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

of a small ball from A. The motion v At inside N c is Lipschitz, but the limitlimt→0+v At − v A‖v At − v A ‖ L pdoes not exist in L p . (Morally, if p = 1, the limit would be the measure δ x ).6.3.3 Riemannian metricLet now p = 2. The set N c may fail to be a smooth submanifold of L 2 ; yetwe will, as much as possible, pretend that it is, in order to induce a sort of“Riemannian metric” on N c from the standard L 2 metric.Definition 6.20 We define the “Riemannian metric” on N c simply by〈h, k〉 := 〈h, k〉 L 2for h, k ∈ T v N c and correspondingly a norm bywhere T v N c is the contingent cone.|h| := √ 〈h, h〉Proposition 6.21 The distance induced by this “Riemannian metric” coincideswith the geodesically induce distance d g .The proof is in 3.22 in Duci and Mennucci [15].To conclude, we propose an explicit computation of the Riemannian Metricfor the case of compact sets in the plane with smooth boundaries; we then pullback the metric to obtain a metric of closed embedded planar curves. We startwith the case of convex sets.We fix p = 2, N = 2.6.3.4 Polar coordinates of smooth convex setsLet Ω ⊂ lR 2 be a convex set with smooth boundary; let y(θ) : [0, L] → ∂Ω bea parameterization of the boundary (by arc parameter), ν(θ) the unit vectornormal to ∂Ω and pointing external to Ω. The following “polar” change ofcoordinates ψ holds:ψ : lR + × [0, L] → lR \ Ω , ψ(ρ, θ) = y(θ) + ρν(θ) (6.3)see figure 8 on the following page. We suppose that y(θ) moves on ∂Ω inanticlockwise direction; so ν = J∂ s y, ∂ ss y = −κν; where J is the rotation matrix(of angle −π/2), κ is the curvature, and ∂ s y is the tangent vector.We can then express a generic integral through this change of coordinates as∫∫f(x) dx = f(ψ(ρ, s))|1 + ρκ(s)| dρ dslR 2 \Ω∫lR + ∂Ωwhere s is arc parameter, and ds is integration in arc parameter.48