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

shape analysis where we study a family of shapes for purposes of statistics,(automatic) cataloging, probabilistic modeling, among others, and possiblycreate an a-priori model for a better shape optimization.1.2 CurvesWe will use formulæ of the form A := B to mean that the formula A is definedby the formula B.S 1 = {x ∈ lR 2 | |x| = 1} is the circle in the plane. S 1 is the template for allpossible closed curves. (Open curves will be called paths, to avoid confusion).Definition 1.1 (Classes of curves) • A C 1 curve is a continuously differentiablemap c : S 1 → lR n such that the derivative c ′ (θ) := ∂ θ c(θ) existsat all points θ ∈ S 1 .• An immersed curve is a C 1 curve c such that c ′ (θ) ≠ 0 at all pointsθ ∈ S 1 .c : S 1 → c(S 1 )✓✏↦→✒✑Note that, in our terminology , the “curve” is the function c, and not just theimage c(S 1 ) inside lR n .Most of the theory following will be developed for curves in lR n , when thisdoes not complicate the math. We will call planar curves those whose imageis in lR 2 .The class of immersed curves is a differentiable manifold. For the purposesof this introduction, we present a simple, intuitive definition.Definition 1.2 The manifold of (parametric) curves M is the set of allclosed immersed curves. Suppose that c ∈ M, c : S 1 → lR n is a closed immersedcurve.• A deformation of c is a function h : S 1 → lR n .• The set of all such h is the tangent space T c M of M at c.• An infinitesimal deformation of the curve c 0 in “direction” h will yield (onfirst order) the curve c 0 (u) + εh(u).• A homotopy C connecting c 0 to c 1 is a continuously differentiablefunction C : [0, 1] × S 1 → lR n such that c 0 (θ) = C(0, θ)and c 1 (θ) = C(1, θ).By defining γ(t) = C(t, ·) we can associate C to a path γ :[0, 1] → M in the space of curves M, connecting c 0 to c 1 .Mhc 0c 1When dealing with homotopies C = C(t, θ), we will use the “prime notation” C ′for the derivative ∂ θ C in the “curve parameter” θ, and the “dot notation” Ċ forthe derivative ∂ t C in the “time parameter” t.3