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Obtaining the curve through envelope?

  1. Apr 14, 2006 #1
    I was looking through how to obtain the curve if you have a family of straight lines.. and I found that you 'obtain the envelope' promptly looking this up i got:The form of envelope treated here is a manifold that manages to be tangent to some point of each member of a family of manifolds. Curves are the usual manifolds involved.

    The simplest formal expression for an envelope of curves in the (x,y)-plane is the pair of equations

    F(x,y,t)=0\qquad\qquad(1)\,

    {\partial F(x,y,t)\over\partial t}=0\qquad\qquad(2)\,

    where the family is implicitly defined by (1). Obviously the family has to be "nicely" -- differentiably -- indexed by t.

    The logic of this form may not be obvious, but in the vulgar: solutions of (2) are places where F(x,y,t), and thus (x,y), are "constant" in t -- ie, where "adjacent" family members intersect, which is another feature of the envelope. [wikipedia]

    any idea on how this could be done? a tip on how enveloping would be helpfull.. thank you
     
  2. jcsd
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