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Math Help - Principal Normal and Bertrand Curves

  1. #1
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    Dec 2007
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    Principal Normal and Bertrand Curves

    I have a problem here, I need it by monday, I'll appreciate any help.

    I'll define first a BERTRAND CURVE, it is a curve \alphain R^{3} such that there exists another curve \beta, distinct from it and a bijection f between \alpha and \beta such that for each corresponding points in f, \alpha and \beta have the same principal normal.

    I need to show this things.
    1. Every plane curve is a Bertrand curve.

    I already know that If \alpha(t) is the plane curve, I can use any of its involute of the form \alpha(t)+ k*N(t). N(t)here is the principal normal of \alpha. The bijection is also obvious. But my problem is, we haven't discussed involutes in class, so I have to show that their principal normals are the same. I'm using the formulas but I can't prove it.

    2. A curve with a nonzero curvature \kappa and a nonzero torsion \tau is a Bertrand curve if and only if a \kappa + b \tau = where a,b are some constants.
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  2. #2
    santos.aag
    Guest

    i also need it by monday

    Hi Diane!
    This is not much but I hope this can help. This is for the first part without using involutions. What I've done so far is the following:
    First represent the curve
     \alpha(t) = \delta^1(t)u^1 + \delta2(t)v^1, \delta^1(t)u^2 + \delta^2(t)v^2
    where u and v span the plane containing the curve and \delta(t) = (\delta^1(t), \delta^2(t), 0).
    It's like a change of basis from { u, v} to {(1, 0, 0), (0, 1, 0)}
    Then we can take the curve
     \beta = (\delta^1(t) + at + c)u + (\delta^2(t) + bt +d)v
    where a, b, c, and d are nonzero constants (this is to ensure that we get a distinct curve). This curve has the same domain as \alpha and observe that if we differentiate this, we get
     \beta'' = \alpha''
    Hence,  N_\alpha = N_\beta as desired.
    If you consider the function
     f((t, \alpha(t)) = (t, \beta(t)) ,
    It can be verified that this is a bijection since \beta is well-defined and since \beta has the same domain as \alpha.
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  3. #3
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    Joined
    Dec 2007
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    Waaaahhhh!!! Gelo's here...(nakakahiya naman that I have to consult in his forum)
    Thanks for the reply, no one here seems to take interest in Advanced Geometry. Well, I posted this days ago, I already have an answer different from yours but it's a little complicated. I don' have anything else. Gosh, I'm really having a hard time in Math 146. Bahala na...
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  4. #4
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    Joined
    Jan 2009
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    1

    bertrand curve

    Let α be a C3 curve with τ ≠ 0.
    Prove that α is a circular helix if and only if α has at least two Bertrand mates.
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