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Math Help - Geometry Question

  1. #1
    Member
    Joined
    Feb 2011
    Posts
    75

    Geometry Question

    If C is the curve parameterised by
    t (cost, sint, cosht):[-pi,pi] reals ^3
    Reparameterise C by arclength?
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  2. #2
    Senior Member
    Joined
    Mar 2010
    Posts
    280
    <br />
s=\int _{- \pi}^t \sqrt{x'^2t)+y'^2(t)+z'^2(t)}dt<br />

    <br />
s=\int _{- \pi}^t \sqrt{sin^2 p+cos^2 p+sinh^2 p} \; dp=\int _{- \pi}^t \sqrt{1+sinh^2 p} \; dp=\int _{- \pi}^t  \; cosh (p) \; dp=<br />

    <br />
=sinh(t)-sinh(- \pi).<br />

    Now we need to express t as function of s

    <br />
t=f(s)<br />

    and the curve becomes

    <br />
( \; cos(f(s)), \; sin(f(s)), \; cosh(f(s)) \; ).<br />
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