### Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”

Introduction

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ using the integral

$\displaystyle u = \int_0^{\phi} \frac{\mathrm{d}\phi}{\sqrt{1-k^2\sin^2\phi}} \,.$

He called ${\phi}$ the amplitude and wrote ${\phi = \mathop\mathrm{am} u}$. It can be difficult to understand what motivated his definitions. We will define the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ in a more intuitive way, as simple ratios associated with an ellipse.

Circular Functions

To set the scene, we recall some key properties of the circular functions. The familiar trigonometric functions ${\sin\theta}$, ${\cos\theta}$ and ${\tan\theta}$ are described effectively by ratios of quantities in a circle. The algebraic formula for a circle is ${x^2 + y^2 = a^2}$, where ${a}$ is the radius of the circle. Then the trigonometric functions are defined as

$\displaystyle \cos\theta = \frac{x}{a} \qquad \sin\theta = \frac{y}{a} \,.$

The parametric description of the circle is

$\displaystyle x = a \cos\theta \qquad y = a \sin\theta \,,$

and the theorem of Pythagoras leads immediately to the identity

$\displaystyle \sin^2\theta + \cos^2\theta =1 \,.$

The circle used in the definition of the circular functions.

The derivatives of the trigonometric functions are related by

$\displaystyle \frac{\mathrm{d}}{\mathrm{d}\theta}\sin\theta = \cos\theta \qquad \frac{\mathrm{d}}{\mathrm{d}\theta}\cos\theta = -\sin\theta \,.$

Moreover, the functions ${\sin\theta}$ and ${\cos\theta}$ both satisfy the nonlinear differential equation

$\displaystyle \left(\frac{\mathrm{d} y}{\mathrm{d}\theta}\right)^2 = 1 - y^2$

with appropriate boundary conditions.

There are numerous trigonometric identities relating the circular functions, for example, the addition formula

$\displaystyle \sin(A+B) = \sin A \cos B + \cos A \sin B \,.$

These have counterparts for the elliptic functions. Introducing the reciprocols and ratios of ${\sin}$ and ${\cos}$, we have in total six trigonometric functions:

$\displaystyle \begin{matrix} \sin & \cos & \tan \\ \csc & \sec & \cot \end{matrix}$

For the elliptic case, we will find a family of twelve functions. Finally, we recall the characteristic forms of ${\sin}$ and ${\cos}$ as periodic functions. Similar behaviour for the elliptic functions ${\mathop\mathrm{sn}}$ and ${\mathop\mathrm{cn}}$ will easily be seen from the definitions as ratios of lengths in an ellipse.

Elliptic Functions

The ellipse with centre at the origin and horizontal major axis has the equation

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \,.$

The eccentricity is usually denoted by ${e}$ or ${\epsilon}$, but here we will write it as ${k}$. It is defined by

$\displaystyle k^2 = \frac{a^2 - b^2}{a^2} \,.$

For the circle, ${a=b}$ and so ${k = 0}$. We assume below that ${a>b}$. For a greatly flattened ellipse, ${k}$ approaches 1. In contrast to the circle, the radius vector from the origin to the ellipse varies in length, and we have the Pythagorean relationship

$\displaystyle x^2 + y^2 = r^2 \,.$

From these two relationships we will derive the main properties of the elliptic functions.

We introduce a variable ${u}$ defined by

$\displaystyle \mathrm{d} u = r\,\mathrm{d}\theta \qquad u = \int r(\theta)\,\mathrm{d}\theta \,.$

We note that in the circular case it is just the arc-length ${u = a\theta}$. In the elliptic case, it allows for the variation of ${r}$ as the representative point progresses around the ellipse. The variable ${u}$ is intimately related to Jacobi’s amplitude function mentioned above. Indeed, using the equation for the ellipse, we can show that

$\displaystyle u = \int_0^{\theta} \frac{\mathrm{d}\theta}{\sqrt{1-k^2\cos^2\theta}} \,.$

If ${b>a}$, we define the eccentricity by ${k = \sqrt{1-a^2/b^2}}$ and the cosine is replaced by a sine, giving the form of Jacobi’s integral that appears at the top of this article.

The ellipse used in the definition of the Jacobi elliptic functions.

“Sun”, “cun” and “dun”

By analogy with the circular functions, we define three elliptic functions:

$\displaystyle \mathop\mathrm{cn} u = \frac{x}{a} \qquad \mathop\mathrm{sn} u = \frac{y}{b} \qquad \mathop\mathrm{dn} u = \frac{r}{a} \,.$

These should be compared to the definitions of ${\sin\theta}$ and ${\cos\theta}$. When reading formulas aloud, we may call the three functions “sun”, “cun” and “dun”. Note that the third function, ${\mathop\mathrm{dn} u}$, has no equivalent in the circular case, as it is identically equal to unity when ${a = b}$.

