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Proof of the FrenetSerret formulae 
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#1
Sep2212, 08:23 AM

PF Gold
P: 2,281

1. The problem statement, all variables and given/known data
Consider the unit tangent vector [itex] T[/itex] unit normal vector [itex] N [/itex] and binormal vector [itex] B [/itex] parametrized in terms of arc length s. 1) Show that [tex] \frac{dT}{ds} = \kappa\,N[/tex] I think this part is fine for me. What I did was: [tex]N(t) = \frac{T'(t)}{T'(t)}[/tex] and said, by the chain rule, [itex] \frac{dT}{ds} \frac{ds}{dt}= T'(t) [/itex] which simplified to [tex] N(s) = \frac{r'(t)}{T'(t)} \frac{dT}{ds} => \frac{dT}{ds} = \kappa N(s) [/tex] Can somebody confirm this is correct? 2) Use a) to show that there exists a scalar [itex] \tau [/itex] such that [tex] \frac{dB}{ds} = \tau\,N [/tex] I was given a hint to try to show that [itex] \frac{dB}{ds} . B = 0 [/itex] I took the derivative [tex]\frac{d}{ds} B = \frac{d}{ds}(T ×N) = T ×\frac{dN}{ds}[/tex] Therefore, [tex] (T × \frac{dN}{ds}) . B = (B ×T) . \frac{dN}{ds} = N . \frac{dN}{ds}. [/tex] Am I correct in assuming the above is equal to 0? Many thanks. 


#2
Sep2212, 10:08 AM

Mentor
P: 18,038




#3
Sep2212, 12:36 PM

PF Gold
P: 2,281




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