Keywords : infinitesimals


A Nonstandard Generalization of Envelopes

Tahir H. Ismail; Ibrahim O. Hamad

AL-Rafidain Journal of Computer Sciences and Mathematics, 2008, Volume 5, Issue 2, Pages 73-83
DOI: 10.33899/csmj.2008.163973

The generalized envelopes are studied by a given nonstandard definition of envelope of a family of lines defined in a projective homogenous coordinates PHC by: u(t)x + v(t)y + w(t)z = 0. The new nonstandard concepts of envelope are applied to conic sections. Our goal in this paper is hat for a given conic section curve f(x,y)=0, we search for the family of lines in which  f  is its envelope.
   
 

On the Generalized Curvature

Tahir H. Ismail; Ibrahim O. Hamad

AL-Rafidain Journal of Computer Sciences and Mathematics, 2006, Volume 3, Issue 2, Pages 83-98
DOI: 10.33899/csmj.2006.164054

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.