Minimization of curve length through energy minimization using finite differences and numerical integration in Euclidean space

We consider the approximation of minimal geodesics between two closed sets in $\mathbb{R}^D$ endowed with a smooth Riemannian metric. The continuous problem is formulated as the minimization of the energy functional over piecewise smooth curves joining the two sets. We study discrete approximations obtained by finite differences together with numerical integration, and reconstruct continuous curves from discrete minimizers by linear interpolation. Our main result is a direct convergence analysis of the trapezoidal-rule discretization. We prove that the energy of the linearly interpolated discrete minimizer converges to the minimum energy, and that the squared length of the reconstructed curve converges to the squared minimal length, both with rate $O(N^{-1/2})$ as the number of subintervals $N$ tends to infinity. We also obtain the corresponding reconstruction estimates for the left-endpoint rule. In addition, we give an explicit example showing that a direct discretization of the length functional does not, in general, converge to the minimal length.