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

9.3 Conformal metricsYezzi and Mennucci [67] proposed to change the metric, from H 0 to a conformalmetric∫〈h 1 , h 2 〉 H 0 = ψ(c) 〈hψ1 , h 2 〉 dscwhere ψ(c) associates to each curve c a positive number. Then the gradientdescent flow of an energy E defined on curves C(t, ·) is∂C∂t = −∇ψ E(C) = − 1ψ(C) ∇E(C)where ∇E(C) is the gradient for the H 0 metric. This is equivalent to a changeof time variable t in the gradient descent flow. So all properties of the flows areunaffected if we switch from a H 0 to a conformal-H 0 metric.Properties 9.2 Consider a conformal metric where ψ(c) depends (monotonically)on the length len(c) of the curve.• If ψ(c) ≥ len(c), the induced metric is non degenerate;• unfortunately, according to a result by Shah [50], when ψ(c) ≡ L(c) veryfew (minimal) geodesics do exist (only “grassfire” geodesics, moving byconstant normal speed).9.4 “Rigidified” normsCharpiat et al.[10] consider norms that favor pre-specified rigid motions. Theydecompose a motion h using the H 0 projection ash = h rigid+ h restwhere h rigidcontains the rigid part of the motion; then they choose λ large, andconstruct the norm‖h‖ 2 rigid = λ‖h rigid ‖2 H 0 + ‖h rest ‖2 H 0.Note that these norms are equivalent to the H 0 -type norm; as a result theinduced distance is (again) degenerate.10 Sobolev type Riemannian metricsIn this part we discuss the Sobolev norms, with applications and experiments.What follows summarizes [55, 57, 58, 35].60