On the generalized curvature

Abstract EN

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 γ (t) at regular points and at points infinitely close to a singular point.

It is known that the radius of curvature of a plane curve γ (t) is the limit of the radius of a circle circumscribed to a triangle ABC, where B and C are points of g infinitely 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 g and B, C are points of g defined by B = γ (t + a) and C = γ (t + b) where a and b are infinitesimals, we intend to calculate the quantity tanA/BC in the cases where A is biregular regular, singular or singular of order p.