The Geometry of the Kiepert Trefoil

Abstract

This article presents a comparative study of Kiepert’s trefoil and its related curves, combining a variety of tools from differential and algebraic geometry, integrable systems, elastica theory, and special functions. While this curve was classically known and well studied in the literature, some related open problems were recently solved, and the goal of this paper is to present and characterize the general solution of the equation that governs this trefoil’s family of curves by involving elliptic functions and elastica theory in the mechanics.

1. Introduction

Folium is a term that comes from Latin, where it means leaf. In mathematics, it represents a closed curve with several lobes. Generally, trifolium means a three-lobed closed curve in plane. The authors are interested in the Kiepert trefoil as a particular case of trifolium from the view points of differential geometry, theory of elastica and flows, geometric PDE and elliptic integrals.

A very interesting paper [1] was published recently by David Singer in this Special Issue, which triggered our interest in the subject and inspired us to provide a classification of the related families of curves and corresponding integrable systems. Surprisingly, it turns out that trifolium is an ambivalent name which refers to several curves. Most commonly, this name denotes the regular trifolium, which represents the quartic algebraic curve:

$${(x^2+z^2)}^2=ax(x^2-3z^2)+(bx+cz)(x^2+z^2),a,b,c\in \mathbb{R}.$$

(1)

This is depicted as a rose with three petals (cf. Figure 1).

This name also denotes one of the Rhodonea curves presented by the following quartic which is a particular case of Equation (1):

$${(x^2+z^2)}^2=ax(x^2-3z^2),a\in \mathbb{R}.$$

(2)

Alternatively, the last curve can be seen in many places presented equivalently by the following equation:

$$({\tilde{x}}^2+{\tilde{z}}^2)(\tilde{x}(\tilde{x}+a)+{\tilde{z}}^2)=4a\tilde{x}{\tilde{z}}^2.$$

(3)

This is accomplished by taking into account that the change $\tilde{x}\mapsto -x$, $\tilde{z}\mapsto z$ transforms Equation (3) into Equation (2). This is illustrated in Figure 2.

The above figures have been drawn through their explicit parameterizations:

$$x=\frac{a}{2}(cos\left(\psi \right)+cos\left(2\psi \right)),z=\frac{a}{2}(sin\left(\psi \right)-sin\left(2\psi \right)),\psi \in [0,2\pi ]$$

(4)

and the intrinsically related to them formulas

$$\tilde{x}=-x,\tilde{z}=z.$$

(5)

What is more, the curve associated with Equation (3) (written without tildes) can be considered a particular case of the Kepler folium (cf. [2] (p. 85))

$$({(x-b)}^2+z^2)(x(x-b)+z^2)=4a(x-b)z^2,a,b\in \mathbb{R}$$

(6)

taken with $b\equiv a$ and translation along the $OX$-axis: $x\mapsto x+a$.

Finally, in many other situations, this notion also refers to Cramer trifolium (see [3] (p. 354) and [4] (p. 463)), specified by a fourth quartic

$$x^4+z^4=ax(x^2-z^2),a\in \mathbb{R}.$$

(7)

The parameterization of the Cramer trifolium is provided by the formulas

$$x=a\frac{2(cos(\psi )+1)cos\left(\psi \right)}{cos\left(2\psi \right)+3},z=a\frac{sin\left(2\psi \right)}{cos\left(2\psi \right)+3},\psi \in [0,2\pi ].$$

(8)

Additionally, its shape is presented by the last graphic in Figure 1.

All these curves arise in various contexts or following concrete ideas and have been considered from different perspectives. As a concept, trifoliums lie at the intersection between algebraic and differential geometry [5]. We are interested in the latter approach. Despite the fact that these curves have been known for centuries, their explicit parameterizations, except for the polar presentations, implicit equations, and exotic description via Dixon functions [6], are missing.

In the next section, we show that the arclength of the Kiepert trefoil is fully determined by an elliptic integral of the first kind, which was the initial impulse for its consideration [7]. Other goals for this paper are the following: (1) to represent the Kiepert trefoil as a special curve in differential geometry, for which the curvature $\kappa$ is a quadratic function of the distance from the origin (see Equation (13)), (2) to show that relative to the curve flow equation (Equation (23)), the Kiepert trefoil is regarded as a time-independent solution in addition to being an elastic curve, and (3) to interpret Kiepert’s trefoil equation (Equation (20)) as the Free Buckled Ring Equation (Equation (28)), thus clarifying why it corresponds to a generalized Euler elastica.

