# Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

**Remark 1.**

## 2. Foci of Real Algebraic Plane Curves

**Definition 1.**

**Proposition 1.**

## 3. Bicircular Quartics: Focal and Elliptic Discriminants

**Theorem 1.**

**Proposition 2.**

## 4. The Confocal Families of Bicircular Quartics ${\mathcal{H}}_{\alpha}^{\mathbf{r}}$

**Remark 2.**

**Remark 3.**

**Definition 2.**

**Remark 4.**

**Remark 5.**

## 5. Bicircular Quartics and Quadratic Differentials

**Lemma 1.**

**Proof.**

**Theorem 2.**

**Proof.**

**Remark 6.**

## 6. Edwards Normal Form and Algebraic Integrability

**Theorem 3.**

## 7. Birational Equivalence of Confocal Curves

**Theorem 4.**

**Proof.**

**Remark 7.**

**Remark 8.**

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Space of Bicircular Quartics ${\mathbf{P}}^{\mathbf{3}}\mathbf{=}\mathbf{\cup}{\mathbf{P}}_{\mathit{j}}^{\mathit{k}}$

- $r=0$ Trinodal: $F={({x}^{2}+{y}^{2})}^{2}+2(p{x}^{2}+q{y}^{2})$: nodes $I,J,O=(0,0)$.
- $s=0$ Conics: $F={F}_{2}{z}^{2}=(2p{x}^{2}+2q{y}^{2}+r{z}^{2}){z}^{2}$.
- $p=q$ Concentric circles: $K={C}_{1}{C}_{2}={u}^{2}{v}^{2}+2puv+r$, center O.
- ${p}^{2}=r$ Circle pairs: $K={C}_{+}{C}_{-};\phantom{\rule{4pt}{0ex}}{C}_{\pm}:=((u\pm \sigma )(v\mp \sigma )+\frac{p+q}{2})$.
- ${q}^{2}=r$ Circle pairs: $K={C}_{+}{C}_{-};\phantom{\rule{4pt}{0ex}}{C}_{\pm}:=((u\pm \sigma )(v\pm \sigma )+\frac{p+q}{2})$.

**Proposition A1.**

**Proof.**

## Appendix B. Concyclic Confocal Families

**Figure A1.**Confocal family of type $r=1$ ($\alpha =1+\sqrt{2}$); the foci are the four red points. Each irreducible quartic in ${\mathcal{H}}_{\alpha}^{1}$ is two-circuited—e.g., see bold blue/green curves (the bold blue one is the unique Cassinian in ${\mathcal{H}}_{\alpha}^{1}$).

**Figure A2.**The $p,q$-plane ${\mathrm{R}}_{1}$ is foliated by hyperbolas: ${\mathcal{H}}_{\alpha}$ represents a confocal family with concyclic foci on the x-axis (blue), y-axis (green), or unit circle (orange). H/V denote (blue) regions whose points represent horizontal/vertical trajectories. Dashed curves are explained in Remark A2.

**Remark A1.**

**Figure A3.**(

**Left**): The confocal family ${\mathcal{H}}_{\alpha}^{r}={\mathcal{H}}_{0}^{1}$; green/blue curves are horizontal/vertical trajectories of ${Q}_{\alpha}^{r}$. (

**Right**): Confocal family of type $r=0$. The blue curves are Booth lemniscates; the bold one is the Bernoulli lemniscate (the unique Cassinian in the family). Green curves are ${S}^{1}$-inverted ellipses.

**Remark A2.**

## Appendix C. Remarks on the Classical Notion of Focus

## Appendix D. Siebeck’s (Other) Theorem

## Appendix E. Circles of Inversion for Circular Cubics

**Figure A4.**A one-circuited circular cubic and its two real circles of inversion $Q,S$ (centers $q,s$); real foci ∘ (two on each circle); asymptote; collinear points $qrs$.

