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Math Help - Curvature

  1. #1
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    Curvature

    Compute explicitly a plane curve  c(s) which is parameterized by arc length and whose curvature is  \kappa(s) = s^{-\frac{1}{2}} .
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  2. #2
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    Quote Originally Posted by mathman88 View Post
    Compute explicitly a plane curve  c(s) which is parameterized by arc length and whose curvature is  \kappa(s) = s^{-\frac{1}{2}} .
    we have c(s)=(x(s), y(s)), where x'(s)=\cos \theta, \ y'(s)=\sin \theta, where \frac{d \theta}{d s} = \kappa = \frac{1}{\sqrt{s}}. thus one solution would be \theta = 2\sqrt{s} and so x'(s)=\cos(2\sqrt{s}), \ y'(s)=\sin(2\sqrt{s}).

    now integrate (substitute 2\sqrt{s}=t and then use by parts) to find x(s) and y(s).
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