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

Abstract

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