Tangent-point energies and ropelength as Gamma-limit 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 1topology, 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 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.