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

Abstract

Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of $\widehat{\mathbb{C}}$ by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in $\mathbb{C}{\mathrm{P}}^2$ is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a parameterized Edwards transformation, providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form.

1. Introduction

The geometric representation of elliptic functions by algebraic plane curves was an important theme in nineteenth century mathematics. Out of a number of threads woven into this theme, we consider two types of elliptic curve parameterization, whose relationship to each other is not obvious. We spell out the connection between the two using quadratic differentials, the birational equivalence of curves, and certain bicircular quartics viewed as a generalization of the Edwards normal form.

For the elementary sine function $x=sint$, the two types of parameterization are almost equally well known:

(A) Since $x=sint$ satisfies the ODE ${\left(\frac{dx}{dt}\right)}^2=g\left(x\right)=1-x^2$, the functions x and $y=x^{\prime }$ together parameterize the circle $y^2=g\left(x\right)$.

(B) The complex function $z=sin(s+it)$ parameterizes ellipses $s\mapsto sin(s+i{t}_0)$ and hyperbolas $t\mapsto sin(s_0+it)$ in $\mathbb{C}\simeq {\mathbb{R}}^2$ with common foci at the critical values of sine—i.e., the zeros of $g\left(z\right)$, $z=\pm 1$.

The most obvious difference between A and B is this: while A parameterizes a single curve, B parameterizes curve families. In fact, the ellipses and hyperbolas in B evidently form an orthogonal system. That they are also confocal is part of what deserves further geometric explanation, though it is often presented as a computational fact: one applies trig identities to $x+iy=sin(s+it)$ to eliminate the parameter s or t and obtain quadric curves $f(x,y)=0$, from which the foci $(x,y)=(\pm 1,0)$ may be read.

Turning to elliptic functions, the story of A is by far the more famous nowadays: “Replace $g\left(x\right)=1-x^2$ in the above ODE by a cubic or quartic $g\left(x\right)$ with distinct roots, use x and $y=x^{\prime }$ to parameterize the elliptic curve $y^2=g\left(x\right)$, …”. Such an idea (if not yet the modern notion of elliptic curve) would have been implicit in early work on the theory of elliptic functions, but according to Stillwell [1], the idea was first explicitly discussed by Clebsch in 1864 [2].

The corresponding curve parameterizations B appear in the 1860 paper of F. H. Siebeck [3]: “On a family of curves of the fourth degree which are related to elliptic functions”. Here, among other things, Siebeck demonstrates that the Jacobi elliptic sine and cosine functions $\mathrm{sn}z=\mathrm{sn}(z,k)$ and $\mathrm{cn}z=\mathrm{cn}(z,k)$ map horizontal and vertical lines to confocal orthogonal systems of quartic plane curves with four (real) foci at the critical values of the elliptic function. Siebeck’s eye-catching discovery is illustrated in Figure 1, which is reproduced from his paper.

Siebeck himself began with the analogy to Euclidean conics, yet he was unaware that his bicircular quartics actually represented non-Euclidean conics: In fact, all three graphics in Figure 1 may be viewed as confocal families of hyperbolic conics [4]. Although spherical conics had already been considered by Chasles (and subsequently Darboux), and there was also a more comprehensive treatment of non-Euclidean conics by 1882 [5], it might have been challenging even then to make a fully satisfactory connection to Siebeck’s work. (Further comments and historical references are included in Appendix D, where we also briefly summarize the contents of [3], which is in German.)

A large part of the issue here is that there are several different natural contexts within which all such Euclidean and non-Euclidean conics may be viewed. The notion of focus itself takes on various—not always equivalent—meanings, depending on the setting. For this reason, we devote two sections to the topic. In Section 2, we have the audacity to give a new definition of focus; it suits the present purposes at least. In Appendix C, we provide the historical context and broader justification.

Yet there is no escaping the fact that the relationship $A\leftrightarrow B$ itself calls for descriptions of relevant objects in two settings at once. We first consider bicircular quartics as real curves in the complex projective plane $\mathbb{C}{\mathrm{P}}^2$—which is the setting of B (that of A being essentially the Riemann sphere $\widehat{\mathbb{C}}\simeq \mathbb{C}{\mathrm{P}}^1$, where the ODE lives). The classical theory of bicircular quartics is a rich subject, and we require several sections (including appendices) to provide sufficient details (Section 3 and Section 4, Appendix A, Appendix B and Appendix E).

This is the case even though Section 3 essentially short-circuits much of the theory; namely, a three-dimensional real projective space of bicircular quartics $P^3\subset P^8$ suffices to represent all bicircular quartics, up to inversive equivalence (as explained in Appendix E). Within $P^3$, we eventually find all the relevant orthogonal families, including the three types in Figure 1 (as well as the Euclidean ellipses/hyperbolas—so Siebeck’s analogy is really more than that).

A simple scaling puts any elliptic curve in $P^3$ in standard position—which fixes the set of four mirrors of inversion symmetry which any such curve is known to possess. (This is not Siebeck’s normalization, which is imposed by the Jacobi elliptic functions, but is essentially the generalized Edwards normal form of Section 6.)

Standard position (SP) sets up the section on quadratic differentials Section 5, where we begin to establish the bridge $A\leftrightarrow B$. Each elliptic curve k in SP determines a focal quadratic differential $Q_{\left\{k\right\}}$, defined on $\widehat{\mathbb{C}}$, with poles at its foci. Meanwhile, the clinant quadratic differential $Q_k$ is defined on k as an elliptic curve. Despite being defined on different Riemann surfaces, $Q_k$ and $Q_{\left\{k\right\}}$ are found to “agree” (along the real locus of k), by virtue of a key identity (Lemma 1). Now the point is that all curves in a given confocal family have the same focal quadratic differential $Q_{\left\{k\right\}}$, which effectively “glues together” all the $Q_k$’s. But in $\widehat{\mathbb{C}}$, the orthogonal family is simply the pair of foliations defined by the horizontal and vertical trajectories of $Q_{\left\{k\right\}}$.

Our use of $Q_k$ may not be essential here, but it has a certain geometric appeal. In effect, we are introducing yet a third classical curve parameterization into the mix (see Remark 1): (C) Arc length parameterization. Namely, $Q_k=d{x}^2+d{y}^2$ may be described as (an analytic continuation of) the usual arc length element along $k_{\mathbb{R}}$, and among its trajectories is the arc-length parameterization of $k_{\mathbb{R}}$ itself. $Q_k$ is intimately related to our subject, already because it has singularities corresponding to the foci of k.

Remark 1. 

Among the geometric representations of elliptic functions, C played perhaps the most central role historically [1]. It was the lemniscate integral that led Euler and Gauss to key insights about elliptic integrals and functions, and Abel to his subdivisibility result for the lemniscate, analogous to Gauss’s result for the circle. The latter results involve ruler and compass constructibility and Fermat primes; some modern analogues involve origami constructibility and Pierpont primes [6,7,8,9,10]. In particular, the Kiepert trefoil admits such a result [11], and the related fact that this exceptional sextic curve has unit speed parameterization by Dixon elliptic functions [12] is explained in [13] using the clinant quadratic differential $Q_k$.

We note that arc length parameterization is almost never defined by elliptic (or elementary) functions—precisely because the analytic continuation of the arc length parameterization of $k_{\mathbb{R}}$ develops branch points at singularities of $Q_k$ [13,14]. A rare exception is provided by the circle (where C happens to coincide with A in the case of $sint$). Another is the Bernoulli lemniscate, which may be parameterized by unit speed via the lemniscate elliptic sine function, $\mathrm{sl}t=\mathrm{sn}(t,i)$.