## References

- Stillwell, J. Mathematics and its history. In Undergraduate Texts in Mathematics; Springer: New York, NY, USA, 1989. [Google Scholar]
- Clebsch, A. Ueber einen Satz von Steiner und einige Punkte der Theorie der Curven dritter Ordnung. J. Reine Angew. Math.
**1864**, 63, 94–121. [Google Scholar] - Siebeck, F.H. Ueber eine Gattung von Curven vierten Grades, welche mit den elliptischen Functionen zusammenhängen. J. Reine Angew. Math.
**1860**, 57, 359–370. [Google Scholar] - Langer, J.C.; Singer, D.A. Confocal families of hyperbolic conics via quadratic differentials. Axioms
**2023**, 12, 507. [Google Scholar] [CrossRef] - Story, W.E. On Non-Euclidean Properties of Conics. Am. J. Math.
**1882**, 5, 358–381. [Google Scholar] [CrossRef] - Alperin, R.C. A mathematical theory of origami constructions and numbers. N. Y. J. Math.
**2000**, 6, 119–133. [Google Scholar] - Cox, D. Galois Theory; Wiley-Interscience: Hoboken, NJ, USA, 2004. [Google Scholar]
- Cox, D.A.; Shurman, J. Geometry and number theory on clovers. Am. Math. Mon.
**2005**, 112, 682–704. [Google Scholar] [CrossRef] - Prasolov, V.; Solovyev, Y. Elliptic functions and elliptic integrals. In Translations of Mathematical Monographs; Leites, D., Translator; American Mathematical Society: Providence, RI, USA, 1997; Volume 170. [Google Scholar]
- Rosen, M. Abel’s theorem on the lemniscate. Am. Math. Mon.
**1981**, 88, 387–395. [Google Scholar] [CrossRef] - Langer, J.C.; Singer, D.A. Subdividing the trefoil by origami. Geometry
**2013**, 2013, 897320. [Google Scholar] [CrossRef] - Langer, J.C.; Singer, D.A. The trefoil. Milan J. Math.
**2014**, 82, 161–182. [Google Scholar] [CrossRef] - Langer, J.C.; Singer, D.A. Flat curves. Mediterr. J. Math.
**2017**, 14, 236. [Google Scholar] [CrossRef] - Langer, J.C. On meromorphic parameterizations of real algebraic curves. J. Geom.
**2011**, 100, 105–128. [Google Scholar] [CrossRef] - Poncelet, J.-V. Traite i. Ann. Math. Pures Appl.
**1818**, 8, 20. [Google Scholar] - Struik, D. Lectures on Analytic and Projective Geometry; Addison-Wesley: Cambridge, UK, 1953. [Google Scholar]
- Plücker, J. Über solche puncte, die bei Curven einer höhern Ordnung als der zweiten den Brennpuncten der Kegelschnitte entsprechen. J. Reine Angew. Math.
**1833**, 10, 84–91. [Google Scholar] - Salmon, G. A Treatise on the Higher Plane Curves: Intended as a Sequel to “A Treatise on Conic Sections”, 3rd ed.; Chelsea Publishing Co.: New York, NY, USA, 1960. [Google Scholar]
- Hilton, H. Plane Algebraic Curves; Oxford University Press: London, UK, 1932. [Google Scholar]
- Coolidge, J.L. A Treatise on Algebraic Plane Curves; Dover Publications, Inc.: New York, NY, USA, 1959. [Google Scholar]
- Brieskorn, E.; Knörrer, H. Plane Algebraic Curves; Stillwell, J., Translator; Modern Birkhäuser Classics, Birkhäuser: Basel, Switzerland; Springer: Basel, Switzerland, 1986. [Google Scholar]
- Kirwan, F. Complex algebraic curves. In London Mathematical Society Student Texts; Cambridge University Press: Cambridge, UK, 1992; Volume 23. [Google Scholar]
- Kollár, J. Lectures on resolution of singularities. In Annals of Mathematics Studies; Princeton University Press: Princeton, NJ, USA, 2007; Volume 166. [Google Scholar]
- Miranda, R. Algebraic curves and Riemann surfaces. In Graduate Studies in Mathematics; American Mathematical Society: Providence, RI, USA, 1995; Volume 5. [Google Scholar]
- Langer, J.C.; Singer, D.A. Foci and foliations of real algebraic curves. Milan J. Math.
**2007**, 75, 225–271. [Google Scholar] [CrossRef] - Davis, P.J. The Schwarz Function and Its Applications; Mathematical Association of America: Buffalo, NY, USA, 1974; Volume 17. [Google Scholar]
- Franklin, F. On Some Applications of Circular Coordinates. Am. J. Math.
**1890**, 12, 161–190. [Google Scholar] [CrossRef] - Darboux, J.G. Note sur une classe de courbes du quatrième ordre et sur l’addition des fonctions elliptiques. Ann. Sci. L’ẽcole Norm. Sup.
**1867**, 4, 81–91. [Google Scholar] [CrossRef] - Edwards, H.M. A normal form for elliptic curves. Bull. Am. Math. Soc.
**2007**, 44, 393–422. [Google Scholar] [CrossRef] - Siegel, C.L. Topics in Complex Function Theory; Wiley-Interscience: New York, NY, USA, 1969; Volume I. [Google Scholar]
- Cayley, A. An Elementary Treatise on Elliptic Functions, 2nd ed.; Dover Publications, Inc.: New York, NY, USA, 1961. [Google Scholar]
