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

2. So the distance can be equivalently computed as the infimum of Len G ⊥ orof Len G ; and distances are equal, d G ⊥ = d G .In practice, when computing minimal geodesic (possibly by numerical methods),it is usually best to minimize E G , since it also penalizes the vertical part of themotion, and hence the minimization will produce a “smoother” geodesic.The proof of the above lemma is based on the validity of “the lifting Lemma”:Lemma 11.18 (lifting Lemma) Given any smooth homotopy C of immersedcurves, there exists a reparameterization given by a parameterized family ofdiffeomorphisms Φ : [0, 1] × S 1 → S 1 , so that setting˜C(t, θ) := C(t, Φ(t, θ))we have that ∂ t ˜C is in the horizontal space WC at all times t.For the metric H 0 , this reduces to lemma 11.3 in this paper; for Sobolev-typemetrics, the proof is in §4.6 in [36].11.9 A geometric gradient flow is horizontalProposition 11.19 If E is a geometric energy of curves, in the sense that Eis invariant w.r.to reparameterizations φ ∈ Diff + (S 1 ); then ∇E is horizontal.Proof. Let c be a smooth curve; suppose for simplicity (and with no loss ofgenerality) that it has len(c) = 2π and that it is parameterized by arc parameter,so that |c ′ | ≡ 1 and c ′ = D s c. Let b : S 1 → lR be smooth, and define φ :lR × S 1 → S 1 by solving the b flow, that is the ODE∂ t φ(t, θ) = b(φ(t, θ))with initial data φ(0, θ) = θ. Note that, for all t, φ(t, ·) ∈ Diff + (S 1 ). LetC(t, θ) = c(φ(t, θ)), note that∂ t C(t, θ) = b(φ(t, θ))c ′ (φ(t, θ))then E(C) = E(c), so that (deriving and setting t = 0)and we conclude by arbitrariness of b.∂ t E(C) = 0 = 〈∇E(c), bc ′ 〉11.10 Horizontality according to H 0Proposition 11.20 The horizontal space W c w.r.to H 0 isW c := {h : h(s) ⊥ c ′ (s)∀s}that is the space of vector fields orthogonal to the curve.97