The Bernoulli lemniscate leads into the subject of Section 6 and Section 7. Each confocal family in Figure 1 contains a Cassini oval—the lemniscate being the Cassinian in the bottom (trinodal) family. Quartics in the Edwards normal form may be interpreted as Cassinians (simply by replacing the affine coordinates $x,y$ in the form by isotropic coordinates $u=x+iy,v=x-iy$). The confocal families may be regarded as variations from Cassinian core curves (Section 7).

Such a variation is what provides the generalized Edwards normal form, where a key element of the Edwards theory carries over by means of a simple generalization (using again Lemma 1). Namely, where Edwards uses a birational transformation $\mathcal{E}$ to relate any ENF quartic to $z^2=(a^2-x^2)(1-a^2{x}^2)$, a parameterized version of the same transformation, ${\mathcal{E}}_{p,q}$, relates all bicircular quartics in a given confocal family to (a minor modification of) the same special quartic. But the latter equation essentially stands in for the confocal quadratic differential $Q_{\left\{k\right\}}$, so ${\mathcal{E}}_{p,q}$ provides an explicit algebraic integrability result for the ODE defining trajectories of $Q_{\left\{k\right\}}$.

Further, although it is unsurprising that a confocal family is made up of equivalent elliptic curves—and there are several ways of seeing this—it follows from ${\mathcal{E}}_{p,q}$ that any two curves in the family are related by a rather simple birational transformation. After observations based on this fact, the discussion wraps up with a summary of the global geometric picture: $P^3$ is stratified via a discriminant $\mathsf{\Delta }k$, whose highest dimensional strata are foliated by confocal families; roughly speaking, the foci fix the family, while the movable pair of singular foci provide the variation through equivalent elliptic curves. To conclude with a metaphor, if $P^3$ is a little galaxy, and foci are slowly drifting stars, the singular foci are asteroids in hyperbolic orbit.

2. Foci of Real Algebraic Plane Curves

If $f(x,y)=\sum c_{\mu \nu }{x}^{\mu }{y}^{\nu }$ is a polynomial with $c_{\mu \nu }\in \mathbb{R}$, then f determines a real algebraic plane curve as the set of solutions to $f(x,y)=0$ in ${\mathbb{R}}^2$ or ${\mathbb{C}}^2$. This may be extended to complex projective space $\mathbb{C}{P}^2$ as the set of solutions to $F(x,y,z)=0$, where $F(x,y,z)=z^{d}f(\frac{x}{z},\frac{y}{z})$ is a homogeneous polynomial. Given a homogeneous real polynomial $F(x,y,z)$, we can of course recover $f(x,y)=F(x,y,1)$. Letting $\overline{f}(x,y)=\overline{f(\overline{x},\overline{y})}=\sum \overline{c_{\mu \nu }}{x}^{\mu }{y}^{\nu }$ denote the complex conjugation of coefficients, we have $\overline{f}=f$ as the reality condition for the curve.

We will also make use of isotropic coordinates $u=x+iy,v=x-iy,w=z$. In these coordinates, the curve is defined by the polynomial $K(u,v,w):=F(\frac{u+v}{2},\frac{u-v}{2i},w)$, or $k(u,v)=K(u,v,1)$. In these coordinates, reality is given by the condition $k(u,v)=\overline{k}(v,u)$.

In what follows, F and f will be used for real curves in ordinary (rectangular) coordinates $(x,y,z)$, and k and K for real curves in isotropic coordinates $(u,v,w)$.

The circular points $I=(1:i:0)$ and $J=(1:-i:0)$ in $\mathbb{C}{P}^2$ are the ideal points which belong to all circles ${(x-x_{0}z)}^2+{(y-y_{0}z)}^2=R^2{z}^2$. A line $ax+by+cz=0$ in $\mathbb{C}{\mathrm{P}}^2$ is isotropic if it contains exactly one of the two points I or J. An isotropic tangent line is a line through I or J tangent to the curve F.

Poncelet was the first to interpret foci of a conic as points of intersection of isotropic tangent lines [15,16]; Plücker used the same idea to define (real and complex) foci for curves of a higher degree [17]. Subsequently, classical texts considered infinite foci [18] and singular foci [19] but generally distinguished the latter from ordinary foci.

Coolidge simply excludes infinite and singular foci by definition [20]: “Definition. A point of intersection of tangents to a curve from the two circular points at infinity, the points of contact being both finite, is called a ‘focus’ of the curve”. He adds that it “would not be wise” to include singular foci, which fail to behave like ordinary foci under inversion (e.g., circle centers do not invert to circle centers).

But a singular focus may be double, triple, etc.; of these, only the double singular foci ought to be excluded. For instance, within confocal families of bicircular quartics sharing four foci, one finds Cassinians—which have two ordinary, and two triple (singular) foci. This only makes sense when the latter two points count as foci.

As just described, the needed definition might appear to be rather ad hoc—but it is not. In fact, we adopt an entirely different definition of the (real) foci of a curve, thus resolving all ambiguities in the classical notion (see Appendix C).

We use the fact that a $\mathbb{C}$-irreducible curve $f(x,y)=0$ may be regarded as a compact Riemann surface $\mathcal{K}={\mathcal{K}}_f$—extending the analytic submanifold structure on the regular set $f_{\mathrm{reg}}\subset {\mathbb{C}}^2$ by the addition of finitely many points. (This amounts to the resolution of singularities [21,22,23]; but for all curves arising here, elementary versions of the result suffice [24].)

Isotropic coordinates define isotropic projections ${\pi }_I,{\pi }_J:{\mathbb{C}}^2\to \mathbb{C}$ by ${\pi }_I(u,v)=u$ and ${\pi }_J(u,v)=v$. These restrict to analytic functions on $f_{\mathrm{reg}}$, and extend meromorphically to ${\pi }_I,{\pi }_J:\mathcal{K}\to \widehat{\mathbb{C}}=\mathbb{C}{\mathrm{P}}^1$. In particular, we consider $\pi ={\pi }_I$ as a branched covering of Riemann surfaces. A ramification point $P\in \mathcal{K}$ is a point of higher multiplicity; a branch point $u_0=\pi \left(P\right)\in \widehat{\mathbb{C}}$ is the image of such a point (usage as in [24]). In other words, $|{\pi }^{-1}\left(u_0\right)|<\mathrm{deg}\left(\pi \right)$, where $|{\pi }^{-1}\left(u_0\right)|$ is the number of distinct points in the inverse image and $\mathrm{deg}\left(\pi \right)={\max }_u\left|{\pi }^{-1}\left(u\right)\right|$ is the degree of the mapping $\pi :\mathcal{K}\to \widehat{\mathbb{C}}$.

Definition 1. 

A focus $u_0\in \widehat{\mathbb{C}}$ of a $\mathbb{C}$-irreducible real curve f is a branch point of extended isotropic projection $\pi :\mathcal{K}\to \widehat{\mathbb{C}}$. If f is reducible, its focal set is the union $\mathrm{foc}\left(f\right)={\cup }_j\mathrm{foc}\left(f_j\right)$.

In particular, the natural place for real foci will not be the real plane, but $\widehat{\mathbb{C}}$; all the more so for foci as poles of quadratic differentials (Section 5). In the meantime, this section presents some preliminaries, including a formula for the number of foci. The focal set itself is considered in Section 3.

Because $I,J$ are swapped by the real symmetry of $\mathbb{C}{\mathrm{P}}^2$, $(x:y:z)\leftrightarrow (\overline{x}:\overline{y}:\overline{z})$, it will suffice to discuss mostly I. An isotropic line L through I has the equation $x+iy=u_{0}z$ for some $u_0\in \mathbb{C}$; using standard identifications $\mathbb{C}\simeq {\mathbb{R}}^2\simeq \{(x:y:1):x,y\in \mathbb{R}\}\subset \mathbb{C}{\mathrm{P}}^2$, the complex number $u_0=x_0+i{y}_0$ is understood to represent the point $(x_0:y_0:1)$ where L meets the real plane. (For heuristic purposes, we sometimes regard the ideal line $\overline{IJ}:z=0$ as isotropic, but it does not represent a point in the real plane $\mathbb{R}{\mathrm{P}}^2$.)

