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

avg c (g) = avg c (f) λL 2 Ds 2 ˜g = − ˜flR n ⊕ D c M .We will now show how to properly solve these coupled equations.We will need to define some useful objects.Definition 10.13 We define the projection operatorΠ c : T c M → D c Mh ↦→ h − ∫ c h ds (10.17)Definition 10.14 When we consider the derivation with respect to the arcparameter as a linear operatorD c : D c M → D c Mh ↦→ D s hthen it admits the inverse, that is the primitive operator(10.18)P c : D c M → D c M (10.19)Proof. The proof is just based on noting that, for any smooth h ∈ T c M, h ∈ D c Miff there is a smooth k ∈ T c M with h = D s k. Vice versa, two primitives differby a constant, hence when h ∈ D c M there is exactly one primitive k in D c Msuch that h = D s k.Example 10.15 The tangent vector field D s c is in D c M, and its primitive isP c D s c = c − avg c (c) .Proposition 10.16 Fix a curve c, let L = len(c). We can extend the primitiveoperator by composing it with the projection; the resulting composite operatorcan be expressed in convolutional form asfor any continuous h ∈ T c M; where the kernel K P cP c Π c h = K P c ⋆ h (10.20)isK P c (s) := − s L + 1 2for s ∈ [0, L) (10.21)and K P c (s) is extended periodically to s ∈ lR (note that K P c (s) jumps at allpoints of the form s = nL, n ∈ Z).The above properties lead to this theorem, that can be used to shed a newlight to what was expressed in 10.10 in the previous section.Theorem 10.17 Let f = ∇ H 0E, g = ∇˜H1E; the solution ofcan be expressed asf = avg c (g) − λL 2 D 2 sgg = avg c (f) − 1λL 2 P cP c Π c f = avg c (f) − 1λL 2 KP c ⋆ K P c ⋆ f .69