Metrics of curves in shape optimization and analysis - Andrea Carlo ...

130−2Figure 10: Examples of curves of different winding number.and α(s) is unique, up to adding the constant k2π with k ∈ Z.Moreover α(s + 2π) − α(s) = 2πI, where I is an integer, known as windingnumber, or rotation index of ξ. This number is unaltered if ξ is homotopicallydeformed in a smooth way.See the examples in Fig. 10.The addition of a real constant to α(s) is equivalent to a rotation of ξ. Wethen understand that we may represent arc parameterized curves ξ(s), up totranslation, scaling, and rotations, by considering a suitable class of liftings α(s)for s ∈ [0, 2π].Two spaces are defined in Klassen et al. [28]; we present the case of “ShapeRepresentation using Direction Functions”;Definition 8.2 let L 2 := L 2 ([0, 2π] → lR); Φ : L 2 → lR 3 is defined byΦ 1 (α) :=∫ 2π0α(s) ds , Φ 2 (α) :=∫ 2π0cos α(s) ds , Φ 3 (α) :=∫ 2πand the space of (pre)-shapes is defined as the closed subset S of L 2 ,S := { α ∈ L 2 | Φ(α) = (2π 2 , 0, 0) } ;0sin α(s) ds .the condition Φ 1 (α) = 2π 2 forces a choice of rotation, while Φ 2 = 0, Φ 3 = 0ensure that α represents a closed curve.For any α ∈ S, it is possible to reconstruct the curve by integratingξ(s) =∫ s0(cos(α(t)), sin(α(t)))dt (8.2)This means that α identifies an unique curve (of length 2π, and arc parameterized)up to rotations and translations, and to the choice of the base point ξ(0); forthis last reason, S is called in [28] a preshape space.Inside the family of arc parameterized curves, the reparameterization 11 of ξis restricted to the transformation ˆξ(u) = ξ(u − s 0 ), that is a different choice ofbase point for the parameterization along the curve. Inside the preshape spaceS, the same operation is encoded by the relation ˆα(s) = α(s − s 0 ).To obtain a model of immersed curves up to reparameterization, rotation,translation, scaling we need to quotient S by the relation α ∼ ˆα, for all possibles 0 ∈ lR. We do not discuss this quotient here.We now prove that S \ Z is a smooth manifold. We first define Z.11 We are considering only reparameterizations in Diff + (S 1 ).53