In homogeneous isotropic coordinates $u=x+iy,v=x-iy,w=z$ on $\mathbb{C}{\mathrm{P}}^2$, the line L has the equation $u=u_{0}w$, but we will mostly use the non-homogeneous equation $u=u_0$ for the affine line $L\setminus \left\{I\right\}$. Isotropic projection $\pi ={\pi }_I$ (“from I onto the real plane”) sends each point of $L\setminus \left\{I\right\}$ to $u_0$.

The Riemann–Hurwitz formula for a branched covering $\pi :\mathcal{K}\to \mathcal{L}$ of compact Riemann surfaces of genus $g\left(\mathcal{K}\right)$ and $g\left(\mathcal{L}\right)$ is

$$g\left(\mathcal{K}\right)=1+\mathrm{deg}\left(\pi \right)(g\left(\mathcal{L}\right)-1)+\frac{1}{2}\sum b_P\left(\pi \right).$$

Here, $b_P\left(\pi \right):=({\mathrm{mult}}_P\left(\pi \right)-1)$ denotes the branching number of $\pi$ at P [24] (so $b_P\left(\pi \right)=0$ for all but finitely many points, $b_P\left(\pi \right)=1$ for simple ramification points, etc.).

The total branching number of $\pi$ is denoted $\mathcal{B}\left(\pi \right):=\sum b_P\left(\pi \right)$. In the case of a meromorphic function, $\pi :\mathcal{K}\to \widehat{\mathbb{C}}$, solving for $\mathcal{B}$ in the Riemann–Hurwitz formula with $g\left(\mathcal{L}\right)=g\left(\widehat{\mathbb{C}}\right)=0$ gives $\mathcal{B}\left(\pi \right)=2\left(\mathrm{deg}\right(\pi )+g(\mathcal{K})-1)$.

Returning to algebraic curves, Definition 1 identifies the foci of $0=f(x,y)=k(u,v)$ with branch points of $\pi :\mathcal{K}\to \widehat{\mathbb{C}}$—projection from the circular point I. Then $\mathcal{B}\left(\pi \right)$ is the number of foci (counted with multiplicity), which the above formula may therefore be used to determine.

For this purpose, we consider the multiplicity $m={\mu }_I\left(K\right)$ of I with respect to the corresponding projective curve K, e.g., K is circular when $m\ge 1$ and bicircular when $m\ge 2$. Then, the projection $\pi$ has degree $\mathrm{deg}\left(\pi \right)={\max }_u\left|{\pi }^{-1}\left(u\right)\right|=\mathrm{deg}\left(K\right)-{\mu }_I\left(K\right)=d-m$.

Putting $\mathrm{deg}\left(\pi \right)=\mathrm{deg}\left(K\right)-{\mu }_I\left(K\right)$ and $g\left(K\right):=g\left(\mathcal{K}\right)$ into the formula for $\mathcal{B}\left(\pi \right)$ gives the number of foci of K:

$$\left|\mathrm{foci}\left(K\right)\right|=B=2(\mathrm{deg}\left(K\right)-{\mu }_I\left(K\right)+g\left(K\right)-1).$$

(1)

In our applications, $\left|\mathrm{foci}\right(K\left)\right|$ will in fact count distinct foci $u_0$ since these will always be simple: $b_P\left(\pi \right)=0$, for all but one point in ${\pi }^{-1}\left(u_0\right)$, and for that point, $b_{P_0}\left(\pi \right)=1$. (Note: the classical term simple focus has a different meaning as is explained in Appendix C.)

To begin with the main example, let K be a bicircular quartic: K is an irreducible real quartic with double points at I and J. In particular, suppose K has $\delta =2$ nodes or $\kappa =2$ cusps, and no other singularities. Then, by the Clebsch formula [21], the curve has genus $g\left(K\right)=\frac{1}{2}(d-1)(d-2)-\delta -\kappa =1$. Thus, K is an elliptic curve. Since ${\mu }_I\left(K\right)=2$, Equation (1) gives $\left|\mathrm{foci}\left(K\right)\right|=2(g\left(K\right)+\mathrm{deg}\left(K\right)-{\mu }_I\left(K\right)-1)=2(1+4-2-1)=4$. Thus, we have the following.

Proposition 1. 

A bicircular quartic has four foci.

Aside from the elliptic curves, we will also have occasion to consider rational bicircular quartics. For example, the Bernoulli lemniscate B is trinodal ($\delta =3$), giving $g=0$ and $\left|\mathrm{foci}\right(B\left)\right|=2(0+4-2-1)=2$. In fact, B is inverse to a (rectangular) hyperbola H, which satisfies $\left|\mathrm{foci}\right(H\left)\right|=2(0+2-0-1)=2$. The fact that $\left|\mathrm{foci}\right(H\left)\right|=\left|\mathrm{foci}\right(B\left)\right|$ is an instance of inversive invariance, to be discussed below.

3. Bicircular Quartics: Focal and Elliptic Discriminants

A bicircular quartic takes the form

$$F(x,y,z)=a{(x^2+y^2)}^2+(bx+cy)(x^2+y^2)z+F_2(x,y,z)z^2=0,$$

with $F_2(x,y,z)$ homogeneous of degree 2. It depends on nine homogeneous coefficients $a,b,c,\cdots \in \mathbb{R}$, and may thus be represented by a point in eight-dimensional projective space $\{a:b:c:\cdots \}\in {\mathrm{P}}^8$.

However, every bicircular quartic is symmetric with respect to a pair of real, mutually orthogonal circles of inversion (or lines of reflection). For brevity, we refer to these as mirrors. In fact, the following result is classical (see, for example [19]):

Theorem 1. 

A bicircular quartic (or circular cubic) is self-inverse with respect to each of four mutually orthogonal circles or lines, and the sixteen (real and complex) foci lie by fours on these four circles. At least two and at most three of these circles are real.

The proof may be found in Appendix E.

Thus, for the purpose of studying confocal families up to inversive equivalence, it will suffice to consider such curves in rectilinear position (symmetric with respect to the $x,y$-axes):

$$F(x,y,z)=s{(x^2+y^2)}^2+2(p{x}^2+q{y}^2)z^2+r{z}^4=0.$$

(2)

In isotropic coordinates, $K(u,v,w)=F(x,y,z)=F(\frac{u+v}{2},\frac{u-v}{2i},w)$:

$$K(u,v,w)=s{u}^2{v}^2+\frac{p-q}{2}(u^2+v^2)w^2+(p+q)uv{w}^2+r{w}^4=0.$$

(3)

Thus, we may represent a bicircular quartic K as a point in the real projective space ${\mathrm{P}}^3=\left\{(p:q:r:s)\right\}$ of rectilinear positions.

Now, we begin to analyze the foci of K. Like any bicircular quartic, K is quadratic in either of the first two isotropic coordinates $u,v$, which simply reflects the fact that isotropic projection $\pi :\mathcal{K}\to \mathbb{C}$ has degree two. In particular, the affine curve $k(u,v)=K(u,v,1)=0$ may be expressed in the form $k=A\left(u\right)v^2+B\left(u\right)v+C\left(u\right)$:

$$k(u,v)=(s{u}^2+\frac{p-q}{2})v^2+(p+q)uv+(\frac{p-q}{2}{u}^2+r).$$

(4)

A point $u_0\in \mathbb{C}$ is regular if $k(u_0,v)$ has two distinct roots. Otherwise, $u_0$ is exceptional: the number of distinct solutions to the equation $k(u_0,v)=0$ is 0 or 1 (or ∞ in case $k(u,v)$ has a linear component $u-u_0$). A focus $u_0\in \mathbb{C}$ is exceptional, but the converse is false in general (as will be seen below).

