# Confocal Families of Hyperbolic Conics via Quadratic Differentials

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## Abstract

**:**

## 1. Introduction

## 2. Confocal Conics in the Hyperbolic Plane

**Remark**

**1.**

**Definition**

**1.**

## 3. Curve Families in the Disc and Meromorphic Differentials

**Remark**

**2.**

- (1)
- $\omega =\frac{\lambda dz}{(z-a)(z-1/\overline{a})},\phantom{\rule{4pt}{0ex}}a\in \mathbb{D}$;
- (2)
- $\omega =\frac{\lambda dz}{{(z-\sigma )}^{2}},\phantom{\rule{4pt}{0ex}}\sigma ={e}^{i\alpha}\in {S}^{1}$; or
- (3)
- $\omega =\frac{\lambda dz}{(z-{\sigma}_{1})(z-{\sigma}_{2})},\phantom{\rule{4pt}{0ex}}{\sigma}_{j}={e}^{i{\alpha}_{j}}\in {S}^{1}$.

## 4. Classification of Confocal Families

- (I)
- Two poles in the disc, and their reflections in the circle (Figure 8).The quadratic differential is equivalent to$$\frac{d{z}^{2}}{({z}^{2}-{a}^{2})({z}^{2}-1/{a}^{2})},$$
- (II)
- One pole in the disc and its reflection, two poles on the circle (Figure 9).The quadratic differential is equivalent to$$\frac{d{z}^{2}}{z({z}^{2}-2cos\left(\alpha \right)z+1)}$$
- (III)
- Four poles at points on the circle (Figure 10).The quadratic differential is equivalent to$$\frac{d{z}^{2}}{{z}^{4}-2cos\left(2\beta \right){z}^{2}+1}$$
- (IV)
- A pole in the disc, and its reflection, and a double pole on the circle (Figure 11).The quadratic differential is equivalent to$$\frac{d{z}^{2}}{z{(z-1)}^{2}}.$$We may place the poles at $z=0$ and at $z=1$: this is a limiting case of an ellipse or hyperbola, with one focus at an ideal point. The trajectories are elliptic parabolas, which reflect rays from the finite focus to rays approaching the ideal point, and convex hyperbolic parabolas, which reflect rays from the finite focus to rays emanating from the ideal point.
- (V)
- One double pole and two single poles on the circle (Figure 12).The quadratic differential is equivalent to$$\frac{d{z}^{2}}{{(z-1)}^{2}({z}^{2}+1)}.$$Rays orthogonal to the line joining the ideal points are reflected to rays to the double ideal point. The mirrors are wide concave hyperbolic parabolas and long concave hyperbolic parabolas.
- (VI)
- Two double poles on the circle (Figure 13).The quadratic differential is equivalent to$$\frac{d{z}^{2}}{{({z}^{2}-1)}^{2}}.$$The trajectories consist of a hyperbolic straight line and the equidistant curves to the line (these being curves of constant curvature less than 1). The orthogonal trajectories are the geodesic field of lines orthogonal to the given line.
- (VII)
- A double pole in the disc, and its reflected double pole (Figure 14).The quadratic differential is equivalent to$$\frac{d{z}^{2}}{{z}^{2}}.$$The trajectories are concentric circles. If the center is chosen to be at the origin, the Euclidean, Poincaré, and Klein models are identical; otherwise, the curves in the Poincarè disc are circles, and the Klein circles are ellipses. The orthogonal trajectories are rays from the center.
- (VIII)
- A triple pole on the circle, and a single pole on the circle (Figure 15).The quadratic differential is equivalent to$$\frac{4d{z}^{2}}{{(z+1)}^{3}(z-1)}\phantom{\rule{4pt}{0ex}}.$$The trajectories are osculating parabolas, as are the orthogonal trajectories. The line ℓ joining the two foci reflects one of the two families to the other. Each parabola reflects rays orthogonal to ℓ to rays from the ideal point.
- (IX)
- A quadruple pole on the circle (Figure 16). The quadratic differential is equivalent to$$\frac{d{z}^{2}}{{(z-1)}^{4}}.$$The trajectories are horocycles. The orthogonal trajectories are the lines meeting at the ideal point, i.e., the hyperbolic parallel lines. This is the limiting case of concentric circles, when the center becomes an ideal point.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Conics and Their Names

