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Math Help - total torsion of closed curve on sphere

  1. #1
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    total torsion of closed curve on sphere

    The total torsion of a unit speed curve γ : [a, b] → R3 is defined to be. ∫ τ(s) ds (bounds a to b), where τ(s) is the torsion of γ at the point γ(s). If γ is a closed curve on the unit sphere S2, show that γ has zero total torsion. I have a couple hints to solve this but I can't figure it out, the hints are use k(T)=K_γ cos θ where T=γ' for a unit speed curve γ, and differentiate cosθ=N · U


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  2. #2
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    Re: total torsion of closed curve on sphere

    For the curve r=r(t), its tangent, normal, and binormal unit vectors, often called T, N, and B, satisfies the Frenet–Serret formulas. So we have:
    \frac{dN}{ds}=-\kappa T+\tau B.
    Let n be the unit normal vector of the sphere, and denote \theta as the angle between n and N such that \cos\theta=\langle N, n \rangle
    Differentiate the above equation against s the arc length parameter we get:
    -\sin\theta \frac{d\theta}{ds}=\langle \frac{dN}{ds}, n \rangle + \langle N, \frac{dn}{ds} \rangle
    For spheres we have n=r the position vector, so \frac{dn}{ds}=\frac{dr}{ds}=T
    Plugin this one and the above Frenet formula we get
    -\sin\theta \frac{d\theta}{ds}=\langle -\kappa T+\tau B, n \rangle + \langle N, T \rangle = \tau \langle  B, n \rangle
    Since B, N, n are all orthogonal to T, their three are in the same plane. And since the angle between N and n is \theta,
    and B, N are orthogonal, the angle between B and n is then \theta-\frac{\pi}{2}.
    So we have \langle B, n \rangle=\sin\theta. Plugin to the above equation we got:
    \tau=-\frac{d\theta}{ds}
    Integrate it we get \int \tau ds = -\int d\theta = 0
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  3. #3
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    Re: total torsion of closed curve on sphere

    why does n=r?
    Last edited by math410; May 5th 2013 at 01:19 PM.
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  4. #4
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    Re: total torsion of closed curve on sphere

    You said it is a unit sphere. so n=r.
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    Re: total torsion of closed curve on sphere

    Quote Originally Posted by xxp9 View Post
    You said it is a unit sphere. so n=r.
    Can you explain more about n=r? i still don't get this part. Thanks
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  6. #6
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    Re: total torsion of closed curve on sphere

    for the unit sphere S, for any point p on it, the unit normal vector n at p has the same direction and length as p. that is, n=p.
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