On the Generalized Curvature

By using methods of nonstandard analysis given by Robinson, A., and axiomatized by Nelson, E., we try in this paper to establish the generalized curvature of a plane curve at regular points and at points infinitely close to a singular point. It is known that the radius of curvature of a plane curve is the limit of the radius of a circle circumscribed to a triangle ABC, where B and C are points ofinfinitely close to A. Our goal is to give a nonstandard proof of this fact. More precisely, if A is a standard point of a standard curve and B, C are points of defined by and where and are infinitesimals, we intend to calculate the quantityin the cases where A is biregular, regular, singular or singular oforder p.