2. Kiepert Trefoil: Methods and Results

(1) The Kiepert trefoil is a well-known curve in algebraic geometry. It represents a sextic curve in the Euclidean plane ${\mathbb{R}}^2$, which is given in Cartesian coordinates $(x,z)$ by the algebraic equation

$$f(x,z)\equiv {(x^2+z^2)}^3-a^{3}x(x^2-3z^2)=0,a\in \mathbb{R}.$$

(9)

The Kiepert trefoil is special curve in differential geometry. In arclength, this trefoil is expressible by first-kind elliptic integrals, in addition to the widely known lemniscates and Serret’s curves. It is named after the German mathematician Friedrich Wilhelm August Ludwig Kiepert (1846–1934), who had written a thesis on the subject [8].

The polar coordinates are expressed as

$$r=\sqrt{x^2+z^2},\phi =arctan\left(\frac{z}{x}\right).$$

(10)

Equation (9) can be rewritten as

$$f(r,\phi )\equiv r^3-a^{3}cos\left(3\phi \right)=0.$$

(11)

Using the well-known expression for the curvature $\kappa$ of an implicitly defined curve in ${\mathbb{R}}^2$, we have

$$\kappa (x,z)=\frac{{\displaystyle \frac{{\partial }^{2}f}{\partial x^2}{\left(\frac{\partial f}{\partial z}\right)}^2-2\frac{{\partial }^{2}f}{\partial x\partial z}\frac{\partial f}{\partial x}\frac{\partial f}{\partial z}+\frac{{\partial }^{2}f}{\partial z^2}{\left(\frac{\partial f}{\partial x}\right)}^2}}{{\displaystyle {\left({\left(\frac{\partial f}{\partial x}\right)}^2+{\left(\frac{\partial f}{\partial z}\right)}^2\right)}^{3/2}}}{}_{|}{}_{{\displaystyle f(x,z)\equiv 0}}·$$

(12)

One can easily verify that the curvature of the trefoil is given by the following formula:

$$\kappa =\frac{4}{a^3}{r}^2.$$

(13)

The components $x\left(s\right),z\left(s\right)$ of the position vector $\mathbf{r}$ of a plane curve whose curvature meets the relation in Equation (13) are determined by the following system of equations (see, e.g., [9,10]):

$$\ddot{x}+\frac{4}{a^3}{r}^2\dot{z}=0,\ddot{z}-\frac{4}{a^3}{r}^2\dot{x}=0$$

(14)

where the dots denote derivatives with respect to the arclength s. It is easy to verify (see [9,10]) that two conservation laws (first integrals) hold for the solutions to the system in Equation (14), which in terms of the polar coordinates r and $\phi$ can be written as

$${\dot{r}}^2=1-\frac{{\left(8r^4-a^6\lambda \right)}^2}{64a^6{r}^2}$$

(15)

and

$$\dot{\phi }=\frac{8r^4-a^6\lambda }{8a^3{r}^2}$$

(16)

where $\lambda$ is an arbitrary real constant.

By differentiating both sides of Equation (13) with respect to the variable s, one obtains the following equation:

$${\dot{\kappa }}^2=\frac{64}{a^6}{r}^2{\dot{r}}^2.$$

(17)

(2) The Kiepert trefoil can be regarded as a time-independent solution of a curve flow.

Theorem 1.

The Kiepert trefoil is a time-independent solution of the curve flow.

Proof. 

Based on Equation (15) of the previous paragraph 1, the above equality takes the following form:

$${\dot{\kappa }}^2=\frac{64}{a^6}\left(1-\frac{{\left(8r^4-a^6\lambda \right)}^2}{64a^6{r}^2}\right)r^2.$$

(18)

If in Equation (18) we set $r^2=\left(a^3/4\right)\kappa$, then as Equation (13) implies, we obtain

$${\dot{\kappa }}^2+\frac{1}{4}{\kappa }^4-\lambda {\kappa }^2-\frac{16}{a^3}\kappa +{\lambda }^2=0$$

