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Math Help - Principal Unit Normal and Curvature

  1. #1
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    Principal Unit Normal and Curvature

    Just wanted to check these... my notes don't have many (read any!) examples, so I'm not sure if what I'm doing is correct.

    Given the arclength parameterisation r(t) of C, find the curvature and principal unit normal of C

     r(t) = (\frac{1}{4}\cos(2t), \frac{1}{2}t - \frac{1}{4}\sin{2t}, \sin{t})


     \kappa(t) = ||r''(t)|| = ||(-\cos{2t}, \sin{2t}, -\sin{t}|| = \sin{t}

    and the principal unit normal  N(t) = \frac{r''(t)}{||r''(t)||}

     N(t) =\frac{1}{\sin{t}} (-\cos{2t}, \sin{2t}, -\sin{t})
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  2. #2
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    Quote Originally Posted by cana View Post
    Just wanted to check these... my notes don't have many (read any!) examples, so I'm not sure if what I'm doing is correct.

    Given the arclength parameterisation r(t) of C, find the curvature and principal unit normal of C

     r(t) = (\frac{1}{4}\cos(2t), \frac{1}{2}t - \frac{1}{4}\sin{2t}, \sin{t})


     \kappa(t) = ||r''(t)|| = ||(-\cos{2t}, \sin{2t}, -\sin{t})|| = \sin{t}
    Should be \color{red}\sqrt{\cos^22t + \sin^22t + \sin^2t} = \sqrt{1+\sin^2 t}.

    and the principal unit normal  N(t) = \frac{r''(t)}{||r''(t)||}

     N(t) =\frac{1}{\sin{t}} (-\cos{2t}, \sin{2t}, -\sin{t})
    Other than that, it's fine.
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