To identify exceptional points, we consider the v-discriminant of k:

$${\mathsf{\Delta }}_{v}k=\left({\mathsf{\Delta }}_{v}k\right)\left(u\right):=B^2-4AC=2(q-p)(s{u}^4+r)+4(pq-rs)u^2,$$

(5)

We will also refer to ${\mathsf{\Delta }}_{v}k$ as the focal discriminant (see Proposition 2). In the generic, elliptic case, $\pi :\mathcal{K}\to \widehat{\mathbb{C}}$ is a branched double cover with four branch points, i.e., foci, and these are the roots of the quartic ${\mathsf{\Delta }}_{v}k$.

To distinguish elliptic cases ($\left|\mathrm{foci}\right(K\left)\right|=4$) from non-elliptic cases ($\left|\mathrm{foci}\right(K\left)\right|<4$), we take the discriminant ${\mathsf{\Delta }}_u$ of the quartic ${\mathsf{\Delta }}_{v}k$:

$$\mathsf{\Delta }K=\mathsf{\Delta }k:=\frac{1}{16384}{\mathsf{\Delta }}_u\left({\mathsf{\Delta }}_{v}k\right)=rs{(p-q)}^2{(p^2-rs)}^2{(q^2-rs)}^2.$$

(6)

This elliptic discriminant gives a natural stratification ${\mathrm{P}}^3=\cup {\mathrm{P}}_j^k$. (Note: we use “elliptic” to distinguish $\mathsf{\Delta }k$ from the usual curve discriminant—which vanishes for any singular quartic; in particular, for genus $g<3$.) The discriminant locus $\mathcal{D}:\mathsf{\Delta }K=0$ consists of lower-dimensional strata defined via factors of $\mathsf{\Delta }K$. The 3-strata ${\mathrm{P}}_j^3$—the four connected components of the complement ${\mathrm{P}}^3\setminus \mathcal{D}={\cup }_{j=0}^3{\mathrm{P}}_j^3$—are foliated by hyperbolas representing confocal families of equivalent elliptic curves (see Section 4 and Appendix B). Several of the 2-strata ${\mathrm{P}}_j^2$ are likewise foliated by hyperbolas representing confocal families of rational (trinodal) quartics and conics, while others represent circle pairs (Appendix A). Finally, the circles/lines of reflection for the bicircular quartics in ${\mathrm{P}}^3$ (Appendix E) belong to the $1/0$-strata (albeit with $p=q=\pm i$ in one case).

To summarize the significance of the discriminants, we state a proposition (proved in Appendix A):

Proposition 2. 

Let k be a quartic in rectilinear position (Equation (4)), with focal discriminant ${\mathsf{\Delta }}_{v}k$ (Equation (5)) and elliptic discriminant $\mathsf{\Delta }k$ (Equation (6)).

(a) $u_0\in \mathbb{C}$ is a focus of k if and only if it is a simple root of ${\mathsf{\Delta }}_{v}k$;

(b) k is an elliptic curve $\iff \mathsf{\Delta }k\ne 0\iff \left|\mathrm{foci}\right(K\left)\right|=4$;

(c) k is rational-non-circular $\iff \left|\mathrm{foci}\right(K\left)\right|=2$;

(d) k is circular or degenerate $\iff \left|\mathrm{foci}\right(K\left)\right|=0$.

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

In this section, we consider the confocal families of curves represented by points in the space ${\mathrm{P}}^3$ of rectilinear positions. For this purpose, we replace the elliptic strata ${\mathrm{P}}_j^3$ by reduced strata ${\widehat{\mathrm{P}}}_j^2$ obtained simply as follows. Since $s\ne 0$ and $r\ne 0$, we may scale coefficients using $s=1$, then scale the affine curves $k(u,v)=0$ (Equation (4)), via $(u,v)\mapsto (\lambda u,\lambda v)$ so that $r=\pm 1$. With this scaling, $f(x,y)=k(u,v)$ is said to be in standard position.

Remark 2. 

Standard position has the following geometric significance. All bicircular quartics are known to be symmetric with respect to four mirrors—circles of inversion (Appendix E). While rectilinear position makes two of these mirrors the x and y axes, standard position fixes the remaining two: for $r=1$, the (real/imaginary) circles $x^2+y^2=\pm 1$; for $r=-1$, the (complex) circles $x^2+y^2=\pm i$.

The reduced strata ${\widehat{\mathrm{P}}}_j^2$ are open subsets of one of the $p,q$-planes ${\mathbb{R}}_r^2:=\left\{\right(p:q:r:1\left)\right\}$, $r=\pm 1$. As will be seen presently, each ${\widehat{\mathrm{P}}}_j^2$ is foliated by hyperbolas ${\mathcal{H}}_{\alpha }$ (Equation (10)); for a fixed value of a parameter $\alpha$, which determines the four foci, ${\mathcal{H}}_{\alpha }$ represents a confocal family of curves. In fact, $\alpha$ is related to the j-invariant for K (Remark 3), and ${\mathcal{H}}_{\alpha }$ consists of equivalent elliptic curves (Section 6). ${\mathcal{H}}_{\alpha }$ itself is “parameterized” by the squared singular focus ${\sigma }^2=\frac{q-p}{2}$ (the singular foci $\pm \sigma$ is discussed in Appendix A).

Thus, we express the standard position:

$$k(u,v)=(u^2-{\sigma }^2)v^2+(p+q)uv+(r-{\sigma }^2{u}^2);{\sigma }^2=\frac{q-p}{2}.$$

(7)

We “normalize” the focal discriminant ${\mathsf{\Delta }}_{v}k$ (Equation (5)) by factoring out $4{\sigma }^2$. Noting that $p\ne q$ on ${\widehat{\mathrm{P}}}_j^2$, we obtain the monic polynomial:

$$\delta \left(u\right)={\delta }_{\alpha }^r\left(u\right):=\frac{{\mathsf{\Delta }}_{v}k}{4{\sigma }^2}=(u^2-b^2)(u^2-\frac{r}{b^2})=u^4-2\alpha u^2+r.$$

(8)

Here, the foci (roots of $\delta \left(u\right)$) are denoted $\pm b,\pm \frac{\sqrt{r}}{b}$ and we introduce the focal parameter $\alpha$:

$$\alpha :=\frac{pq-r}{p-q}=\frac{1}{2}(b^2+\frac{r}{b^2}).$$

(9)

The foci are distinct from the singular foci $\pm \sigma$ (except in the case of Cassinians, where $q=-p={\sigma }^2$ and $\alpha =\frac{1}{2}({\sigma }^2+\frac{r}{{\sigma }^2})$). Finally, we note that these formulas are meaningful even when $r=0$, and will thus include the trinodal quartics as a limiting case.

Remark 3. 

The focal parameter α is related to the cross ratio of the four foci (or the “pencil of isotropic tangents”). Namely, $\lambda =[p_0,p_1,p_2,p_3]:=\frac{(p_0-p_1)(p_2-p_3)}{(p_1-p_2)(p_3-p_0)}$, becmes $\lambda =[b,-\frac{\sqrt{r}}{b},\frac{\sqrt{r}}{b},-b]=\frac{\alpha }{2\sqrt{r}}+\frac{1}{2}$, i.e., $\alpha =\sqrt{r}(2\lambda -1)$. Ultimately, this implies equivalence of confocal bicircular quartics, since λ may be shown to determine the j-invariant of k as an elliptic curve. But we do not require the j-invariant for this conclusion since we have more direct arguments in Section 6.