(19)

which has to be recognized as the intrinsic equation of any plane curve whose curvature is specified by Equation (13). In the special case of the Kiepert trefoil, Equation (11) implies $\lambda \equiv 0$. In other words, we have

$${\dot{\kappa }}^2+\frac{1}{4}{\kappa }^4-\frac{16}{a^3}\kappa =0.$$

(20)

Additionally we have the fundamental relation

$$\dot{\phi }=\frac{\kappa }{4}=\frac{\dot{\theta }}{4}$$

(21)

in which $\theta$ denotes the slope angle of the tangent line, and after integration, we end up with the remarkable formula

$$\theta =4\phi +\mathrm{const}.$$

(22)

Let a regular plane curve parameterized by its arclength s evolve smoothly and without extension in time t according to the following curve flow equation:

$$\frac{\partial \mathbf{r}(s,t)}{\partial t}=-\frac{\partial \kappa (s,t)}{\partial s}\mathbf{n}(s,t)-\frac{1}{2}{\kappa }^2(s,t)\mathbf{t}(s,t)$$

(23)

where $\mathbf{r}(s,t)$ is the position vector of the curve, $\kappa (s,t)$ is its curvature, and $\mathbf{t}\left(s,t\right)$ and $\mathbf{n}\left(s,t\right)$ are the unit tangent and inward unit normal vectors of the curve, respectively. Then, the curvature of this curve satisfies the focusing mKdV equation:

$$\frac{\partial \kappa (s,t)}{\partial t}+\frac{{\partial }^3\kappa (s,t)}{\partial s^3}+\frac{3}{2}{\kappa }^2(s,t)\frac{\partial \kappa (s,t)}{\partial s}=0.$$

(24)

This represents the well-known result of Goldstein and Petrich [11] and Nakayama et al. [12] (see also [13]), where the plane curve flow of the special form in Equation (23) is an mKdV flow; that is, if a curve evolves according to Equation (23), then the curvature of this curve satisfies the focusing mKdV equation (Equation (24)).

Evidently, the stationary solutions of Equation (24) also satisfy the equation

$$\frac{{\mathrm{d}}^3\kappa \left(s\right)}{\mathrm{d}{s}^3}+\frac{3}{2}{\kappa }^2\left(s\right)\frac{\mathrm{d}\kappa \left(s\right)}{\mathrm{d}s}=0$$

(25)

which can be integrated twice to obtain

$${\left(\frac{\mathrm{d}\kappa }{\mathrm{d}s}\right)}^2+\frac{1}{4}{\kappa }^4+2c_1\kappa +c_2=0$$

(26)

where $c_1$ and $c_2$ are the constants of integration. Thus, by setting $c_1=-8/a^3$ and $c_2\equiv 0$ here, one obtains the intrinsic equation of the Kiepert trefoil (Equation (20)). This means that the Kiepert trefoil is a time-independent solution of the curve flow specified by Equation (23), which finalizes the proof of the theorem. □

(3) The Kiepert trefoils can also be viewed as generalized Euler’s elasticas, and thus their arclength representations via elliptic integrals. Singer [1] had recognized the Kiepert trefoil as a particular solution of the Buckled Rings Equation, which is nothing more than the Generalized Elastica Equation [14]:

$${\kappa }_{ss}+\frac{1}{2}{\kappa }^3-\mu \kappa -\sigma =0⟺{\kappa }_s^2+\frac{1}{4}{\kappa }^4-\mu {\kappa }^2-2\sigma \kappa -\epsilon =0$$

(27)

where $\mu$ (tension), $\sigma$ (pressure), and $\epsilon$ (energy) are the physical characteristics of the elastica or the elastic ring (${\kappa }_s\equiv \mathrm{d}\kappa /\mathrm{d}s$, ${\kappa }_{ss}\equiv {\mathrm{d}}^2\kappa /\mathrm{d}{s}^2$). What is more, the Kiepert’s trefoil equation (Equation (20)) actually represents the Free Buckled Ring Equation ($\mu =0$, $\sigma =8/a^3$, $\epsilon =0$):

$${\kappa }_{ss}+\frac{1}{2}{\kappa }^3-\sigma =0⟺{\kappa }_s^2+\frac{1}{4}{\kappa }^4-2\sigma \kappa =0.$$

(28)

