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Frenet-Serre Frames and circles.

  1. Dec 17, 2007 #1

    WWGD

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    Science Advisor
    Gold Member

    Hi, everyone:

    I am trying to show this:

    Given C(t) a unit-speed curve, using the usual Frenet-Serre frames T,N,B. Define the normal lines to C(t) to be the lines extending N, i.e, line segments containing N.
    Then:

    If all normal lines meet at a common point,
    C(t) must be part of a circle.

    Problem is that for curves, unlike for surfaces,
    tangent plane is not defined , meaning that there
    is no unique direction, no unique plane containing all
    derivatives thru a point ; I know N is perpen-
    dicular to T , by the first F-S equation, together
    with the fact that <T,T>=1 (by unit speed; k is
    curvature, assume k =/0):

    T'=kN

    <T,T>=1-> 2<T',T>=0


    But there is a planeful worth of perpendiculars
    to T', so I don't know how to tell which plane
    contains N.

    I tried to shift the axes so that the normals
    meet at the origin, but I still cannot see it.

    I thought I had found a counterexample, by
    embedding a curve in S^2 , so that the tangent
    space of the curve is a subspace of the tangent
    space to S^2. Then I used the fact that the
    normals to S^2 meet at the origin.



    I showed this to a friend, who told me only
    that my counterexample was worth 2 Kourics
    (S.Park), and I am back to the drawing board.

    Thanks for any ideas.
     
  2. jcsd
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