Reformulation of the Generalized Curvature by Using Decomposition Technique

The aim of this paper is to establish a new formula of generalized curvature discussed with the help of decomposition theorem which is used to obtain generalized curvature for algebraic curves by using some concepts of nonstandard analysis given by Robinson A. and axiomatized by Nelson E. 1-Introduction: Among the difference expressions of the generalized curvature, there is one which can be expressed by using only one point infinitely close to a singular point of a standard curve and this will be our consideration in this paper. This is derived from generalized curvature form [10], with the help of decomposition theorem. The following definitions of nonstandard analysis will be needed throughout this paper. [4], [5], [9], [11], [12], [15], [16], [17] A real number x is called unlimited if and only if r x > for all positive standard real numbers, otherwise it is called limited. The set of all unlimited real numbers denoted byR , and the set of all limited real numbers denoted by R A real number x is called infinitesimal if and only if r x < for all positive standard real numbers r . Reformulation of the Generalized.... 46 Two real numbers x and y are said to be infinitely close if and only if y x − is infinitesimal and denoted by y x ≅ . If x is a limited number in R , then it is infinitely close to a unique standard real number, this unique number is called the standard part of x or shadow of x denoted by ) (x st or x 0 . Every set or element defined in a classical mathematics is called standard. Any set or formula which does not involve new predicates “standard, infinitesimals, limited,...etc” is called internal, otherwise is called external. Let α : I → R , I =(a,b) , be a curve parametrized by an arc length s, and its tangent vector be α ' which has a unit length, then the measure of the ratio of change of the angle with neighboring tangents made with the tangent at s is known as a curvature of the curve α at s ,and it is given by ( ) ′′ s g and denoted by ( ) s κ Theorem 1.1 [10] Let A ( ) = t γ be a standard point on the planar curve γ , and let B and C be two points infinitely close to the point A, then