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

7.3 L 1 metric and Plateau problemIf we wish to define a geometric norm on T c M that is modeled on the norm ofthe Banach space L 1 (S 1 → lR N ), we may define the metric∫F 1 (c, h) = ‖π N h‖ L 1 = |π N h(θ)||c ′ (θ)|dθ ;the length of a homotopy is then∫∫Len(C) = |π N ∂ t C(t, θ)||C ′ (t, θ)| dt dθwhich coincides with∫∫Len(C) =|∂ t C(t, θ) × ∂ θ C(t, θ)| dθ dtThis last is easily recognizable as the surface area of the homotopy (up tomultiplicity); the problem of finding a minimal geodesic connecting c 0 and c 1 inthe F 1 metric may be reconducted to the Plateau problem of finding a surfacewhich is an immersion of I = S 1 × [0, 1] and which has fixed borders to thecurves c 0 and c 1 . The Plateau problem is a wide and well studied subject uponwhich Fomenko expounds in the monograph [20].8 Riemannian metrics of curves up to poseA particular approach to the study of shapes is to define a shape to be a curve upto reparameterization, rotation, translation, scaling; to abbreviate, we call thisthe shape space of curves up to pose. We present two examples of Riemannianmetrics.8.1 Shape Representation using direction functionsKlassen, Mio, Srivastava et al in [28, 41] represent a planar curve c by a pair ofvelocity-angle functions (φ, α) through the identityc ′ (u) = exp(φ(u) + iα(u))(identifying lR 2 = IC), and then defining a metric on the velocity-angle functionspace. They propose models of spaces of curves where the metrics involve higherorder derivatives in [28].We review here the simplest such model, where φ = 0. We consider in thefollowing planar curves ξ : S 1 → lR 2 of length 2π and parameterized by arclength. (Note that such curves are automatically Lipschitz continuous).Proposition 8.1 (Continuous lifting, winding number) If ξ ∈ C 1 and parameterizedby arc parameter, then ξ ′ is continuous and |ξ ′ | = 1, so there existsa continuous function α : lR → lR satisfyingξ ′ (s) = ( cos(α(s)), sin(α(s)) ) (8.1)52