Applications of fractional calculus in equiaffine geometry: plane curves with fractional order

Abstract

In this paper, we introduce the notions of equiaffine arclength and curvature with fractional order for a plane curve and compare them with the standard ones. By using these definitions, we obtain an equiaffine Frenet formula and then construct an analogue of the fundamental theorem. The plane curves of constant equiaffine curvature with fractional order are classified. Several examples are also illustrated.