I am trying to approximate (what I believe to be) an elliptic function in a modeling program for engineering work. Unfortunately I'm a bit out of my depth, and it's been quite a while since I did complex analysis.
The function is : $$ (y/k)^2 = -2ax(x-1)^3 -bx^2(x-1)^2 + x^2(3x^2-8x+6), x\in[0,1] $$ The modeling program I'm using can approximate the curve using a 5th order Hermite interpolation, which means that I need the first and second derivatives at 0,1 as well as the arc length to normalize the curve. Both the derivatives can be easily found by inspection, but the arc-length is tricky (and apparently impossible). From what I've read it is apparently an elliptic function.
This will be used for a parametric model, so while a Riemann sum is possible, it would be much better to have an algebraic approximation.