- Bonifant, A.; Milnor, J. On real and complex cubic curves. L’Enseign. Math.
**2017**, 63, 21–61. [Google Scholar] [CrossRef] - Strebel, K. Quadratic differentials. In Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]; Springer: Berlin, Germany, 1984; Volume 5. [Google Scholar]
- Basset, A.B. An Elementary Treatise on Cubic and Quartic Curves; Deighton Bell and Co.: Cambridge, UK, 1901. [Google Scholar]
- Marden, M. The Geometry of the Zeros of a Polynomial in a Complex Variable; American Mathematical Society: New York, NY, USA, 1949; Volume 3. [Google Scholar]
- Siebeck, F.H. Ueber eine neue analytische Behandlungsweise der Brennpunkte. J. Reine Angew. Math.
**1865**, 64, 175–182. [Google Scholar] - Singer, D.A. The location of critical points of finite Blaschke products. Conform. Geom. Dyn.
**2006**, 10, 117–124. [Google Scholar] [CrossRef] - Siebeck, F.H. Ueber die graphische Darstellung imaginärer Funktionen. J. Reine Angew. Math.
**1858**, 55, 221–253. [Google Scholar] - Bowman, F. Introduction to Elliptic Functions with Applications; Dover: New York, NY, USA, 1961. [Google Scholar]
- Darboux, J.G. Remarques sur la théorie des surfaces orthogonales. Comptes Rendus Acad. Sci. Paris
**1864**, 59, 240–242. [Google Scholar] - Darboux, J.G. Sur les sections du tore. Nouv. Ann. Math.
**1864**, 3, 156–165. [Google Scholar] - Darboux, J.G. Théorèmes sur l’intersection d’une sphère et d’une surface du second degré. Nouv. Ann. Math.
**1864**, 3, 199–202. [Google Scholar] - Eisenhart, L. Darboux’s contribution to geometry. Bull. Am. Math. Soc.
**1918**, 24, 227–237. [Google Scholar] [CrossRef] - Sym, A. Darboux’s greatest love. J. Phys. A
**2009**, 42, 404001. [Google Scholar] [CrossRef] - Izmestiev, I. Spherical and hyperbolic conics. In Eighteen Essays in Non-Euclidean Geometry; IRMA Lectures in Mathematics and Theoretical Physics; European Mathematical Society: Zürich, Switzerland, 2019; Volume 29, pp. 263–320. [Google Scholar]
- Schilling, F. Die Brennpunktseigenschaften der eigentlichen Ellipse in der ebenen nichteuklidischen hyperbolischen Geometrie. Math. Ann.
**1950**, 121, 415–426. [Google Scholar] [CrossRef] - Veselov, A.P. Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space. J. Geom. Phys.
**1990**, 7, 81–107. [Google Scholar] [CrossRef] - Chasles, M. Construction géométric des amplitudes dans les fonctions elliptiques et propriétés nouvelles des sections coniques. Comptes Rendus Acad. Sci. Paris
**1844**, 19, 1239–1261. [Google Scholar] - Kummer, E. Ueber Systeme von Curven, welche einander überall rechtwinklig durchschneiden. J. Reine Angew. Math.
**1847**, 35, 5–12. [Google Scholar] - Bacon, C.L. The Cartesian Oval and the Elliptic Functions p and σ. Am. J. Math.
**1913**, 35, 261–280. [Google Scholar] [CrossRef] - Casey, J. On bicircular quartics. Trans. Royal Irish Acad.
**1871**, 24, 457–569. [Google Scholar] - Cayley, A. Note on the geometrical representation of the integral. Philos. Mag.
**1853**, 5, 281–284. [Google Scholar] [CrossRef] - Franklin, F. Note on the Double Periodicity of the Elliptic Functions. Am. J. Math.
**1889**, 11, 283–292. [Google Scholar] [CrossRef] - Franklin, F. On Confocal Bicircular Quartics. Am. J. Math.
**1890**, 12, 323–336. [Google Scholar] [CrossRef] - Greenhill, A.G. The Applications of Elliptic Functions; Macmillan: New York, NY, USA, 1892. [Google Scholar]
- Laguerre, E. Sur quelques applications de la géométrie au calcul intégral. Bull. Soc. Philomath.
**1867**, 35–40. Available online: https://gallica.bnf.fr/ark:/12148/bpt6k90212c/f44 (accessed on 13 August 2023). - Serret, J.-A. Propriétés géométriques relatives à la théorie des fonctions elliptiques. J. Math. Pures Appl.
**1843**, 8, 495–501. [Google Scholar] - Barbin, E.; Guitart, R. Algèbre des fonctions elliptiques et géométrie des ovales cartésiennes. Rev. Histoire Math.
**2001**, 7, 161–205. [Google Scholar] - Langer, J.C.; Singer, D.A. On the geometric mean of a pair of oriented, meromorphic foliations, Part I. Complex Anal. Synerg.
**2018**, 4, 4. [Google Scholar] [CrossRef] - Kunle, H.; Fladt, K. Erlanger program in higher geometry. In Fundamentals of Mathematics; MIT Press: Cambridge, UK, 1974; Volume II. [Google Scholar]

