You're really trying to nerd snipe, aren't you? I spent some quality time with elliptic functions as the solutions to the elastica, and also with the Fresnel integral which of course generates the Euler spiral. So yes, it wouldn't be surprising if there were a closed form solution, and that it might even be more efficient to compute.
So, this would be a good topic for someone else to show off their math skills :)
So, this would be a good topic for someone else to show off their math skills :)