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Math Help - Frenet-Serret Formulas

  1. #1
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    Frenet-Serret Formulas

    Let \kappa=\frac{\delta \overline T}{\delta s}\cdot \overline N and \tau=\frac{\delta \overline N}{\delta s}\cdot \overline B. Assume in turn that each of the intrinsic derivatives of \overline T, \overline N, \overline B are some linear combination of the unit vectors \overline T, \overline N, \overline B and hence derive the Frenet-Serret formulas of differential geometry.

    I am sure thus must be easy but I cannot see it!

    If \frac{\delta \overline T}{\delta s}=a \overline T + b \overline N + c \overline B then taking the dot product of both sides with N and using the first equation given produces \frac{\delta \overline T}{\delta s}=a \overline T + \kappa \overline N + c \overline B If I claim to know that N is orthogonal to T then I can get the first equation, that is \frac{\delta \overline T}{\delta s}=\kappa \overline N

    But when I move onto the next equation \frac{\delta \overline N}{\delta s}=a_2 \overline T + b_2 \overline N + c_2 \overline B then taking the dot product of both sides with B and using the second equation given produces \frac{\delta \overline N}{\delta s}=a_2 \overline T + b_2 \overline N + \tau \overline B.

    I cannot see how to get from here to the required equation:
    \frac{\delta \overline N}{\delta s}=-\kappa \overline T + \tau \overline B

    Or the third equation:
    \frac{\delta \overline B}{\delta s}=-\tau \overline N
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  2. #2
    Super Member Rebesques's Avatar
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    I am sure thus must be easy but I cannot see it!


    What happens when you differentiate a unit vector?
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  3. #3
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    Differentiating a unit vector we get:

    \frac{\delta \overline T}{\delta s}\cdot \overline T=\frac{\delta \overline N}{\delta s}\cdot \overline N=\frac{\delta \overline B}{\delta s}\cdot \overline B=0

    So this helps me to reduce my three equations to:

    \frac{\delta \overline T}{\delta s}=\kappa \overline N + c_1 \overline B

    \frac{\delta \overline N}{\delta s}=a_2 \overline T + \tau \overline B.

    \frac{\delta \overline B}{\delta s}=a_3 \overline T + b_3 \overline N

    But how do I show that c_1=a_3=0 and a_2=-\kappa, b_3=-\tau?
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  4. #4
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    Quote Originally Posted by Kiwi_Dave View Post
    Let \kappa=\frac{\delta \overline T}{\delta s}\cdot \overline N and \tau=\frac{\delta \overline N}{\delta s}\cdot \overline B. Assume in turn that each of the intrinsic derivatives of \overline T, \overline N, \overline B are some linear combination of the unit vectors \overline T, \overline N, \overline B and hence derive the Frenet-Serret formulas of differential geometry.

    I am sure thus must be easy but I cannot see it!

    If \frac{\delta \overline T}{\delta s}=a \overline T + b \overline N + c \overline B then taking the dot product of both sides with N and using the first equation given produces \frac{\delta \overline T}{\delta s}=a \overline T + \kappa \overline N + c \overline B If I claim to know that N is orthogonal to T then I can get the first equation, that is \frac{\delta \overline T}{\delta s}=\kappa \overline N

    But when I move onto the next equation \frac{\delta \overline N}{\delta s}=a_2 \overline T + b_2 \overline N + c_2 \overline B then taking the dot product of both sides with B and using the second equation given produces \frac{\delta \overline N}{\delta s}=a_2 \overline T + b_2 \overline N + \tau \overline B.

    I cannot see how to get from here to the required equation:
    \frac{\delta \overline N}{\delta s}=-\kappa \overline T + \tau \overline B

    Or the third equation:
    \frac{\delta \overline B}{\delta s}=-\tau \overline N
    Given \frac{d\vec{T}}{ds} = \kappa \vec{N} and \frac{d \vec{B}}{ds} = - \tau \vec{N},

    if we consider

    \vec{N} = \vec{B} \times \vec{T} then

    \frac{d \vec{N}}{ds} = \frac{d \vec{B}}{ds} \times \vec{T} + \vec{B} \times \frac{d \vec{T}}{ds} = - \tau \vec{N} \times \vec{T} + \vec{B} \times (\kappa \vec{N})<br />
= \tau \vec{B} - \kappa \vec{T}<br />

    as required.
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