I was reading the wiki article about Euler spirals and I reached this passage:
Unaware of the solution of the geometry by Leonhard Euler, Rankine cited the cubic curve (a polynomial curve of degree 3), which is an approximation of the Euler spiral for small angular changes in the same way that a parabola is an approximation to a circular curve.
I then tried to find out more, but I have been unable to find any source or detailed explanation of how the cubic can approximate a (well delimited section of a) Euler spiral.
Given two points on a spiral, with their local curvature, how can the coefficients of a cubic curve can be computed so to optimally approximate the spiral?