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

Using interpolation with biarc curves we prove $Γ$-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 $γ$, 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.