyle \frac{2d_{e1}}{1-d_{e1}^2}}\right)$$

(15)

$$d_{h2}=\mathrm{arccosh}\left(1+{\displaystyle \frac{2d_{e2}}{1-d_{e2}^2}}\right)$$

(16)

The hyperbolic geodesic distance between points $P_1$ and $P_2$ is computed by the integration of the Poincaré metric over the shortest path represented as a segment of a circle orthogonal to the boundary of the disk in the Poincaré model.

$${{\displaystyle \int }}_{P_1}^{P_2}{\displaystyle \frac{2\left|dz\right|}{1-{\left|z\right|}^2}}$$

(17)

To approximate the complex integral, a Taylor series expansion is employed around the shortest path between $P_1$ and $P_2$. The optimal path corresponds to the minima of the hyperbolic distance. The optimal hyperbolic path is then translated back to the Mader model using the inverse mapping function ${\mathsf{\Psi }}^{-1}$ and subsequently to the Euclidean model to provide a real-world trajectory for objects such as airplanes, ships, and the like.

6. Diagrams

For a system modeling shortest-path trajectories in Euclidean space, the metric utilized is the Euclidean distance, given by

$$\begin{array}{r}\hfill d_E=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}.\end{array}$$

The transformation function, $\mathsf{\Phi }$, transforms the system’s trajectories from Euclidean space ${ℝ}^2$ into the unit disk $\mathbb{D}$, following the condition $\begin{array}{r}\hfill \mathsf{\Phi }\left(ℝ\right)=z\mathrm{in}\mathbb{D},\mathrm{with}z=x+yi.\end{array}$

The geodesic distance in the Poincaré disk, representing the shortest path between two points in hyperbolic space is

$$d_H=\mathrm{arccosh}\left(1+{\displaystyle \frac{2{\left|z_2-z_1\right|}^2}{\left(1-{\left|z_1\right|}^2\right)\left(1-{\left|z_2\right|}^2\right)}}\right)$$

(18)

• Original trajectory (Euclidean): Figure 1a displays the shortest path between two points using the standard Euclidean metric, represented as a straight line;
• Transformed trajectory (Poincaré disk model): In Figure 1b, the direct line in Euclidean space transforms, resulting in a curved path in the Poincaré disk called a geodesic. This geodesic demonstrates the hyperbolic shortest path, emphasizing that the Poincaré distance is strictly less than the corresponding Euclidean distance for the same two points on the bounded unit circle.

Overall, for our diagrams, we employed mathematical transformations to analyze different physical systems from a hyperbolic perspective. We used the numpy, matplotlib, and hypertiling packages in Google Colab to aid with visualizations of our models. Using the Poincaré disk model, we defined a mapping function $\mathsf{\Psi }$ to transform Euclidean spaces into hyperbolic spaces. Hyperbolic geodesic distances were calculated using a definite integral over the Poincaré metric. Taylor series expansions were utilized in approximating complex integrals, especially when computing distances. For thermodynamic systems, a 6n-dimensional phase space was transformed using the $\mathsf{\Psi }$ function. The pendulum system’s dynamics, characterized in a 2D phase space, were explored through a transformation relation capturing angular position and momentum.

7. Conclusions

We established a novel application of transcendental equations for modeling and optimization in hyperbolic space. Utilizing the Poincaré disk model, we defined the boundary function $\mathsf{\lambda }\left(r\right)={\displaystyle \frac{1-r^2}{1+r^2}}$ and transformation functions $\mathsf{\Phi }$ and $\mathsf{\Psi }$, with specific expressions for hyperbolic distance calculations. A key limitation of the proposed model in this study lies in the Taylor series approximations and assumptions of underlying geometries. However, the error remains negligibly small within the model’s defined constraints. Future research is necessary to apply this model to higher-dimensional non-Euclidean surfaces, improving geometric analysis in differential geometry and topological mapping.