I'm not sure if I fully understand the answer to Determining the angle degree of an arc in ellipse?, and my main question is: Is there a way, and what would be the best route to take to learn how to extend the answer from Determining the angle degree of an arc in ellipse? in order to derive a direct formula for what that question asks, or at very least, be able to make the computation with access only to a free language such as Gnu Octave, JavaScript, Python, Java, or just a calculator? Or, should I use Mathematica's Wolfram Engine? https://www.wolfram.com/engine/ And I'd like to be able to derive the formula a) to see how that would be done and b) to be able to use the formula for other parameter sets, not just for the parameters given. In other words, I would like to learn to derive a working formula for this, without using Mathematica's FindRoot. And if I do need FindRoot or a similar tool, is there a way to use free software instead. If it helps, if it's simpler, I would also be happy deriving $x$ or $y$ instead of $ϕ$ starting from arc length, $a$ and $b$.
Also if it helps, what I'm leading up to with this is that I'm trying to derive an equation that would define the edge of a lune of a sphere, ie. tracing a beach ball segment. If I can get $x$, $y$ or $ϕ$ in terms of arc length, $a$ and $b$, then I will know the radius to that edge from inside the sphere and hence the cord length across the lune, which will give me hopefully the lune width in terms of this arc length, which is the lune length. In short, I'm hoping for my end result to be an equation that lets me precisely create perfect beach ball segments.
I see this: http://1.bp.blogspot.com/_uuCqsyqQObY/STtDEfSyV7I/AAAAAAAAATU/OnElGYpwdaU/s1600-h/dissect_the_sphere.png and I'd like to be able to derive the formulas leading up to it for three reasons: curiosity, flexibility, and correctness checking.
