Formula omitted] quintic polynomial approximation for Generalised Cornu Spiral segments

Abstract

Within Computer Aided Design (CAD) there is a need to construct fair curves. The Generalised Cornu Spirals (GCSs) are a set of curves with a monotonic curvature profile and are hence considered fair but implementation in current CAD systems is not straightforward, partly due to not being in the usual polynomial form.A method to approximate a GCS using a quintic polynomial curve is presented. The method seeks to interpolate the GCS to satisfy the G3 constraints at the end points with a quintic Bézier, leaving two degrees of freedom.An initial approximation is shown to be effective for the majority of GCS curves. Moreover, it is possible to determine when an initial approximation is likely to be poor. If this approximation does not meet the tolerance required, a search involving two parameters is performed. Characteristics of the search domain are used to establish a suitable starting value.