Tangent-point energies and ropelength as Gamma-limit of discrete tangent-point energies on biarc curves

TL;DR

This paper proves that discretized tangent-point energies on biarc curves converge to their continuous counterparts and to ropelength, ensuring that discrete minimizers approximate continuous minimizers.

Contribution

It establishes Gamma-convergence of discretized tangent-point energies to continuous energies and ropelength, with explicit convergence rates for interpolated data.

Findings

Discrete almost minimizing biarc curves converge to ropelength minimizers.

Discrete tangent-point energies converge to continuous energies in the $C^1$-topology.

Explicit convergence rates are provided for energies evaluated on interpolated curves.

Abstract

Using interpolation with biarc curves we prove $\Gamma$-convergence of discretized tangent-point energies to the continuous tangent-point energies in the $C^1$-topology, as well as to the ropelength functional. As a consequence discrete almost minimizing biarc curves converge to ropelength minimizers, and to minimizers of the continuous tangent-point energies. In addition, taking point-tangent data from a given $C^{1,1}$-curve $\gamma$, we establish convergence of the discrete energies evaluated on biarc curves interpolating these data, to the continuous tangent-point energy of $\gamma$, together with an explicit convergence rate.