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

• F (c) = F (c ◦ φ) for all curves c and for all φ ∈ Diff(S 1 ).• For all curves c and all φ, if φ ∈ Diff + (S 1 ) then F (c) = F (c ◦ φ), whereasif φ ∈ Diff − (S 1 ) then F (c) = −F (c ◦ φ)In the first case, F can be “projected” to B, that is, it may be considered as afunction F : B → lR. In the second case, F can be “projected” to M/Diff + (S 1 ).It is important to remark that “geometric” theories have often provided thebest results in computer vision.1.4 Curve–related quantitiesA good way to specify the design goal for shape optimization is to define anobjective function (a.k.a. energy) F : M → lR that is minimum in the curvethat is most fit for the task.When designing our F , we will want it to be geometric; this is easily accomplishedif we use geometric quantities to start from. We now list the mostimportant such quantities.In the following, given v, w ∈ lR n we will write |v| for the standard Euclideannorm, and 〈v, w〉 or (v · w) for the standard scalar product. We will againwrite c ′ (θ) := ∂ θ c(θ).Definition 1.6 (Derivations) If the curve c is immersed, we can define thederivation with respect to the arc parameter∂ s = 1|c ′ | ∂ θ .(We will sometimes also write D s instead of ∂ s .)Definition 1.7 (Tangent) At all points where c ′ (θ) ≠ 0, we define the tangentvectorT (θ) = c′ (θ)|c ′ (θ)| = ∂ sc(θ) .(At the points where c ′ = 0 we may define T = 0).It is easy to prove (and quite natural for our geometric intuition) that D s and Tare geometric quantities (according to the second Definition 1.5).Definition 1.8 (Length) The length of the curve c is∫len(c) := |c ′ (θ)| dθ . (1.1)S 1We can define formally the arc parameter s byds := |c ′ (θ)| dθ ;we use it only in integration, as follows.5