**Figure 1.**Figure from Siebeck’s 1860 paper [3]: three types of confocal families of bicircular quartics.

**Figure 2.**(

**Left**): The orthogonal (green/blue) foliations of the confocal family ${\mathcal{H}}_{\alpha}^{r}$ with $\alpha =0$, $r=-1$; the foci are the four red points. The (green/blue) bold curves are the two Cassinians in ${\mathcal{H}}_{0}^{-1}$. (

**Right**): The $p,q$-plane for $r=-1$. The region ${\widehat{\mathrm{P}}}_{0}^{2}=\{p\ne q\}$ is filled by hyperbolas ${\mathcal{H}}_{\alpha}$ representing confocal families for each $\alpha \in \mathbb{R}$. The (green/blue) points and their branches represent the Cassinians and corresponding foliations of ${\mathcal{H}}_{0}^{-1}$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Langer, J.C.; Singer, D.A.
Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form. *Axioms* **2023**, *12*, 870.
https://doi.org/10.3390/axioms12090870

**AMA Style**

Langer JC, Singer DA.
Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form. *Axioms*. 2023; 12(9):870.
https://doi.org/10.3390/axioms12090870

**Chicago/Turabian Style**

Langer, Joel C., and David A. Singer.
2023. "Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form" *Axioms* 12, no. 9: 870.
https://doi.org/10.3390/axioms12090870