I was working on the math involved with easements while I was trying to fall asleep the other night. I was hoping to find a closed form solution using the integral of dK/dO and the equation of K(O). However, I was not able to achieve separation of variables in polar coordinates but I could see the solution of the curve in my mind as well as the separate function describing the effective radius for the angle (the osculating circle). I was aware that I had seen that curve before. It's not just a plain spiral and it has double quadrant mirrored symmetry.
It is a "Euler Spiral" and the solution is most likely based on the Euler functions. Check out (https://en.wikipedia.org/wiki/Euler_spiral) and the section on Track Transition Curve. Given boundary conditions (angle of transition (eg. 45 or 90) and the final curvature (eg. 1/36 <= 1/(O-72/2)), you can calculate the exact points for the easement.
Anthony
