Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function

Abstract

We present superspirals, a new and very general family of fair curves, whose radius of curvature is given in terms of a completely monotonic Gauss hypergeometric function. The superspirals are generalizations of log-aesthetic curves, as well as other curves whose radius of curvature is a particular case of a completely monotonic Gauss hypergeometric function. High-accuracy computation of a superspiral segment is performed by the Gauss–Kronrod integration method. The proposed curves, despite their complexity, are the candidates for generating G2, and G3 non-linear superspiral splines.