We formally define elliptic confocal families ${\mathcal{H}}_{\alpha }^r\subset {\mathrm{R}}_r^2$ of a given type $r=\pm 1$ by fixing the focal parameter $\alpha$, while allowing $p,q$ to vary.

Definition 2. 

For $r=1$ and real $\alpha \ne \pm 1$, or for $r=-1,\alpha \in \mathbb{R}$, we define ${\mathcal{H}}_{\alpha }^r=\{K=(p:q:r:1)\in {\mathrm{P}}^3:p\ne q,\frac{pq-r}{p-q}=\alpha \}$.

For a given $\alpha =\frac{pq-r}{p-q}$, we see that indeed $\mathcal{H}={\mathcal{H}}_{\alpha }^r$ consists of points $(p,q)$ on a rectangular hyperbola in the $p,q$-plane ${\mathrm{R}}_r^2$:

$$\mathcal{H}:(p+\alpha )(q-\alpha )+{\alpha }^2-r=0.$$

(10)

For variable $\alpha$, $\mathcal{H}$ in fact describes a pencil of hyperbolas, each of which represents a confocal family of elliptic curves.

In case $r=-1$, each quartic is one-circuited, and the mirrors are the x and y axes and the complex circles $x^2+y^2=\pm i$ (Appendix E). The left side of Figure 2 shows the orthogonal pair of foliations for the particular choice $\alpha =0$.

On the right side of the figure, the complement of the diagonal in the $p,q$-plane is foliated by the hyperbolas ${\mathcal{H}}_{\alpha },\alpha \in \mathbb{R}.$ The upper branch of ${\mathcal{H}}_{\alpha }$ represents all the (blue) curves enclosing the foci $\pm b$; the lower branch represents (green) curves enclosing the foci $\pm ib$. Points on the anti-diagonal $p=-q$ represent Cassinians. These play a distinguished role in Section 6 and Section 7; each represents the “half-way” curve in its family (see Remark 8).

Remark 4. 

It is easy to explain why two curves in the confocal family pass through a general point $(x_0,y_0)\in {\mathbb{R}}^2$. (Orthogonality will follow from Theorem 2.) Consider the line in the $p,q$-plane:

$${\mathcal{L}}_{(x_0,y_0)}:f(x_0,y_0)={(x_0^2+y_0^2)}^2+2x_0^{2}p+2y_0^{2}q+r=0.$$

(11)

Since ${\mathcal{L}}_{(x_0,y_0)}$ is a line of negative slope (for $x_0,y_0\ne 0$), and the hyperbola $\mathcal{H}$ has a positive slope, there is a point of intersection on each branch. (For points on the x or y axes, the horizontal/vertical lines ${\mathcal{L}}_{(x_0,y_0)}$ meet $\mathcal{H}$ in only one finite point, and determine just one irreducible quartic; the other quartic is either $y^2{z}^2$ or $x^2{z}^2$.)

Remark 5. 

The confocal family plotted in the figure is atypical of the case $r=-1$ in one respect; it belongs to the exceptional case $\alpha =0$, and consequently the foci $\pm 1,\pm i$ happen to be concyclic (with the harmonic cross ratio). The other values of α result in a rhombic focal set. We note that the case $r=1$ also includes a confocal family with foci $\pm 1,\pm i$ (but its curves are two-circuited). Since the picture is more complicated to describe in detail for $r=1$, we defer this discussion to Appendix B.

5. Bicircular Quartics and Quadratic Differentials

Though the constructions of this section apply, in principle, to bicircular quartics in general, we usually assume a standard (or at least rectilinear) position. We define the focal quadratic differential of $k(u,v)$,

$$Q_{\left\{k\right\}}=Q_{\alpha }^r=\frac{d{u}^2}{\delta \left(u\right)}=\frac{d{u}^2}{u^4-2\alpha u^2+r}=\frac{d{u}^2}{(u^2-b^2)(u^2-\frac{r}{b^2})},$$

(12)

a quadratic differential on $\widehat{\mathbb{C}}$, whose horizontal/vertical trajectories are parameterized by solutions to the ODE: ${\left(\frac{du}{dt}\right)}^2=\pm \delta \left(u\right)$.

Since the poles of $Q_{\left\{k\right\}}$ are exactly the foci of the confocal family ${\mathcal{H}}_{\alpha }^r$, it is already plausible that such trajectories of $Q_{\alpha }^r$ represent the two confocal foliations defining ${\mathcal{H}}_{\alpha }^r$. Of course, we depend on the identification of the real plane ${\mathbb{R}}^2=\left\{(x,y)\right\}$ with $\mathbb{C}=\{u=x+iy\}$ to relate the curves in the two settings. To be quite precise, we make a distinction between the confocal families $\mathcal{H}$ in $\mathbb{R}{\mathrm{P}}^2$ and the corresponding orthogonal families $\mathcal{F}={\mathcal{F}}_{\alpha }^r$ in $\widehat{\mathbb{C}}$, which differ also in the following respect. In order for $\mathcal{F}$ to define a pair of foliations on the complement of the focal set in $\widehat{\mathbb{C}}$, the leaves of $\mathcal{F}$ are understood to include the real mirrors of $\mathcal{H}$ (or arcs between foci on such the circles/lines). The notion of the orthogonal family is discussed in greater detail in Appendix B, where the case $r=1$ more fully illustrates the difference between $\mathcal{H}$ and $\mathcal{F}$.

The divide between the two settings $\mathbb{R}{\mathrm{P}}^2$ (or $\mathbb{C}{\mathrm{P}}^2$) and $\widehat{\mathbb{C}}$ is undoubtedly reflected in the difficulty of identifying curves in $\mathcal{H}$ with those in $\mathcal{F}$ by direct computation. For instance, one may integrate the ODE for trajectories $u\left(t\right)=x\left(t\right)+iy\left(t\right)$ of the quadratic differential $Q_{\alpha }^r$ in terms of elliptic functions, but the process of eliminating t to obtain quartics $f(x,y)$ (or $k(u,v)$) requires extensive computations involving various elliptic function identities (as in [3]). But since the ODE and the required implicit solution are both specified by polynomials—$\delta \left(u\right)$ and $k(u,v)$—one may naturally seek to bypass the elliptic functions altogether.

This is, in fact, not hard to do, with the help of quadratic differentials. For this purpose, we first introduce an auxiliary quadratic differential $Q_k$ defined on a given real, irreducible curve $f(x,y)=k(u,v)=0$. Heuristically, $Q_k$ may be obtained via the analytic continuation of the real arc length element $d{s}^2=d{x}^2+d{y}^2=dudv$ along the curve in ${\mathbb{R}}^2$ to the complex curve $\Gamma \subset {\mathbb{C}}^2$. Along a real arc $\gamma \subset \Gamma \cap {\mathbb{R}}^2$, $Q_k$ is positive, i.e., $\gamma$ is horizontal.

More formally, isotropic coordinates $u=x+iy,v=x-iy$ on $k_{\mathrm{reg}}$ extend to meromorphic functions on the Riemann surface $\mathcal{K}$ as in Section 2. The corresponding meromorphic differentials $du=(dx+idy),dv=(dx-idy)$ satisfy the equation $dk=k_{u}du+k_{v}dv=0$. Then, we may express $Q_k$, locally, in various ways:

$$Q_k=dudv=\frac{dv}{du}d{u}^2=-\frac{k_u}{k_v}d{u}^2=-\frac{k_v}{k_u}d{v}^2.$$

(13)

We call $Q_k$ the clinant quadratic differential of k, for reasons to be explained below (see also [11,14,25] for geometric applications of $Q_k$).

