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

We will use this lifting to obtain, as a first step, a representation of curves upto rotation, translation, scaling. To this end, let ξ be an immersed planar closedcurve of len(ξ) = 2 (not necessarily parameterized by arc parameter); let α bethe square root lifting of the derivative ξ ′ . Let e, f be the real and imaginarypart of α, that is, α = e + if. The condition that ξ is a closed curve translatesinto ∫ 2π0 (e + if)2 dθ = 0 (where equality is in IC), hence we have two equalities(for real and imaginary part)∫ 2π0e 2 − f 2 dθ = 0,∫ 2πwhile the condition that len(ξ) = 2 translates into∫ 2π0|ξ ′ | dθ =∫ 2π0|e + if| 2 dθ =0∫ 2π0ef dθ = 0(e 2 + f 2 ) dθ = 2 .With some algebra we obtain that the above conditions are equivalent to∫ 2π0e 2 dθ = 1,∫ 2π0f 2 dθ = 1,∫ 2π0ef dθ = 0.Let L 2 = L 2 ([0, 2π] → lR); let S ⊂ L 2 × L 2 be defined byS :={(e, f) |∫ 2π0e 2 dθ = 1 =∫ 2π0f 2 dθ,∫ 2π0}ef dθ = 0.S is the Stiefel manifold of orthonormal pairs in L 2 . S is a smooth manifold,and inherits the (flat) metric of L 2 × L 2 . What is most surprising is thatTheorem 8.10 (2.2 in [39]) Let δξ be a small deformation of ξ, and δe, δfbe the corresponding small deformation of the representation e, f. Then∫∫ 2π|D s δξ| 2 ds = (δe) 2 + (δf) 2 dθξ0that is, the (geometric) Riemannian metric in M is mapped into the (flat andparametric) metric in S.It is then natural to “embed” closed planar curves (up to translation andscaling) into S.8.2.1 Curves up to rotation represented as a GrassmanianWe note that rotation of ξ by an angle τ is equivalent to rotation of the frame(e, f) by an angle τ/2. So the orbit of all rotations of ξ is associated to the planegenerated by (e, f) in L 2 : the space of curves up to rotation/translation/scalingis represented by the Grassmanian manifold of 2-planes in L 2 . A theoremby Neretin then applies, that provides a closed form formula for critical andminimal geodesics. See section 4 in [39].57