An arc spline approximation to a clothoid

The clothoid is a spiral used in highway and railway route design. Clothoids are transcendental functions and so have been approximated by polynomials, by power series and continued fractions, and by rational functions. Here the clothoid is approximated by an arc spline. The chief advantage in doing so is that arc splines are very easy to lay out and to offset. Examples show that the approximation is of extremely high accuracy. It is proved that if the arc spline has n arcs, then the error in the approximation is of order O(1/n2).