In particular, one of u or v will be used as the local coordinate near a regular point of the curve, e.g., if $v=S\left(u\right)$ is one of the locally defined analytic functions satisfying $k(u,S(u\left)\right)=0$, then $Q_k=S^{\prime }\left(u\right)d{u}^2$. Here, S denotes the Schwarz function, as defined by Davis [26]. Given an analytic arc $\gamma \subset \mathbb{C}$, this is the unique analytic function $w=S\left(z\right)$, defined in a suitable neighborhood of $\gamma$, satisfying $\overline{z}=S\left(z\right)$ along $\gamma$.

The clinant (or inclination function) along $\gamma$ is the complex unit $S^{\prime }\left(z\right)=\frac{d\overline{z}}{dz}=e^{-2i\theta }$, where $\theta$ measures the angle that the tangent to $\gamma$ makes with the real axis. (This explains our name for $Q_k=S^{\prime }\left(u\right)d{u}^2$. The word clinant was defined as such by Davis [26] but appears to date back to Franklin [27], who explains the usage for the reciprocal, $e^{2i\theta }$. Franklin may have been familiar with Darboux’s earlier use of the quantity $\frac{dx+idy}{dx-idy}=e^{2i\theta }$ [28].) Now observe that if $\gamma \left(\tau \right)=x+iy=u$ parameterizes a real arc of k, we may write ${\gamma }^{\prime }=\rho e^{i\theta }$, and thus verify the required condition ${\left({\gamma }^{\prime }\right)}^2{S}^{\prime }\left(\gamma \right)={\rho }^2>0$ for $\gamma$ to be a horizontal arc of $Q_k=S^{\prime }\left(u\right)d{u}^2$.

Now to relate the trajectories of $Q_k$ and $Q_{\left\{k\right\}}$, we require the following lemma (which will also be used in Section 6):

Lemma 1. 

Let $k(u,v)$ be a bicircular quartic (Equation (4)) and let ${\mathsf{\Delta }}_{v}k$ be its focal discriminant (Equation (5)). Then the polynomial ${\left(\frac{\partial k}{\partial v}\right)}^2-{\mathsf{\Delta }}_{v}k$ belongs to the ideal $\left(k\right)\subset \mathbb{R}[u,v]$. In fact, the following identity holds (introducing the “operator" ${\square }_v$)

$${\square }_{v}k(u,v):={\left(\frac{\partial k}{\partial v}\right)}^2-{\mathsf{\Delta }}_{v}k=4(u^2-{\sigma }^2)k(u,v),$$

(14)

where ${\sigma }^2=\frac{q-p}{2s}$ is the squared singular focus of k. Interchanging the roles of $u,v$, the corresponding statement holds for ${\square }_{u}k(u,v)$.

Proof. 

Equation (14) may be verified by a straightforward computation, which we omit. But the main consequence—that ${\square }_{u}k$ and ${\square }_{v}k$ belong to the ideal $\left(k\right)$—may be shown more simply, as follows. We note that the discriminant of a quadratic polynomial $P\left(x\right)=a{x}^2+bx+c$ is the square of its derivative at a root $x_0$; that is, $\mathsf{\Delta }P=b^2-4ac={(2a{x}_0+b)}^2={\left(P^{\prime }\left(x_0\right)\right)}^2$. This applies to $k(u,v)$, which can be regarded as a quadratic polynomial in either variable, u or v. Thus, along the curve $k=0$, the corresponding partial derivatives and discriminants are related by ${\mathsf{\Delta }}_{v}k\left(u\right)=b^2-4ac=k_v^2,{\mathsf{\Delta }}_{u}k\left(v\right)=B^2-4AC=k_u^2.$ □

The proof of the following theorem uses the lemma to compare the two quadratic differentials $Q_{\left\{k\right\}}$ and $Q_k$ along the real locus of k:

Theorem 2. 

A circuit of a bicircular quartic k in standard position is a horizontal or vertical trajectory of its focal quadratic differential $Q_{\left\{k\right\}}=\frac{d{u}^2}{\delta \left(u\right)}.$ Thus, the curves comprising the orthogonal family of algebraic curves ${\mathcal{F}}_{\alpha }^r$ are exactly the trajectories of $Q_{\alpha }^r=Q_{\left\{k\right\}}$.

Proof. 

Combining Lemma 1 with $dk=k_{u}du+k_{v}dv=0$, we conclude that the following separable ODE holds along k (see Remark 6):

$${\left(\frac{dv}{du}\right)}^2={\left(-\frac{k_u}{k_v}\right)}^2=\frac{{\mathsf{\Delta }}_{u}k}{{\mathsf{\Delta }}_{v}k}=\frac{\delta \left(v\right)}{\delta \left(u\right)}.$$

Here, ${\mathsf{\Delta }}_{u}k=4{\sigma }^2\delta \left(v\right)$ and ${\mathsf{\Delta }}_{v}k=4{\sigma }^2\delta \left(u\right)$ involve the same monic polynomial $\delta$ for reasons of symmetry. Namely, bicircular quartics in rectilinear position have the reflection symmetry $f(x,-y)=f(x,y)=\sum a_{rs}{x}^r{y}^{2s}$. In this case, $k(u,v)$ and $k(v,u)$ have identical, real coefficients, and consequently so do ${\mathsf{\Delta }}_{u}k$ and ${\mathsf{\Delta }}_{v}k$. (This symmetry is convenient, but our argument generalizes to all bicircular quartics.)

In particular, at a real point of the curve, we have $\delta \left(v\right)=\delta \left(\overline{u}\right)=\overline{\delta \left(u\right)},$ so the above equation gives ${\left(\frac{dv}{du}\right)}^2={\left(\frac{\left|\delta \right(u\left)\right|}{\delta \left(u\right)}\right)}^2$. Taking the square roots, we can write this as $\frac{dv}{du}=\frac{\pm \lambda }{\delta \left(u\right)},$ with $\lambda =\left|\delta \right(u\left)\right|>0$. Since a real arc of k is an arc of its clinant quadratic differential $Q_k=\frac{dv}{du}d{u}^2$, it follows that it is also a horizontal or vertical arc of $Q_{\left\{k\right\}}=d{u}^2/\delta \left(u\right)$.

Next, we use this conclusion (the first claim) to prove its converse. Namely, for given r and $\alpha$, we wish to show that all horizontal and vertical trajectories of the quadratic differential $Q=Q_{\alpha }^r$ correspond to curves in the orthogonal family $\mathcal{F}={\mathcal{F}}_{\alpha }^r$.

So let $u_0=x_0+i{y}_0\in \mathbb{C}$ be any point other than a pole of Q, and let ${\gamma }_{\pm }$ be the horizontal and vertical trajectories of Q through $u_0$. As discussed in Remark 4, there are also two distinct curves $k_{\pm }\in \mathcal{F}$ containing the point $u_0$. For most $u_0$, these are both irreducible bicircular quartics $f_{\pm }(x,y)=k_{\pm }(u,v)$. By the first claim, each agrees with one or the other trajectory ${\gamma }_{\pm }$. But $k_{\pm }$ cannot both agree with the same trajectory; so, ${\gamma }_{+}$ and ${\gamma }_{-}$ must each correspond to one of the curves $k_{\pm }\in \mathcal{F}$. The exceptional trajectories of $Q_{\alpha }$ are easily identified with (arcs of) a line/circle of reflection. □

Not only are all (horizontal/vertical) trajectories of $Q_{\alpha }=\frac{d{u}^2}{\delta \left(u\right)}$ algebraic but $Q_{\alpha }$ will be seen to be algebraically integrable: the (elliptic) trajectories of $Q_{\alpha }$ may be produced directly from the elliptic curve $w^2=\delta \left(u\right)$ via explicit birational transformation (Theorem 3).

Remark 6. 

