Three Near Points Method for Calculating Generalized Curvature and Torsion

The aim of this paper is to establish a new method for calculating generalized curvature and generalized torsion. This method depends on at least three points infinitely close to the considered point. The established method is characterize from other methods by using only dot and cross product of vectors combined from this points, without any parameter that interacts in the defined curve by using some concepts of nonstandard analysis given by Robinson A. and axiomatized by Nelson E. 1Introduction: In this paper the problem of obtaining the generalized curvature[6], [7], and generalized torsion[8], is considered and studied by using three points infinitely close to the given point. This method is different from the two methods given in [7], [8] and [9], and it has an advantage that it does not depend on any parameter which interact with the curve.

The axioms of IST (Internal Set Theory) which given by Nelson E. [11] are the axioms of ZFC (Zermelo-Fraenckel Set Theory with Axiom of Choice) together with three additional axioms, which are called the Transfer principle (T ), the Idealization principle ( I ), and the Standardization principle ( S ), they are stated in the following

2-Idealization principle ( I ):
Let be an internal formula with the free variables x , y and with possibly other free variables. Then

3-Standardization principle ( S ):
Let ) (z C be a formula, internal or external, with the free variable z and with possibly other free variables. Then 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, unlimited…etc" is called internal, otherwise it is called external.
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 is denoted by R , and the set of all limited real numbers is denoted by R A real number x is called infinitesimal if and only if r x < for all positive standard real numbers r .
A real number x is called appreciable if it is neither unlimited nor infinitesimal, and the set of all positive appreciable numbers is denoted by A + .
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 Necessary definitions in classical and non classical geometry can be found in [4], [5], and [12].
By a parameterized differentiable curve, we mean a differentiable map : I  R 3 of an open interval I =(a,b) of the real line R in to R such that: 3 , and x, y, and z are differentiable at t ; it is also called spherical curve.

y(t), z(t)) = x(t)e 1 + y(t)e 2 + z(t)e
Let : I  R 3 , I =(a,b), be a curve parameterized by an arc length s, and its tangent vector be ' 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 at s ,and it is given by ( ) The torsion of the curve : Let γ be a standard curve of order n C and A be a standard singular point of order p-1 on γ ; and let B and C be two points infernally close to the point A, then the generalized curvature of γ at the point A is given as follows and denoted by G where q is the order of the first vector derivative of γ not collinear with Theorem [8] Let γ be a standard curve of order n C and A be a standard singular point of order p-1 on γ ; and let B and C be two points infernally close to the point A, then the generalized torsion of the curve γ is given as follows where q is the order of the first vector derivative of γ not collinear with