How a Lemniscate Became an Elliptic Curve

 

The lemniscate of Bernoulli

The Lemniscate of Bernoulli is the locus of points  whose distances from the two fixed points  and  satisfy

Thus,

Squaring both sides gives

Expanding and simplifying,

Switching to polar coordinates , ;

so

Assuming  (the origin is handled separately), we obtain

The origin does lie on the lemniscate (it satisfies the original equation). So  is also a solution.

Now set , and the polar equation becomes

Renaming  back to  for simplicity yields the standard form

Note that this equation is defined only for angles

The curve takes its name from the lemniscus, a narrow woolen ribbon whose looped shape closely resembles the figure. In antiquity, such ribbons were used in ceremonial crowns awarded to victors and were also worn by initiates of the Kabeirian Mysteries. The lemniscus was traditionally regarded as a protective emblem, believed to possess sacred power and to safeguard its wearer from grave danger.

 

Arc Length

For a polar curve

 the arc length is

Indeed, from ,  and the Cartesian arc-length element

we have:

Therefore,

Hence

Since  then

so

and the formula

holds.

Now if  

 and

Thus,

 

We’ll use this result to find the arc length of the lemniscate.

From the polar form of the lemniscate

we have

or

so

Thus,

The arc length of the lemniscate is

Recall the complete elliptic integral of the first kind (see previous post)

Set

For   and

Recall (see Analytic Continuation of Gamma function in The Riemann Hypothesis Revealed) that the Beta function is defined by

So

Additionally

Thus,

The functional formula for the Gamma function states that

so  hence  and .

Therefore,

It follows that the arc length of the lemniscate is

 

 

Elliptic Curves

In general, an elliptic integral is an integral of the form

where  is a rational function and  is a cubic or quadratic polynomial in . The integral is called complete if  and  are roots of .

Such integrals arise naturally in problems involving arc lengths of curves and cannot, in general, be expressed in terms of elementary functions.

The lemniscate provides one of the classical examples leading to elliptic integrals and, ultimately, to elliptic functions. These functions, in turn, play a fundamental role in the theory of elliptic curves, which today have important applications in number theory and cryptography.

The arc length of the lemniscate involves the integrand .

If we introduce   then . Although this is already an algebraic curve, it is not yet in the standard Weierstrass form of an elliptic curve,

One possible transformation is obtained by setting

So

and after a simple sign/coordinate adjustment

Thus, the geometry of the lemniscate leads naturally to the elliptic curve , illustrating the deep connection between arc-length problems, elliptic integrals, elliptic functions, and elliptic curves.

Epilogue

What begins as a simple question about the length of a curve leads naturally to infinite processes, special functions, and deep geometric structures.

The lemniscate provides a particularly striking example. Its arc length gives rise to an elliptic integral, and the underlying algebraic curve leads to an elliptic curve. What starts as a problem in geometry ultimately connects to some of the most profound ideas in modern mathematics, including number theory and cryptography.

 

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