Minimization of Curve Length through Energy Minimization using Finite Difference and Numerical Integration in Real Coordinate Space

The problem of determining the minimal length is garnering attention in various fields such as computer vision, robotics, and machine learning. One solution to this problem involves linearly interpolating the solution of a nonlinear optimization problem that approximates the curve's energy minimization problem using finite differences and numerical integration. This method tends to be easier to implement compared to others. However, it was previously unknown whether this approach successfully minimizes the curve's length under the Riemannian metric in real coordinate spaces. In this paper, we prove that the length of a curve obtained by linear interpolation of the solution to an optimization problem, where the energy of the curve is approximated using finite differences and the trapezoidal rule, converges to the minimal curve length at a rate of $1/2$ in terms of the number of points used in the numerical integration. Similarly, we prove that when using the left-point rule, the approximated curve's length likewise converges to the minimal curve length at a rate of $1/2$ in terms of the number of points used in the numerical integration.