This is a counterpart of the so-called Free Euler’s Elastica Equation ($\mu =0$, $\sigma =0$, $\epsilon =0$) [15] (p. 114):

$${\kappa }_{ss}+\frac{1}{2}{\kappa }^3=0⟺{\kappa }_s^2+\frac{1}{4}{\kappa }^4=0.$$

(29)

The solutions to Equations (27) and (28) are documented in [14,15], while their periodicity is discussed in [16]. Here, we choose to follow the scheme developed in [17] which is based on Cartan’s repère mobile (moving frame). In technical terms, the curve specified by its curvature $\kappa \left(r\right)$ is reconstructed through the formulas

$$x\left(s\right)=\xi \left(s\right)cos\theta \left(s\right)+\eta \left(s\right)sin\theta \left(s\right),z\left(s\right)=\xi \left(s\right)sin\theta \left(s\right)-\eta \left(s\right)cos\theta \left(s\right)$$

(30)

in which $(\xi ,\eta )$ are the coordinates of the point in the moving frame and $\theta$ is the tangential angle. They are defined by another set of formulas:

$$\xi =r\dot{r},\eta =-\int r\kappa \left(r\right)\mathrm{d}r,\theta =\int \kappa \left(s\right)\mathrm{d}s.$$

(31)

Effectively, this scheme is realized via integration of the following intrinsic equation (more details can be found in [15] (pp. 47–49)):

$${\xi }^2+{\eta }^2=r^2.$$

(32)

Proposition 1.

The arclength of the Kiepert trefoil is given by an elliptic integral of the first kind.

Proof. 

One finds an explicit expression for r as a function of the arclength s, which via the fundamental relation (13) produces the required expression for the curvature:

$$\kappa \left(s\right)=\frac{4\left(\mathrm{cn}\right(\tau s,k)-1)}{a(\sqrt{3}+1)(\gamma \mathrm{cn}(\tau s,k)+1)},\tau =\frac{2\sqrt[4]{3}}{a},k=\frac{\sqrt{2-\sqrt{3}}}{2},\gamma =\frac{\sqrt{3}-1}{\sqrt{3}+1}$$

(33)

where $\mathrm{cn}(·,·)$ denotes the cosine Jacobian elliptic function and k is the so-called elliptic modulus. The formulas above enable us to find $\theta \left(s\right)$ in the form

$$\theta \left(s\right)=\frac{1}{\sqrt[4]{3}\gamma }\left(\Pi (\mathrm{am}(\tau s,k),n,k)-2\sqrt[4]{3}\gamma arctan\left(\frac{\mathrm{sd}(\tau s,k)}{2\sqrt[4]{3}}\right)-\tau (1-\gamma )s\right),n=\frac{{\gamma }^2}{{\gamma }^2-1}$$

(34)

while the computation of $\xi \left(s\right)$ and $\eta \left(s\right)$ is almost straightforward and produces

$$\xi \left(s\right)=a\frac{\sqrt[4]{27}\mathrm{sn}(\tau s,k)\mathrm{dn}(\tau s,k)}{(\sqrt{3}+2){(\gamma \mathrm{cn}(\tau s,k)+1)}^2},\eta \left(s\right)=a\frac{{(\mathrm{cn}(\tau s,k)-1)}^2}{{(\sqrt{3}+1)}^2{(\gamma \mathrm{cn}(\tau s,k)+1)}^2}·$$

(35)

By combining the obtained results for Equations (34) and (35) through the formulas in Equation (30), we find the sought explicit parameterization of the Kiepert trefoil. It is clear that the length L of a single leaf coincides with the lowest period T of the curvature $\kappa \left(s\right)$, which is given by the formula

$$L\equiv T=\frac{2K\left(k\right)}{\sqrt[4]{3}}a$$

(36)

where $K\left(k\right)$ denotes the complete elliptic integral of the first kind, and obviously, the total perimeter $L^{\mathrm{tot}}$ of the trefoil is exactly

$$L^{\mathrm{tot}}=3L.$$

(37)

Equation (36) represents a direct confirmation of the fact that the arclength of the Kiepert trefoil is given by the elliptic integral of the first kind, thus proving the proposition. □

