Chapter 2: Space Curves
Section 2.8: Resolution of R″ along T and N
Derive the decomposition R″p=ρ′T+κ ρ2N.
A derivation of the decomposition formula R″p=ρ′T+κ ρ2N follows from the Frenet formulas
Tp=R′/ρ and N=1κ dTds=1κ ρ T′p
if they are written as
R′=ρ T and dTds=k N
Differentiating, with respect to p, the first of these, gives
R″p=ρ T′=ρ′pT+ρ T′p
By the chain rule, the second term on the right becomes
Hence, R″p becomes
R″p=ρ′pT+ρ dTds ρ
which, upon replacing dTds with κ N, and dropping the explicit display of the independent variable p on the right, becomes the desired result R″p=ρ′T+κ ρ2N.
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