Type | Story | Coolidge | Ismestiev |
---|---|---|---|

I | ellipse | ellipse | ellipse |

I | hyperbola | convex hyperbola | convex hyperbola |

II | semihyperbola | semihyperbola | semihyperbola |

III | hyperbola | concave hyperbola | concave hyperbola |

IV | elliptic parabola | elliptic parabola | elliptic parabola |

IV | $\mathbb{H}$ parabola | convex $\mathbb{H}$ parabola | convex $\mathbb{H}$ parabola |

V | $\mathbb{H}$ parabola | concave $\mathbb{H}$ parabola | wide/long parabola |

VI | circle | equidistant curve | hypercycle |

VII | circle | circle | circle |

VIII | semicircular parabola | osculating parabola | osculating parabola |

IX | circular parabola | horocycle | horocycle |

## Appendix B. Computation

## References

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**Figure 1.**Semihyperbolic mirror in Poincaré disc (

**left**); in Klein disc (

**right**). The pink arcs in the figure on the left represent light rays, which are reflected off the (orange) mirror to the blue light rays that converge at the focus. In the Klein model, these rays are represented by black straight lines.

**Figure 2.**

**Left**: The two models superimposed.

**Right**: Locating the ultra-ideal point in the Klein model. Red = mirror; pink and gray = incident rays, blue = reflected rays.

**Figure 3.**One focus at the origin, the other moving right, ellipses to parabolas to semihyperbolas, in the Poincaré disc model. Red = mirror; pink = incident rays, blue = reflected rays.

**Figure 4.**One focus at the origin, the other moving right, in the Klein model. Red = mirror; black = incident rays, blue = reflected rays.

**Figure 7.**Pole patterns for the quadratic differentials. Pole key: ● = simple; $\circ =$ double; $\odot =$ triple; $\oplus =$ quadruple. Poles at $z=0$ are paired with identical poles at $z=\infty $—out of view. Hyperbolic–geometric parameters for types I, II, and III: pole separation $0<2\rho <\infty $; angle between geodesic rays $0<2\alpha <\pi $; angle between geodesics $0<2\beta \le \pi /2$.

**Figure 8.**I. Ellipse and hyperbola. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 9.**II. Semihyperbola. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 10.**III. Concave hyperbolas.The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 11.**IV. Elliptic and convex hyperbolic parabola. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 12.**V. Wide and long concave hyperbolic parabolas. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 13.**VI. Equidistant curves. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 14.**VII. Circles. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 15.**VIII. Osculating parabolas. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 16.**IX. Horocycles. The left two parts show the horizontal (red) and vertical (magenta) trajectories in the Poincaré disc model; the right two parts show the horizontal (blue) and vertical (pink) trajectories in the Klein model.

**Figure 17.**The nine orthogonal families of conics. The red curves are horizontal trajectories and the magenta curves are vertical trajectories.

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**MDPI and ACS Style**

Langer, J.; Singer, D.
Confocal Families of Hyperbolic Conics via Quadratic Differentials. *Axioms* **2023**, *12*, 507.
https://doi.org/10.3390/axioms12060507

**AMA Style**

Langer J, Singer D.
Confocal Families of Hyperbolic Conics via Quadratic Differentials. *Axioms*. 2023; 12(6):507.
https://doi.org/10.3390/axioms12060507

**Chicago/Turabian Style**

Langer, Joel, and David Singer.
2023. "Confocal Families of Hyperbolic Conics via Quadratic Differentials" *Axioms* 12, no. 6: 507.
https://doi.org/10.3390/axioms12060507