Separable ODEs of the form ${\left(\frac{dy}{dx}\right)}^2=\frac{f\left(y\right)}{f\left(x\right)}$, i.e., $\frac{dy}{\sqrt{f\left(y\right)}}=\pm \frac{dx}{\sqrt{f\left(x\right)}}$, feature prominently in the classical literature. This is especially so in the case of quartics $f\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+e$, where the ODE is intimately related to the addition theorem for elliptic integrals [1,29,30]. See also [31], Ch. XIV, which begins with a method of algebraic integration of Lagrange. (This is not exactly algebraic integrability in the sense of Theorem 3.)

6. Edwards Normal Form and Algebraic Integrability

In 2007, Harold Edwards [29] presented an alternative to the elliptic curve normal forms associated with Jacobi or Weierstrass elliptic functions. Any elliptic curve can be represented as a quartic in the Edwards normal form (ENF): $a^2(x^2{y}^2+1)=x^2+y^2,a^5\ne a$. One advantage of this normal form is that the addition $p+p^{\prime }=p^{″}$ on the elliptic curve may be expressed in a very symmetrical form: $x^{″}=\frac{1}{a}\frac{x{y}^{\prime }+y{x}^{\prime }}{1+xy{x}^{\prime }{y}^{\prime }},y^{″}=\frac{1}{a}\frac{y{y}^{\prime }-x{x}^{\prime }}{1-xy{x}^{\prime }{y}^{\prime }}$ (see Theorem 3.1 in [29]).

Edwards notes the close resemblance to the original Euler–Gauss addition law for the special elliptic curve $x^2+y^2+x^2{y}^2=1$, which famously generalized the addition law on the circle $x^2+y^2=1$—the addition formulas for $x=sint$ and $y=cost$. He also implicitly raises a historical question: “I have not been able to find [this addition law] in the literature. If it is not new, it is certainly not as well known as it deserves to be”. We will not attempt to address this historical question, but we believe that the best answer is to be found in the investigations of bicircular quartics and their connections to elliptic functions and integrals (see Appendix D for references).

Without going too far afield here, we now proceed to interpret ENF quartics (in the real case) as Cassinians, then relate the Edwards theory more generally to the confocal families of bicircular quartics and their focal quadratic differentials. In fact, we will regard the confocal families as a kind of generalized ENF—obtained by “variation of Cassinians” (or in one case Cayley Cassinians) by a one-parameter family of birational equivalences.

To begin with the ovals of Cassini, we consider Equation (7) in the special case $q=-p={\sigma }^2$ (so $\alpha =\frac{1}{2}(q+\frac{r}{q})=\frac{1}{2}({\sigma }^2+\frac{r}{{\sigma }^2})$):

$$f:={(x^2+y^2)}^2-2{\sigma }^2(x^2-y^2)+r;k:=u^2{v}^2-{\sigma }^2(u^2+v^2)+r.$$

(15)

Observe that $k(u,v)$ is exactly ENF after substitutions $x\to u,y\to v,a\to 1/\sigma$—provided $r=1$; here, we allow $r=-1$ as well. (Note: To put Cassinians with $r=-1$ exactly into ENF has certain advantages, but this would ultimately have a modified standard position—with awkward consequences for our discussion of the non-concyclic case, mirrors, Siebeck’s result, etc.).

To develop his normal form, Edwards makes essential use of an auxiliary quartic curve which is birationally equivalent to ENF. In our notation for the Cassinian $k(u,v)$, the equivalent curve is $h(u,w):=w^2-({\sigma }^2-u^2)(r-{\sigma }^2{u}^2)$—roughly, the Legendre normal form. The required birational equivalence has the simple form $(u,v)\stackrel{\mathcal{E}}{⟷}(u,w)$, and is given by: $w:=v(u^2-{\sigma }^2)$. In fact, substitution into $h(u,w)$ results in $k(u,v)$ (modulo an extra factor $(u^2-{\sigma }^2)$, which can be neglected). Likewise, $v=w/(u^2-{\sigma }^2)$ turns $k(u,v)$ into $h(u,w)$.

Where does this birational equivalence come from, and can it be generalized? In view of Lemma 1, the fact that $w=v(u^2-{\sigma }^2)=\frac{1}{2}\frac{\partial k}{\partial v}$ is the key. This (and normalization by $\sigma$) yields the following generalization of the above equivalence $(u,v)\stackrel{\mathcal{E}}{⟷}(u,w)$ to bicircular quartics $k(u,v)$ in standard position (Equation (7)):

$$2\sigma w:=\frac{\partial k}{\partial v}=2(u^2-{\sigma }^2)v+(p+q)u;v=\frac{2\sigma w-(p+q)u}{2(u^2-{\sigma }^2)},$$

(16)

where ${\sigma }^2=\frac{q-p}{2}$. Writing Lemma 1 as ${\left(\frac{\partial k}{\partial v}\right)}^2-{\mathsf{\Delta }}_{v}k={\left(\frac{\partial k}{\partial v}\right)}^2-4{\sigma }^2\delta \left(u\right)=4(u^2-{\sigma }^2)k$, we find that $4{\sigma }^2(w^2-\delta \left(u\right))=4(u^2-{\sigma }^2)k$, so the bicircular quartic k has the equivalent normal form $h(u,w):=w^2-\delta \left(u\right)$:

$$h:w^2=\delta \left(u\right)=(u^2-b^2)(u^2-\frac{r}{b^2})=u^4-2\alpha u^2+r.$$

(17)

The resulting “new curve” h is now independent of $p,q$—the reason we normalized via $\sigma$.

For $r=\pm 1$, let ${\mathcal{E}}_{p,q}$ denote the parameterized Edwards transformation defined by Equation (16). Trajectories of the quadratic differential $Q_{\alpha }^r=\frac{d{u}^2}{\delta \left(u\right)}$ satisfy the ODE ${\left(\frac{du}{dt}\right)}^2=\delta \left(u\right)$, to which we associate the elliptic curve $h:w^2=\delta \left(u\right)$. By applying ${\mathcal{E}}_{p,q}$ to the latter, for points $(p,q)$ on the hyperbola ${\mathcal{H}}_{\alpha }$, all but finitely many trajectories of $Q_{\alpha }^r$ are obtained (the exceptions being linear or circular arcs). In fact, for a given initial condition $u_0=x_0+i{y}_0$, the pair of trajectories through $u_0$ are thus explicitly produced (the required $p,q$ are given in terms of $u_0$ by the quadratic formula). This is what we mean by the following:

Theorem 3. 

The ODE ${\left(\frac{du}{dt}\right)}^2=\delta \left(u\right)$ is algebraically integrable.

7. Birational Equivalence of Confocal Curves

In this section, we consider how two curves in a confocal family are related to each other. Namely, if k and $k^{\prime }$ are two such bicircular quartics, we can use ${\mathcal{E}}_{p,q}$ and ${\mathcal{E}}_{p^{\prime },q^{\prime }}$ to relate both to h, and hence to each other: $k(u,v)\leftrightarrow h(u,w)\leftrightarrow k^{\prime }(u,v)$. To be clear, the resulting equivalence $k\simeq k^{\prime }$ is a complex birational equivalence of real curves.

More explicitly, if k and $k^{\prime }$ are represented by two points $(p,q)$ and $(p^{\prime },q^{\prime })$ on ${\mathcal{H}}_{\alpha }$, the two formulas of Equation (16) may be combined to prove the following.

Theorem 4. 

If k and $k^{\prime }$ are confocal bicircular quartics in standard position, they are related by the following birational equivalence:

$$\frac{u^2-{{\sigma }^{\prime }}^2}{{\sigma }^{\prime }}{v}^{\prime }+\frac{p^{\prime }+q^{\prime }}{2{\sigma }^{\prime }}u=\frac{u^2-{\sigma }^2}{\sigma }v+\frac{p+q}{2\sigma }u$$

(18)

$$\frac{u^2-{{\sigma }^{\prime }}^2}{{{\sigma }^{\prime }}^2}{k}^{\prime }(u,v^{\prime })=\frac{u^2-{\sigma }^2}{{\sigma }^2}k(u,v).$$

(19)

Proof. 

