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

y eqn. (11.3), where the terms RHS are evaluated at (t, ϕ(t, ·)); since G ⊥ iscurve-wise parameterization invariant, then‖∂ t ˜C(t, ·)‖G ⊥ = ‖∂ t C(t, ·)‖ G ⊥ .Consequently,Corollary 11.15 G is homotopy-wise parameterization invariant if and only iffor all b.‖h‖ G = ‖h + bc ′ ‖ GSummarizing all above theory, we obtain a method to understand/study/designthe metric G on B, following these steps:1. choose a metric G on M that is curve-wise parameterization invariant2. G generates the horizontal space W = V ⊥3. project G on W to define G ⊥ that is homotopy-wise parameterizationinvariant.11.8.1 Horizontal G ⊥ as length minimizerAs we defined in 11.9, to define the distance we minimize the length of the pathsγ : [0, 1] → M that connect a curve c 0 to a reparameterization c 1 ◦ φ of the curvec 1 . Consider the following sketchy example.Example 11.16 Consider two paths γ 1 and γ 2 connectingc 0 to reparameterizations of c 1 . Recall that the spaceW c is orthogonal to the orbits, the space V c is tangent tothe orbits. The path γ 1 (that moves in some tracts alongthe orbits, with ˙γ 1 ∈ V γ ) is longer than the path γ 2 .The above example explains the following lemma.W cc 0cVcγγ12Lemma 11.17 Let us choose G to be a curve-wise parameterization invariantmetric (so, by Prop. 11.8, it cannot be homotopy-wise parameterizationinvariant).1. Let ˜C(t, θ) = C(t, ϕ(t, θ)) where ϕ(t, ·) is a diffeomorphism for all fixed t.Minimizemin E G ( ˜C)ϕThen at the minimum ˜C ∗ , ∂ t ˜C∗ is horizontal at every point, andE G ( ˜C ∗ ) = E G ⊥( ˜C ∗ ) .O ccc 11 o φo φ’96