On the Generalized Curvature - Tahir Ismail; Ibrahim Hamad (Al-Rafidain Journal of Computer Sciences and Mathematics)
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 quantity in the cases where A is biregular, regular, singular or singular oforder p.