Some math for gravity effect of hairstrands

Many thanks to Manuel who installed watchmath in our Blogger site so I could type in some math about hairs :)

In hair_properties.py, hairstrands are first created with a fixed length along normal of the head. We may denote the length as $l$. When gravity is applied the strand should fall, the shape of this fall looks like the graph of (for simplicity we consider our curves in 2 dimensions) \[y = -cx^4\]

where $c$ is a positive constant dependent on the gravity factor. It has the formula (I got this after working out some equations on Young's elasticity formula, considering hairstrand an elastic material)
\[
c=\frac{{2^{2g/3}}}{2^6}
\]

But this is only half of the complexity. We need to keep the hairlength as $l$, whatever our gravity factor is. We basically need to calculate $x$ and $y$'s for a given length $l$ and a given gravity factor g. Getting arc-lengths of $y=-cx^4$ is a brachistrochrone problem. Using the equation of the arc-length we want to know $x_0$ such that 
\[
\int_0^{x_0} \sqrt{ 1 + 16c^2x^6}dx = l
\]

The above integral can be converted to an elliptic integral of the first kind. And the answer to our question is the "inverse" of the above integral, meaning we need Jacobi elliptic functions .. (more to come)