Mathematical Generalization of Log-Aesthetic Curves and Aesthetic Curves Hierarchy Via Similarity Geometry

Abstract

This article deals with mathematical generalization of log-aesthetic curves (LAC). We generalized LAC in terms of similarity geometry. This generalized family of curves called quasi-aesthetic curves (QAC) contains LACs, parabolic arcs, typical curves of Mineur and some well-known plane curves in differential geometry. However, the well-used curves in the field of industrial shape design, for exsample, elliptical arcs and hyperbolic arcs are not contained in the family of QACs. As our viewpoint, we observed the similarity curvature of QAC. The similarity curvature of QAC satisfies the 2-th order Burgers equation. In the applied mathematical context, the 2-th order Burgers equation is a member of the family of the n-th order Burgers equations (Burgers hierarchy). By interpreting this result, we present aesthetic curve hierarchy.