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Math Help - tangent lines to curves

  1. #1
    Member
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    tangent lines to curves

    Hello,

    i consider a (smooth) curve in euclidean space c:I->\mathbb{R}^3
    and i want to prove:
    if all the tangent lines to the curve meet up in a single point, then the curve must be a straight line.

    Can you please help me to proove this fact?
    Lets say, the point where all the tangent lines meet is  P\in\mathbb{R}^3.
    Then all the tangent lines look like this:
    c(t_0)+t*(P-c(t_0)).

    But how can i show that c'(t) is constant?

    Regards
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  2. #2
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    Re: tangent lines to curves

    Let s be the unit parameter, c(s)-P = f(s)c'(s), f is some unknow function of s. So
    c'(s) = f'(s)c'(s) + f(s)c''(s)
    That is (1-f'(s))c'(s) = f(s)c''(s)
    since c'(s) is unit, their inner product <c'(s), c''(s)>=0, so we must have
    1-f'(s)=0, that is f(s)=f(0)+s. f(s) is not 0 when s is not -f(0).
    So c''(s)=0 when s is not -f(0)
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