Reparameterization and the Tools of Frenet-Serret📂Geometry
Reparameterization and the Tools of Frenet-Serret
Definition
Let’s call β:[a,b]→R3 a regular curve. The arc length reparametrizationt=t(s) satisfies s(t)=∫at∣β′(t)∣dt, and the Frenet-Serre apparatus of the unit speed curve α(s):=β(t(s)){κα(s(t)),τα(s(t)),Tα(s(t)),Nα(s(t)),Bα(s(t))}
is defined as the Frenet-Serre apparatus of β.
Explanation
The definition of the Frenet-Serre apparatus was generalized for regular curves. This is a common method used throughout mathematics, where the unit speed is enforced if not already present. For a bijective arc length reparametrization, one can consider α, which is not β itself but incorporates its geometry.
Notation
dsdf=f′anddtdf=f˙
Dot ˙ and prime ′ both indicate differentiation, but in the context of differential geometry, symbols are distinguished as above. Usually, s represents the parameter of a unit speed curve, and t=t(s) is the parameter of the curve after arc length reparametrization. Additionally, for representing the scalar triple product, the following ternary symbols are used for cross product× and dot product⟨⋅,⋅⟩.
[u,v,w]:=⟨u×v,w⟩
Frenet-Serre Formulas: If α is a unit speed curve such that
T′(s)=N′(s)=B′(s)=κ(s)N(s)−κ(s)T(s)+τ(s)B(s)−τ(s)N(s)
Following the Notation introduced, we can easily deduce:
dtds=s˙=β˙=dtd∫at∣β′(t)∣dt
Especially, T˙ is according to the Frenet-Serre formula
T˙=dtdT=dsdTstds=κNs˙
For convenience, we first prove (d).
β˙=s˙T
Differentiating both sides by t gives
β¨===s¨T+s˙T˙s¨T+s˙2T˙′s¨T+κs˙2N
Thus
β˙×β¨===s˙T×(s¨T+κs˙2N)κs˙3Bκs˙3
As s˙=β˙, then
κ=β˙3β˙×β¨
(b)
From the proof of (d), if κ=0 then
B==κs˙3β˙×β¨β˙×β¨β˙×β¨
(c)
It’s obvious.
(e)
Differentiating β¨=s¨T+s˙T˙ once more with t gives
β⋅⋅⋅====s⋅⋅⋅T+s¨T˙+(κs˙2)⋅N+κs˙2N˙s⋅⋅⋅T+s˙s¨T′+(κs˙2)⋅N+κs˙3N′s⋅⋅⋅T+κs˙s¨N+(κs˙2)⋅N−κ2s˙3T+κτs˙3B(s⋅⋅⋅−κ2s˙3)T+(κs˙s¨+(κs˙2)⋅)N+κτs˙3B
As the scalar triple product is [β˙,β¨,β⋅⋅⋅]=⟨β˙×β¨,β⋅⋅⋅⟩ and with T⊥B, N⊥B, one finds
[β˙,β¨,β⋅⋅⋅]====⟨β˙×β¨,β⋅⋅⋅⟩⟨κs˙3B,β⋅⋅⋅⟩τ(κs˙3)2τβ˙×β¨2
In conclusion
τ=β˙×β¨2[β˙,β¨,β⋅⋅⋅]
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Millman. (1977). Elements of Differential Geometry: p46~47. ↩︎