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

We consider a geometrical problem of existence of an equidistant set of N points, under the Euclidean metric, on non-closed and closed curves. For the 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 existence of the set if N is taken sufficiently large. We develop numerical algorithms that build solutions for both types of curves.