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Math Help - Finding unit speed reparametrisations of regular curves

  1. #1
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    Finding unit speed reparametrisations of regular curves

    How can i find unit speed reparametrisations of these regular curves?

    1) y(t) = (cos^2(t),sin^2(t)) -infinity < t < infinity (cos squared t, sine squared t)

    2) y(t) = (t, cosh(t) -infinity < t < infinity

    thanks
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  2. #2
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    Quote Originally Posted by millwallcrazy View Post
    How can i find unit speed reparametrisations of these regular curves?

    1) y(t) = (cos^2(t),sin^2(t)) -infinity < t < infinity (cos squared t, sine squared t)

    2) y(t) = (t, cosh(t) -infinity < t < infinity
    I'll show how to do 2). Then you try 1). (Though why someone from Leeds should want to help a Millwall fan I can't imagine. )

    In 2), the coordinates are x=t, y=cosh(t). The arc length is given by u(s) = \int_0^s\sqrt{\bigl(\tfrac{dx}{dt}\bigr)^2 + \bigl(\tfrac{dy}{dt}\bigr)^2}\,dt = \int_0^s\sqrt{1+\sinh^2t}\,dt = \sinh s. The inverse function is given by s = \sinh^{-1}u , and the unit speed parametrisation (with parameter u) is given by

    \bigl(x(s(u)),y(s(u))\bigr) = \bigl(\sinh^{-1}u,\cosh(\sinh^{-1}u\bigr) = \bigl(\sinh^{-1}u,\sqrt{1+u^2}\bigr).
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