Existence of set-interpolating and energy-minimizing curves

We consider existence of curves c : [0, 1] Rn which minimize an energy of the form c(k) p (k = 1, 2, . . . , 1 < p < ) under side-conditions of the form Gj(c(t1,j), . . . , c(k-1) (tk,j)) Mj, where Gj is a continuous function, ti,j [0, 1], Mj is some closed set, and the indices j range in some index set J. This includes the problem of finding energy minimizing interpolants restricted to surfaces, and also variational near-interpolating problems. The norm used for vectors does not have to be Euclidean. It is shown that such an energy minimizer exists if there exists a curve satisfying the side conditions at all, and if among the interpolation conditions there are at least k points to be interpolated. In the case k = 1, some relations to arc length are shown.

• Type: Journal
• Year: 2004
• Where: CAGD
• Authors: Johannes Wallner
• Keywords: CAGD 2004, Continuous Function, Energy Minimizing Interpolants, Form Gj