Frenet-Serret Formulas

  • #1
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I want to substitute ds by dt in the Frenet-Serret Formulas where κ is the curvature and is the torsion:
Tangential:[tex]\frac{d\vec{T}}{ds} = κ*\vec{N}[/tex]
Normal:[tex]\frac{d\vec{N}}{ds} = -κ*\vec{T}+τ*\vec{B}[/tex]
Binormal:[tex]\frac{d\vec{B}}{ds} =- τ*\vec{N}[/tex]
I want to substitute [tex]\frac{d\vec{T}}{ds} → \frac{d}{dt} T(t)[/tex] N(t), B(t) and solve for κ and τ.
 
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Answers and Replies

  • #2
vanhees71
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Just use the chain rule and the fact that
[tex]\dot{s}=\frac{\mathrm{d} s}{\mathrm{d} t}=\left | \frac{\mathrm{d} \vec{r}}{\mathrm{d} t} \right|=\left |\dot{\vec{r}} \right|,[/tex]
where the function [itex]\vec{r}(t)[/itex][/itex] is the parametrization of the curve. Then you have
[tex]\vec{T}=\frac{\dot{\vec{r}}}{\dot{s}}[/tex]
etc.
 

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