On the Generalized Curvature

• Main Authors: Tahir Ismail, Ibrahim Hamad
• Format: Article
• Language: Arabic
• Published: Mosul University 2006-12-01
• Series: Al-Rafidain Journal of Computer Sciences and Mathematics
• Subjects: infinitesimals; curvature; torsion; singularity
• Online Access: https://csmj.mosuljournals.com/article_164054_46d6bf9bd2c8d4597b059d5c795cc6d0.pdf

By using methods of nonstandard analysis given by <strong>Robinson, A.</strong>, and axiomatized by <strong>Nelson, E.</strong>, 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 <strong>curvature</strong> of a plane curve is the limit of the radius of a circle circumscribed to a triangle <strong><em>ABC</em></strong>, where <strong><em>B</em></strong> and <strong><em>C</em></strong> are points ofinfinitely close to <strong><em>A</em></strong>. Our goal is to give a nonstandard proof of this fact. More precisely, if <strong><em>A</em></strong> is a standard point of a standard curve  and <strong><em>B</em></strong>, <strong><em>C</em></strong> are points of  defined by  and where  and  are infinitesimals, we intend to calculate the quantity in the cases where <strong><em>A</em></strong> is <strong>biregular</strong>, <strong>regular</strong>, <strong>singular</strong> or <strong>singular </strong>oforder <strong><em>p.</em></strong>