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

2.4.5 Centroid energyWe will now propose another simple example where the above phenomenon isagain evident.Example 2.8 (Centroid-based energy) Let us fix a target point v ∈ lR n .We recall that avg c (c) is the center of mass of the curve. The energyE(c) := 1 2 |avg c(c) − v| 2 (2.9)penalizes the distance from the center of mass to v. Let in the following c =avg c (c) for simplicity. The directional derivative of E in direction h is〈〉DE(c; h) = c − v, D(c)(c; h)where in turn (by eqn. (2.7) and (2.8))∫D(c)(c; h) = h + (c − c)(D s h · D s c) ds = (2.10)c∫= h − D s c (h · D s c) − (c − c)(h · Dsc) 2 dssupposing that the curve is planar, thench − D s c (h · D s c) = N(h · N)soDE(c; h) =∫〈c − v, N〉(h · N) − 〈c − v, c − c〉κ(h · N) ds .The H 0 gradient descent flow is then∂c∂t = −∇ H 0E(c) = 〈(v − c), N〉N − κN〈 (c − c), (v − c) 〉 .The first term 〈(v − c) · N〉N in this gradient descentflow moves the whole curve towards v.PvLet P := {w : 〈(w − c) · (v − c) 〉 ≥ 0} be the half plane“on the v side” . The second term −κN 〈 (c − c) · (v − c) 〉in this gradient descent flow tries to decrease the curvelength out of P and increase the curve length in P , andthis is ill posed.ccWe will come back to this example in Proposition 10.20.17