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I Bounded Curves

  1. Aug 3, 2016 #1
    It is well known that a curve in ##\mathbb{R}^3## is uniquely (up to a position in the space) defined by its curvature ##\kappa(s)## and torsion ##\tau(s)##, here ##s## is the arc-length parameter. We will consider ##\kappa(s),\tau(s)\in C[0,\infty)##

    Thus a natural problem arises: to restore features of the curve from given functions ##\kappa(s),\tau(s)##. This is not a simple problem: the Frenet-Serret equations are not in general integrable explicitly.


    A simplest property of the curve is boundedness. We shall say that a curve is bounded iff its radius-vector is bounded: $$\sup_{s\ge 0}|\boldsymbol r(s)|<\infty.$$

    For example, if a function ##\kappa(s)/\tau(s)## is monotone and
    $$\lim_{s\to\infty}\frac{\kappa(s)}{s\cdot\tau(s)}=0$$ then the curve is unbounded. This is an almost trivial fact, it follows from some another almost trivial theorem, for details see http://www.ma.utexas.edu/mp_arc/c/16/16-63.pdf

    The comments are welcome. Particularly is there a geometrical interpretation of brought above proposition
     
    Last edited: Aug 3, 2016
  2. jcsd
  3. Aug 4, 2016 #2
    I find it amazing. But will you explain me the restore features of curve. Actually I am going to read about curves in detail but I need little guidance. Can you name some of the books which I can buy.
     
  4. Aug 4, 2016 #3
    I was saying that why you have used word bounded curves. Will you tell me its precise meaning.
     
  5. Aug 4, 2016 #4
    Be more specific what you are saying
     
  6. Aug 4, 2016 #5
    I have already done that
     
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