A Nonstandard Generalization of Envelopes

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.

Every concept concerning sets or elements defined in the classical mathematics is called standard [7].
Any set or formula which does not involve new predicates "standard, infinitesimals, limited, unlimited…etc" is called internal, otherwise it is called external. [7] A real number x is called unlimited if and only if r x  for all positive standard real numbers; otherwise it is called limited [4].
A real number x is called infinitesimal if and only if r x  for all positive standard real numbers r [6], [4].
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  [8]. 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 [6], [8]. [3] Let X and Y be two standard sets, s X and s Y be the subsets constitute of the standard elements of X and Y, respectively. If we can associate with every x s X a unique y =f(x) s Y then there exists a unique standard y *  Y such that st  x , y * =f(x) Let  and be any two infinitesimal numbers and 0  r is a limited real number, then:

1.
r is an infinitesimal.

3.
r is limited.

4.
is an infinitesimal (in general the sum of any arbitrary finite number of infinitesimal numbers is infinitesimal) [6].
The projective plane over R , denoted by 2 R P is the set in this sense the projective homogeneous coordinates of any point is not unique [2].
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 [2].
By a parameterized differentiable curve, we mean a differentiable map : I 3 R of an open interval =(a,b) of the real line R in to 3 R such that:

An Envelope of a Family of Lines in a Plane
We consider 2 R as a subset of PHP, let   t L be a family of lines in PHC space defined by: , and suppose that the ordered pairs (u(t),v(t)),(u(t),w(t)),(v(t),w(t)) ,where u, v, w are standard functions defined on an interval sub set of R .
The purpose is to associate a standard curve which is coincident with the envelope to the family   t L .
Then the intersection point of   L t and   + L t in PHC is given by: Suppose that the functions u, v, and w are differentiable functions each of order at least n , then by expanding each of u(t+ ), v(t+ ), and w(t+ ) using Taylor development, we get 3 are infinitesimals. In general, put The following cases are related to the last assumption Case1. If q1(t)  0 and p1(t)and r1(t) are not both zero, then the PHC points of envelope curve γ(t), (p1(t), r1(t), q1(t) ), are independent on , and the triple (p1(t), r1(t), q1(t) ) represents the classical definition of an envelope curve.

v'(t)w(t)-w'(t)v(t))
In the same way, we have

t)w(t)-w'(t)v(t)), (w'(t)u(t)-u'(t)w(t)), (u'(t)v(t)-v'(t)u(t)))
Now, using the properties of the PHC, we deduce that any point of the form ( a, b, c) is equivalent with the point (a, b, c) for any parameter .

w t w t v t w t u t u t w t u t v t v t u t u t v t v t u t
This is the classical form of the envelope curve of the family of straight lines.

Case4. 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 . Thus, we get the generalized nonclassical form of the envelope curve (t) as follows:

Case5. If pk(t)= rk(t)= qk(t)
=0 for any value of k, then we can not say any thing about the generalization of the envelope curve. In the following sections, by () pt , () rt and () qt we mean p1(t), r1(t) and q1(t) respectively.

Applications to Conic Sections
We restrict our study on a family of straight lines only, other studies on envelops and singularity of envelops, for example can be found in [1]. Our goal is that for a given conic section curve f(x,y)=0, we search for a family of lines in which f is its envelope.

Proof:
Obvious If   t L is a family of lines defined by:

Proof:
Consider the families of lines   t L and   + t L By using Lemma 3.1 , we get that: :

Therefore,
This is a general form of second degree equation in two variables x and y.
Conversely, assuming that we have a second degree equation of two variables x and y such as: Completing the square for each uncompleted square related to the variables x and y in Equation (3.2.4) and simplifying the result, we get:  Hence we get the required result.

Remark 3.3
If the last conditions of the previous theorem are not valid or The circle x 2 + y 2 =1 is an envelope curve of the family of lines (1-t 2 )x + (2t)y + (t 2 +1) =0 , such as shown in the (1/2-x/2)t 2 + yt + 1/2 + x/2=0, which is an equation of a family of lines.  Figure 3.1 Note that we can show that the given circle equation x 2 + y 2 =1 is an envelope equation of the founded family classically or by nonstandard tools. In the following, we give a nonstandard method for such purpose.

Example 3.5
Consider the curve x + y 2 -1=0 By applying Theorem 3.2 to the given equation, we get y 2 -4(1/4)(1-x) =0 , now use Lemma 3.1we get 1/4t 2 + yt + 1-x =0 which is a family of lines whose envelope is the given equation, such as shown in the