Efficient Robust Approximation of the GCS

A Generalised Cornu Spiral (GCS) is a planar curve defined to have a monotonic rational linear curvature profile and as such these curves are considered fair. However, their implementation in current CAD systems is not straight forward partly due to not being in the usual polynomial form. A GCS cannot be expressed exactly using a finite polynomial and so a compromise can be achieved by instead approximating the GCS with a suitable polynomial. An efficient robust approximation of the GCS using quintic polynomials is presented. The approximation satisfies the G2 continuity conditions at the end points and the remaining four degrees of freedom are argued for by looking at G3 approximations. The method begins by reparamterising the GCS in terms of more intuitive geometric descriptions; the winding angle, change in curvature and a shape factor. The G3 approximations provide insight to help define values for the free parameters, and the new geometric form allows for the shortcomings in the G3 approximations to be controlled. The efficiency of the approximation is improved compared to earlier methods which required a numerical search. Also, there is strong evidence that the method guarantees a satisfactory approximation when the GCS lies within certain identified bounds.