Blaga P. Lectures on the differential geometry of - tiera.ru

48 Chapter 1. Space curves
1.9 Oriented curves. The Frenet frame of an oriented curve
As we saw before, the Frenet frame of a parameterized curve is not invariant at a parameter
change (well, at least not at any parameter change). Therefore, in order to be able
to use this apparatus also for regular curve, we need to make it invariant. The idea is to
modify a little bit the definition of the regular curve, imposing some further condition on
the local parameterization, to make sure that the parameter changes will not modify the
Frenet frames.
Definition 1.9.1. Two parameterized curves are (I, r = r(t)) and (J, ρ = ρ(u)) are called
positively equivalent if there is a parameter change λ : I → J, u = λ(t), with λ ′ (t) > 0,
∀t ∈ I.
Definition 1.9.2. An orientation of a regular curve C ⊂ R 3 is a family of local parameterizations
{(Iα, rα = rα(t))}α∈A such that
a) C = �
rα(Iα),
α∈A
b) For any connected component C b αβ of the intersection Cαβ = rα(Iα) ∩ rβ(Iβ) with
α, β ∈ A the parameterized curves (I b α, r b α) and (I b β , rb β ) with Ib α = r −1
α (C b αβ ), rb α = rα| I b α ,
I b β
= r−1
β (C0 αβ ), rb β = rβ| I b β are positively equivalent.
Example. For the unit circle S 1 the following parameterizations:
and
(I1 = (0, 2π), r1(t) = (cos t, sin t, 0))
(I2 = (−π, π), r2(t) = (cos t, sin t, 0))
give an orientation of S 1 . C12 = r1(I1)∩r2(I2) has two connected components (the upper
and the lower half circles).
Starting with the upper component, C1 12 , we have
I 1 1
I 1 2
= r−1
1 (C1 12 ) = (0, π),
= r−1
2 (C1 12 ) = (0, π)
and the parameter change is the identity, λ : (0, π) → (0, π), λ(t) = t, ∀t ∈ (0, π),
therefore the two parameterized curves are, clearly, positively equivalent.