Tyrone Ferguson Jr.

In my journey in taming the ellipse, through reviewing the connections curves similar to elliptic functions have with each other, I realized that while it may be impossible to derive the ellipse through brute force, an indirect method is possible, Most of the approximations derived through curve parameterization yielded a form similar to this:

Integral of sqrt(49sec^2(x)+169cot^2(x)) {integral-calculator}

The form follows a pair of mirrored logarithms. Upon further investigation, the derivatives of each of its components:

Combining like fractions yields a simpler pair:

NOTE: The second fraction can be rewritten using the same denominator as the first, with 1352tanxcot^2x in the numerator.

I postulate that it’s possible map out how the pieces fit into place, and then constructing from scratch an equation for an elliptic function. I already have a few candidate functions that I’m working on using a combination of factorization and partial fraction decomposition to coax into a differentiable and integrable form for at least the first quadrant.

Tyrone Ferguson Jr.