With ${{\sigma }^{\prime }}^2=\frac{q^{\prime }-p^{\prime }}{2}$, Equation (18) is obtained straightforwardly from Equation (16) as described above. Solving for

$$v^{\prime }=\frac{\frac{{\sigma }^{\prime }}{\sigma }(2(u^2-{\sigma }^2)v+(p+q)u)-(p^{\prime }+q^{\prime })u}{2(u^2-{{\sigma }^{\prime }}^2)}$$

and inserting this expression into $k^{\prime }(u,v^{\prime })$ leads to the equation

$$\frac{u^2-{{\sigma }^{\prime }}^2}{{{\sigma }^{\prime }}^2}{k}^{\prime }(u,v^{\prime })-2{\alpha }^{\prime }{u}^2=\frac{u^2-{\sigma }^2}{{\sigma }^2}k(u,v)-2\alpha u^2.$$

Under the assumption $\alpha ={\alpha }^{\prime }:=\frac{p^{\prime }{q}^{\prime }-r}{p^{\prime }-q^{\prime }}$, this gives Equation (19), expressing the equivalence of the two curves $k^{\prime }(u,v)$ and $k(u,v)$. □

Remark 7. 

As the theorem illustrates, birational transformations are to binodal quartics as projective transformations are to nonsingular cubics. Namely, if two of the latter plane curves $c,c^{\prime }$ are equivalent to the elliptic curves ($j\left(c\right)=j\left(c^{\prime }\right)$), they are in fact projectively equivalent (though not necessarily real projectively equivalent, in the case of real curves [32]). But projective transformations do not suffice for binodal quartics, e.g., if $k,k^{\prime }$ are confocal bicircular quartics, no projective transformation preserves the shared isotropic tangents to $k,k^{\prime }$ while appropriately moving the isotropic tangents at circular points. The fact that birational transformations fill the gap is not surprising, given that the two classes of curves are themselves related by inversive equivalence (Appendix E), but this is quite awkward to exploit directly.

Equations (18) and (19) may also be used, e.g., to transfer the group structure from Cassians (ENF) to bicircular quartics in standard position, or to show that such confocal $k(u,v)$ and $k^{\prime }(u,v)$ are rationally equivalent, say, when $p,q,p^{\prime },q^{\prime }\in \mathbb{Q}$ and ${\left(\frac{{\sigma }^{\prime }}{\sigma }\right)}^2=\frac{q^{\prime }-p^{\prime }}{q-p}=\frac{m^2}{n^2}$, for integers $m,n$.

The simplest case of the latter arises from reflection symmetry of the hyperbola ${\mathcal{H}}_{\alpha }^r$ across the anti-diagonal: $(p,q)\leftrightarrow (-q,-p)$. In this case, $\frac{{\sigma }^{\prime }}{\sigma }=1$, and Equation (18) reduces simply to $v^{\prime }=v+\frac{(p+q)u}{u^2-{\sigma }^2}$. Introducing self-evident notation, Equation (19) becomes

$$k(u,v;p,q)=k(u,v+\frac{(p+q)u}{u^2-{\sigma }^2};-q,-p)$$

(20)

The involution $k(u,v;p,q)\stackrel{\mathcal{R}}{⟷}k(u,v;-q,-p)$ fixes Cassinians, and pairs the remaining curves in a confocal family with the same singular foci $\pm \sigma$. In fact, $\mathcal{R}$ corresponds to the Cassinian reflection as explained in the following remark.

Remark 8. 

For an analytic arc $\gamma \subset \mathbb{C}=\{u=x+iy\}$ with Schwarz function $S\left(u\right)$, the Schwarz reflection in γ is the antiholomorphic involution ${\mathcal{R}}_{\gamma }\left(u\right):=\overline{S\left(u\right)}$ fixing γ [26], e.g., the Schwarz function of the circle $\left|u\right|=\rho$, is $S\left(u\right)={\rho }^2/u$ and ${\mathcal{R}}_{\gamma }\left(u\right):={\rho }^2/\overline{u}$ gives circle inversion.

Turning now to Cassinians—say, one of those in Figure 2—such a curve γ determines a maximal ring domain [33]: D is foliated by closed trajectories of $Q=\frac{d{u}^2}{\delta \left(u\right)}$, including γ, and is maximal with respect to this property. In fact, γ is the core curve of D—it is the unique curve fixed by an antiholomorphic involution of D.

To check this, we solve for v in Equation (15), obtaining the two-valued function $S\left(u\right)=\sqrt{\frac{{\sigma }^2{u}^2-r}{u^2-{\sigma }^2}}$. The branch points of $S\left(u\right)$ are the foci $\pm \sigma ,\sqrt{r}/\sigma$. These, together with $0,\infty$ lie on the boundary of D, and they correspond under reflection ${\mathcal{R}}_{\gamma }$ according to: $\pm \sigma \stackrel{{\mathcal{R}}_{\gamma }}{⟷}\infty$, $\pm \sqrt{r}/\sigma \stackrel{{\mathcal{R}}_{\gamma }}{⟷}0$. Evidently, Schwarz reflection gives antiholomorphic involution ${\mathcal{R}}_{\gamma }:D\to D$ fixing γ. (In case $r=1$, similar comments apply to the left and right halves of Figure A1.)

Now it turns out that $\mathcal{R}$, as defined after Equation (20), agrees with ${\mathcal{R}}_{\gamma }$ in the sense that it swaps the same pairs of curves foliating D. Yet this is rather curious; ${\mathcal{R}}_{\gamma }$ is a fixed antiholomorphic involution on D, while $\mathcal{R}$ is realized by complex birational transformations defining Riemann surface equivalences for each of the paired real curves.

In view of the pairing $\mathcal{R}$ of confocal curves with shared singular foci, it is reasonable to regard the squared singular focus ${\sigma }^2=\frac{q-p}{2}$ as the “geometric parameter” for a confocal family. (Of course, p or q may be used to rationally parameterize ${\mathcal{H}}_{\alpha }^r$, but for most purposes, there does not appear to be a great advantage to doing so.)

Thus, the global geometric picture may be described roughly as follows. Returning to the projective space of bicircular quartics in rectilinear position, ${\mathrm{P}}^3=\left\{(p:q:r:s)\right\}$, the complement of the discriminant locus ($\mathsf{\Delta }k\ne 0$) is foliated by confocal families; the foci determine the confocal family, while the singular foci fix the position (essentially) on a given confocal family.

8. Conclusions

We considered real bicircular quartics in two different contexts: as algebraic curves in $\mathbb{C}{\mathrm{P}}^2$, and as members of confocal families of curves in $\widehat{\mathbb{C}}$. The relationship between these two points of view was established by the use of quadratic differentials $Q_k$ defined on each elliptic curve k in a confocal family $\left\{k\right\}$, and a quadratic differential $Q_{\left\{k\right\}}$ defined on $\widehat{\mathbb{C}}$. A moduli space of bicircular quartics (w.r.t. inversive equivalence) was presented as a stratified real projective space $P^3$, whose points in the top-dimensional strata represent elliptic curves with four ordinary foci and two singular foci. A generic quartic k was essentially specified by its focal discriminant ${\mathsf{\Delta }}_{v}k$, a quartic polynomial in a variable u whose roots are the foci of k, and a parameter ${\sigma }^2$, the squared singular focus, which parametrized the family of confocal quartics. We showed explicitly that curves in each family are all birationally equivalent via a parameterized Edwards transformation.