A Generalized Curvature of a Generalized Envelope

Tahir H. Ismail Ibrahim O. Hamad tahir_hsis@uomosul.edu.iq ibrahim.hamad@su.edu.krd College of computers Sciences College of Sciences and Mathematics University of Mosul Salahaddin University Received on: 27/6/2007 Accepted on: 4/11/2007 ABSTRACT In this paper we study one of the applications of a generalized curvature [3] on the generalized envelope of a family of lines given in [7], [8], using some concepts of nonstandard analysis given by Robinson, A. [5] and axiomatized by Nelson, E..


1-Introduction:
The following definitions and notations are needed throughout this paper.
Every concept concerning sets or elements defined in classical mathematics is called standard [4].
Any set or formula which does not involve new predicates "standard, infinitesimals, limited, unlimited…etc" is called internal, otherwise it is called external [2], [4].
A real number x is called unlimited if and only if r x  for all positive standard real numbers, otherwise it is called limited [2].
A real number x is called infinitesimal if and only if r x  for all positive standard real numbers r [2].
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  [2], [6]. 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 [2], [4]. If x is a real limited number, then the set of all numbers, which are infinitely close to x, is called the monad of x and denoted by () mx [2], [3].
A curve  is called envelope of a family of curves {   } depending on a parameter , if at each of its points, it is tangent to at least one curve of the family {   }, and if each of its segments is tangent to an infinite set of these curves [1]. The projective homogenous plane over R , denoted by 2 R P is the set: In this sense the projective homogeneous coordinates of any point is not unique. [1] By a parameterized differentiable curve, we mean a differentiable map  : I → 3 R of an open interval I =(a,b) of the real line R into 3 R such that:  (t)=(x(t), y(t), z(t)) = x(t)e1+ y(t)e2+ z(t)e3, and x, y, and z are differentiable at t ; it is also called spherical curve [2].

Definition 1.1 [7]
Let A= ) t (  be a standard point on the curve  , then the following cases occur for the point A with the existence of the order of derivatives of  : then the point is called biregular point.
then the point is called only regular point, and we say that the point is only regular point of order p-1 if 0   and . In this case we say that p is the order of the first vector derivative not collinear with  4-If 0 =  then the point is called singular point. In general if ,then the point is called singular point of order p.

Theorem 1.2 [7]
Let  be a standard curve of order n C and A be a standard singular point of order p-1 on  ; and let B and C be two points infinitely close to the point A, then the generalized curvature of  at the point denoted by G K and given by where q is the order of the first vector derivative of  not collinear with If pk(t)= rk(t)= qk(t)=0 for  1 k n (n standard) and pn(t), rn(t), qn(t) are not all zeros, then the PHC points of  (t) are of the form (pn(t), rn(t), qn(t)) which does not depend on e . Thus, we get the generalized nonclassical form of the envelope curve  (t) as follows:

2-A Generalized Curvature of the Envelope of a Family of Lines
Throughout this section, we give a curvature formula for the envelope of a family of lines

Apply the generalized curvature formula given in Theorem 1.2 at the points A(to), B(to+  ) and C(to+ ).
Where  and  are infinitesimal numbers.
The following theorems will give a new formula of the generalized curvature of the envelope of a family of lines.

Theorem 2.1
Let A= (to) be a regular point of the envelope curve  of the family Since, the point A is regular, then Theorem 1.3 for n=1 is satisfied, and therefore  (t)= (p1(t), r1(t), q1(t)) Using the spherical case of the generalized curvature given in Theorem 1.2 for a curve ( )

Theorem 2.2
Let A=  (to) be a singular point of the envelope curve  of order n-1, and let m be the order of the first nonzero derivative which is not Then, the generalized curvature G K of the envelope curve  at the points of the monad of A is given by  where n and m are positive integer numbers.

Proof:
First, applying the spherical case of the generalized curvature given in Theorem 1.2 at x =p1(t), y =r1(t) and z =q1(t), we get the generalize curvature formula (2.2.1). Since the point (p1(t), r1(t), q1(t)) in PHC is equivalent to the point (p1(t)/q1(t),r1(t)/q1(t),1), so again, applying the spherical case of generalized curvature, we get Moreover, let the coefficient vector (u(t), v(t), w(t)) of the envelope curve has a singularity of order n-1, then the generalized curvature G K of the envelope curve  at points in the monad of A is given by