From Arc length of an Ellipse to Elliptic Curves – The Amazing journey of Mathematics!

In geometry, an ellipse is described as a curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. In the following figure, F1 and F2 are called the foci of the ellipse.

Ellipse has two types of axis- Major Axis and Minor Axis. The longest chord of the ellipse is the major axis. The perpendicular chord to the major axis which bisects the major axis at the center is the minor axis.

The area of the ellipse

is given by the formula A=πab. We recognize this as a simple generalization of the formula for the area of a circle of radius a given by A=πa². What can we say about the perimeter of the ellipse? The circle of radius a has circumference C=2πa, so is the perimeter of the ellipse something simple like P = 2π(a+b)/2, where we have used the average of a and b in place if the radius of the circle? The answer is no!

The differential arc length for a curve given by parametric equations x=x(θ) and y=y(θ) is

The ellipse given by the parametric equations x=a cos θ and y=b sin θ has differential arc length

Therefore, the perimeter of the ellipse is given by the integral

In which we have quadrupled the arc length found in the first quadrant. Replacing sin² θ by 1-cos² θ we get

if we let

We can write (1) as

Since cos θ = sin(π/2- θ), for any function f(x) we can write

With the substitution t= π/2- θ we convert this to

Therefore, we can replace the cosine in (3) by a sine to obtain

The reason for this conversion is to express the perimeter in terms of the named function

Which is called a complete elliptic integral of the second kind. Thus we can now write

The complete elliptic integral of the first kind defined by the integral

Occurs in the analysis of the motion of the pendulum with large amplitude

Elliptic integrals led to the discovery of Elliptic functions which had applications in mechanics and relativity in physics. They were generalizations of singly periodic trigonometric functions.

Below we show an elliptic function, a function named sinus amplitudinis (the sine of the amplitude) by Jacobi in his famous work -New foundations of the theory of the elliptic functions- from 1829. It is denoted by sn(z,k) and depends on a parameter k which is called the ``modulus''. (Note: sometimes m = k² is called the modulus instead.)

Usually k is taken between 0 and 1, but the function can also be defined for complex values of k. When k = 0, sn(z,k) simply reduces to sin(z), so it can be thought of as a ``distorted sine function''.

Here is what the function looks like when k = 1/4, on the square with corners at z = ±6±6i 

Along the real axis (the middle horizontal line in the plot), this function looks much like the sine function; it oscillates periodically between −1 and 1. The period depends on k and is actually larger than 2π, although for k = 1/4 the difference is so small that it's barely noticable in the picture.

In the complex plane, however, something new has happened: the vertical edges of the half-strips have bent inward to enclose rectangles instead. Where the tips meet, the function has simple poles. The function is doubly periodic; it repeats itself not only in the right-left direction, but in the up-down direction as well

The general definition of an elliptic function is a f(z) that is monomorphic and doubly periodic. Monomorphic means analytic except possibly at isolated poles, and doubly periodic  means that there are two periods A and B (complex numbers) such that F(z+nA+mB)=F(z) for all zϵC and all integers n and m. The periods span a parallelogram in the complex plane. If F(z) is known in this parallelogram, then it is known everywhere because of the periodicity. In case of sn(z,k) with K between 0 and 1, A is real, B is purely imaginary, and the period is made up of two blue and two red rectangles.

Elliptic functions led mathematicians to Elliptic curves. They have shown important applications in applied math (Cryptography) and pure math in the proof of Fermat’s Last theorem by Sir Andrew Wiles in 1995 and in the formulation of Birch Swinnerton-Dyer conjecture in the 1960’s.

It has been quite amazing to observe how the question of a tidy formula for the arc length of an ellipse led over time into important math far removed from the original setting of ellipses. 

Acknowledgement

I would like to thank Prof Keith Conrad for suggesting this discussion. All inaccuracies are mine.

Reference
1)Elliptic Function, Wikipedia
2) Hans Lundmark's complex analysis page
3) Mathematics and its History by John stillwell
4) Elliptic integral, Wikipedia