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<p>We consider a geometrical problem of the existence of an equidistant set of <i>N</i> points, under the Euclidean metric, on non-closed and closed curves. For a non-closed curve, we give a new version of the geometrical proof of the existence based on the continuous mapping argument. For a closed curve, we construct a counterexample if the number of points <i>N</i> is arbitrarily chosen, and we give a proof of the existence of the set if <i>N</i> is taken sufficiently large. We develop numerical algorithms that build solutions for both types of curves.</p>

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Construction of a Euclidean metric based equidistant grid on an arbitrary curve

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • William J Ceely,
  • Marina Chugunova,
  • Raj Sawhney,
  • Hans Volkmer

摘要

We consider a geometrical problem of the existence of an equidistant set of N points, under the Euclidean metric, on non-closed and closed curves. For a non-closed curve, we give a new version of the geometrical proof of the existence based on the continuous mapping argument. For a closed curve, we construct a counterexample if the number of points N is arbitrarily chosen, and we give a proof of the existence of the set if N is taken sufficiently large. We develop numerical algorithms that build solutions for both types of curves.