For completeness, we list below the rescaled (${\displaystyle \alpha =\frac{a}{\sqrt[3]{2}\sqrt{3}}}$) Kiepert’s parameterization of the Trefoil in terms of the Weierstrass ℘ function, which is borrowed from his dissertation [8]:

$$x\left(u\right)=\frac{3{\alpha }^2\wp \left(u\right){\wp }^{\prime }\left(u\right)}{2({\wp }^3\left(u\right)+{\alpha }^6)},z\left(u\right)=\frac{3\sqrt{3}{\alpha }^5\wp \left(u\right)}{2({\wp }^3\left(u\right)+{\alpha }^6)},\wp \left(u\right)\equiv \wp (u;g_2,g_3),{\wp }^{\prime }\left(u\right)\equiv \frac{\mathrm{d}\wp }{\mathrm{d}u},g_2=0,g_3=-{\alpha }^6.$$

(38)

Graphics of the Kiepert trefoil and the Kepler folium (Equation (3)) with the same parameter a are presented and compared below in Figure 3.

3. Discussion, Open Problems, and Future Directions

By solving the algebraic Equation (9), one finds another interesting parameterization by means of circular functions, namely

$$\begin{array}{ccc}\hfill x\left(u\right)& =& a\frac{1-\sqrt{3}-(3-\sqrt{3})cosu+(3+\sqrt{3}){cos}^{2}u-(1+\sqrt{3}){cos}^{3}u}{2\left(\sqrt{3}-1-(3-\sqrt{3})cosu+(3+\sqrt{3}){cos}^{2}u+(1+\sqrt{3}){cos}^{3}u\right)},u\in [0,2\pi ]\hfill \\ & & \\ \hfill z\left(u\right)& =& a\frac{\sqrt[4]{3}\left(\sqrt{3}-1+(1+\sqrt{3})cosu\right)sinu\sqrt{4-(2+\sqrt{3}){sin}^{2}u}}{2\left(\sqrt{3}-1-(3-\sqrt{3})cosu+(3+\sqrt{3}){cos}^{2}u+(1+\sqrt{3}){cos}^{3}u\right)}·\hfill \end{array}$$

(39)

The radical which appears in the expression of the z coordinate suggests use of the Jacobian elliptic functions, and this leads to an alternative parameterization in the form of

$$\begin{array}{ccc}\hfill x\left(u\right)& =& a\frac{1-\sqrt{3}-(3-\sqrt{3})\mathrm{cn}(u,k_{{}_{+}})+(3+\sqrt{3}){\mathrm{cn}}^2(u,k_{{}_{+}})-(1+\sqrt{3}){\mathrm{cn}}^3(u,k_{{}_{+}})}{2\left(\sqrt{3}-1-(3-\sqrt{3})\mathrm{cn}(u,k_{{}_{+}})+(3+\sqrt{3}){\mathrm{cn}}^2(u,k_{{}_{+}})+(1+\sqrt{3}){\mathrm{cn}}^3(u,k_{{}_{+}})\right)},u\in [0,4K\left(k_{{}_{+}}\right)]\hfill \\ & & \\ \hfill z\left(u\right)& =& a\frac{\sqrt[4]{3}\mathrm{sn}(u,k_{{}_{+}})\left(\sqrt{3}-1+(1+\sqrt{3})\mathrm{cn}(u,k_{{}_{+}})\right)\mathrm{dn}(u,k_{{}_{+}})}{\sqrt{3}-1-(3-\sqrt{3})\mathrm{cn}(u,k_{{}_{+}})+(3+\sqrt{3}){\mathrm{cn}}^2(u,k_{{}_{+}})+(1+\sqrt{3}){\mathrm{cn}}^3(u,k_{{}_{+}})},k_{{}_{+}}=\frac{\sqrt{2+\sqrt{3}}}{2}·\hfill \end{array}$$

(40)

More details about elliptic functions and integrals can be found in [18,19]. Here, we can mention that it would be very interesting to interpret other n foliums (i.e., $n=4,5,6,\dots$) as generalized elasticas and obtain their explicit parameterizations in terms of, for example, (hyper)elliptic functions. This represents one of our further directions of research. Similarly, it would also be interesting to compare the PDE-based and analytic approaches with the n foliums (cf. [20,21]) that were studied in algebraic geometry. Another interesting area of research lies in studying foliums on non-planar surfaces in particular on spheres, which is a current project for some of the authors.