From the equation of the ellipse, it follows immediately that

$\displaystyle {\mathop\mathrm{sn}}^2 u + {\mathop\mathrm{cn}}^2 u = 1 \,.$

The Pythagorean relationship ${x^2+y^2=r^2}$ gives rise to another identity:

$\displaystyle {\mathop\mathrm{dn}}^2 u + k^2 {\mathop\mathrm{sn}}^2 u = 1 \,.$

The definitions of the elliptic functions, together with the equation of the ellipse and the Pythagorean relationship, allow us to obtain expressions for the derivatives:

$\displaystyle \frac{\mathrm{d}}{\mathrm{d} u}\mathop\mathrm{sn} u = \mathop\mathrm{cn} u \mathop\mathrm{dn} u \qquad \frac{\mathrm{d}}{\mathrm{d} u}\mathop\mathrm{cn} u = - \mathop\mathrm{sn} u \mathop\mathrm{dn} u \,.$

It follows from these that the elliptic functions satisfy nonlinear differential equations, for example

$\displaystyle \left(\frac{\mathrm{d} \mathop\mathrm{sn} u}{\mathrm{d} u}\right)^2 = (1 - {\mathop\mathrm{sn}}^2 u)(1-k^2{\mathop\mathrm{sn}}^2 u) \,.$

The elliptic functions satisfy a wealth of identities. For example, the addition formula for the ${\mathop\mathrm{sn}}$ function is

$\displaystyle \mathop\mathrm{sn}(A+B) = \frac{\mathop\mathrm{sn} A \mathop\mathrm{cn} B \mathop\mathrm{dn} B + \mathop\mathrm{sn} B \mathop\mathrm{cn} A \mathop\mathrm{dn} A} {1-k^2 \mathop\mathrm{sn}^2A\mathop\mathrm{sn}^2B} \,,$

which clearly reduces to the addition formula for ${\sin(A+B)}$ when ${k\rightarrow 0}$.

We introduce the reciprocals of ${\mathop\mathrm{sn}}$, ${\mathop\mathrm{cn}}$ and ${\mathop\mathrm{dn}}$, denoting them by ${\mathop\mathrm{ns}}$, ${\mathop\mathrm{nc}}$ and ${\mathop\mathrm{nd}}$. We also define the ratio ${\mathop\mathrm{sc}=\mathop\mathrm{sn}/\mathop\mathrm{cn}}$, and two other ratios in an obvious way. Finally, we denote the reciprocal ${\mathop\mathrm{cs}=1/\mathop\mathrm{sc}}$, and so on. Thus, we have in total twelve distinct elliptic functions:

$\displaystyle \begin{matrix} \mathop\mathrm{sn} & \quad\mathop\mathrm{ns} & \quad\mathop\mathrm{sc} & \quad\mathop\mathrm{cs} \\ \mathop\mathrm{cn} & \quad\mathop\mathrm{nc} & \quad\mathop\mathrm{sd} & \quad\mathop\mathrm{ds} \\ \mathop\mathrm{dn} & \quad\mathop\mathrm{dn} & \quad\mathop\mathrm{cd} & \quad\mathop\mathrm{dc} \end{matrix}$

What do the elliptic functions look like?

The representative point ${P}$ on the ellipse may be represented as ${(x,y)=(a\cos\theta, b\sin\theta)}$ or, from the definitions of ${\mathop\mathrm{sn}}$ and ${\mathop\mathrm{cn}}$, as ${(x,y)=(a\mathop\mathrm{cn} u, b\mathop\mathrm{sn} u)}$. Thus, we have

$\displaystyle \mathop\mathrm{sn} u = \sin\theta \qquad\mbox{and}\qquad \mathop\mathrm{cn} u = \cos\theta \,.$

The Jacobi elliptic functions sn (blue), cn (black) and dn (red) for ${k^2=0.9}$.

Now let us consider how ${\mathop\mathrm{sn} u = y/b}$ changes as the point ${P}$ moves around the ellipse, starting at ${(x,y)=(1,0)}$. For descriptive purposes, we can approximate ${u}$ by the polar angle ${\theta}$ if ${k}$ is small. Clearly, ${\mathop\mathrm{sn} u}$ varies periodically, starting from ${\mathop\mathrm{sn}(0) = 0}$ and oscillating in the interval ${[-1,+1]}$ as ${y}$ varies between ${-b}$ and ${+b}$. The function ${\mathop\mathrm{cn} u}$ starts from ${\mathop\mathrm{cn}(0) = 1}$ and varies periodically, oscillating in the interval ${[-1,+1]}$ as ${x}$ varies between ${-a}$ and ${+a}$. Finally, the function ${\mathop\mathrm{dn} u}$ starts from ${\mathop\mathrm{dn}(0) = 1}$ and varies periodically, but it remains positive, in the range ${[b/a,1]}$ with period half that of ${\mathop\mathrm{sn}}$ and ${\mathop\mathrm{cn}}$.

There are numerous interrelationships between the twelve elliptic functions. For example, the function ${\mathop\mathrm{cd} = \mathop\mathrm{cn} / \mathop\mathrm{dn}}$ is just ${\mathop\mathrm{sn}}$ shifted by a quarter-period, reflecting the relationship between the sine and cosine (see Fig. below).

The Jacobi elliptic functions sn (blue) and cd (magenta) for ${k^2=0.9}$.

Applications

Nonlinear differential equations for ${y(x)}$ of the form

$\displaystyle \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^2 = c_4 y^4 + c_3 y^3 + c_2 y^2 + c_1 y + c_0$

arise in a wide range of problems in physics. By a judicious transformation of variables, such equations can be solved by one of the standard Jacobi elliptic functions.

Thus, the Jacobi elliptic functions occur in many applications, such as celestial mechanics, the dynamics of spinning tops, fluid dynamics, electromagnetic theory, evaluation of integrals and the solution of differential equations.

Sources

${\bullet}$ William A. Schwalm, 2018: Elliptic Functions and Elliptic Integrals. PDF. See also four 1-hour lectures on YouTube

${\bullet}$ Wikipedia article Jacobi elliptic functions.